Available online at www.sciencedirect.com ScienceDirect Journal of Differential Equations 332 (2022) 123–201 www.elsevier.com/locate/jde The criticality of reversible quadratic centers at the outer boundary of its period annulus ✩ D. Marín a,b , J. Villadelprat c,∗ a Departament de Matemàtiques, Edifici Cc, Universitat Autònoma de Barcelona, 08193 Cerdanyola del Vallès, Barcelona, Spain b Centre de Recerca Matemàtica, Edifici Cc, Campus de Bellaterra, 08193 Cerdanyola del Vallès, Barcelona, Spain c Departament d’Enginyeria Informàtica i Matemàtiques, ETSE, Universitat Rovira i Virgili, 43007 Tarragona, Spain Received 26 November 2021; revised 12 May 2022; accepted 30 May 2022 Available online 9 June 2022 Abstract This paper deals with the period function of the reversible quadratic centers Xν = −y(1 − x)∂x + (x + Dx 2 + F y 2 )∂y , where ν = (D, F ) ∈ R2 . Compactifying the vector field to S2 , the boundary of the period annulus has two connected components, the center itself and a polycycle. We call them the inner and outer boundary of the period annulus, respectively. We are interested in the bifurcation of critical periodic orbits from the polycycle ν at the outer boundary. A critical period is an isolated critical point of the period function. The criticality of the period function at the outer boundary is the maximal number of critical periodic orbits of Xν that tend to ν0 in the Hausdorff sense as ν → ν0 . This notion is akin to the cyclicity in Hilbert’s 16th Problem. Our main result (Theorem A) shows that the criticality at the outer boundary is at most 2 for all ν = (D, F ) ∈ R2 outside the segments {−1} × [0, 1] and {0} × [0, 2]. With regard to the bifurcation from the inner boundary, Chicone and Jacobs proved in their seminal paper on the issue that the upper bound is 2 for all ν ∈ R2 . In this paper the techniques are different because, while the period function extends analytically to the center, it has no smooth extension to the polycycle. We show that the period function has an asymptotic expansion near the polycycle with the remainder being uniformly flat with respect to ν ✩ This work has been partially funded by the Ministry of Science, Innovation and Universities of Spain through the grants PGC2018-095998-B-I00 and MTM2017-86795-C3-2-P and by the Agency for Management of University and Research Grants of Catalonia through the grants 2017SGR1725 and 2017SGR1617. * Corresponding author. E-mail addresses: davidmp@mat.uab.cat (D. Marín), jordi.villadelprat@urv.cat (J. Villadelprat). https://doi.org/10.1016/j.jde.2022.05.026 0022-0396/© 2022 The Author(s). Published by Elsevier Inc. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). D. Marín and J. Villadelprat Journal of Differential Equations 332 (2022) 123–201 and where the principal part is given in a monomial scale containing a deformation of the logarithm, the so-called Écalle-Roussarie compensator. More precisely, Theorem A follows by obtaining the asymptotic expansion to fourth order and computing its coefficients, which are not polynomial in ν but transcendental. Theorem A covers two of the four quadratic isochrones, which are the most delicate parameters to study because its period function is constant. The criticality at the inner boundary in the isochronous case is bounded by the number of generators of the ideal of all the period constants but there is no such approach for the criticality at the outer boundary. A crucial point to study it in the isochronous case is that the flatness of the remainder in the asymptotic expansion is preserved after the derivation with respect to parameters. We think that this constitutes a novelty that is of particular interest also in the study of similar problems for limit cycles in the context of Hilbert’s 16th Problem. Theorem A also reinforces the validity of a long standing conjecture by Chicone claiming that the quadratic centers have at most two critical periodic orbits. A less ambitious goal is to prove the existence of a uniform upper bound for the number of critical periodic orbits in the family of quadratic centers. By a compactness argument this would follow if one can prove that the criticality of the period function at the outer boundary of any quadratic center is finite. Theorem A leaves us very close to this existential result. © 2022 The Author(s). Published by Elsevier Inc. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). MSC: 34C07; 34C20; 34C23 Keywords: Criticality; Period function; Asymptotic expansion; Critical periodic orbit; Hilbert’s 16th problem Contents 1. Introduction and main results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 2. Criticality vs bifurcation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132 3. Asymptotic expansion of the period function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 3.1. Study of {D = −1/2, F ∈ (1/2, 1)} and {F = 1/2, D ∈ (−1, 0)} . . . . . . . . . . . . . . . . 144 3.2. Study of {F = 2, D ∈ (−2, 0)} and {D = G(F ) : F ∈ (1, 3/2)} . . . . . . . . . . . . . . . . . . 146 3.3. Study of {F = 1, D ∈ (−1, 0)} . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148 4. Distinguished cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153 5. Proof of Theorem A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165 Appendix A. Coefficient formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170 A.1. Previous results about the Dulac time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170 A.2. Proof of Proposition 3.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174 A.3. Proof of Proposition 3.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179 Appendix B. Beta and hypergeometric functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189 Appendix C. A technical result for the proof of Proposition 3.3 . . . . . . . . . . . . . . . . . . . . . . . . . 193 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200 1. Introduction and main results A singular point p ∈ R2 of a planar differential system  ẋ = f (x, y), ẏ = g(x, y), 124 D. Marín and J. Villadelprat Journal of Differential Equations 332 (2022) 123–201 is a center if it has a punctured neighborhood that consists entirely of periodic orbits surround- ing p. The period annulus of the center is the largest punctured neighborhood with this property and we denote it by P. The period annulus is an open subset of R2 that may be unbounded. For this reason we embed P in RP 2 and, abusing notation, we denote the boundary of the re- sulting set by ∂P. Clearly the center p belongs to ∂P and in what follows we call it the inner boundary of the period annulus. We also define the outer boundary of the period annulus to be  := ∂P \ {p}, which is a nonempty compact subset of RP 2 . The subject of our study is the period function of the center, that assigns to each periodic orbit in P its period. Since the period function is defined on the set of periodic orbits in P, in order to study its qualitative properties we need to parametrize this set. This can be done by taking a transverse section to the vector field X = f (x, y)∂x + g(x, y)∂y on P, for instance an orbit of the orthogonal vector field X ⊥ . To fix ideas let us suppose that {γs }s∈(0,1) is such a parametrization where s ≈ 0 corresponds to the periodic orbits near p and s ≈ 1 to the ones near . Then the map P : (0, 1) −→ (0, +∞) defined by P (s) := {period of γs } provides the qualitative properties of the period function that we are concerned with and one can readily show by using the Implicit Function Theorem that it is as smooth as X. It is also well-known that if X is analytic and the center p is non-degenerate then P extends analytically to s = 0. Let us advance that, on the contrary, P does not extend smoothly to s = 1. The critical periods are the isolated critical points of P , i.e. ŝ ∈ (0, 1) such that P  (ŝ) = 0 and P  (s) = 0 if 0 < |s − ŝ| < ε. In this case, more geometrically, we shall say that γŝ is a critical periodic orbit of X. One can easily see that the property of being a criti- cal periodic orbit does not depend on the particular parametrization of the set of periodic orbits used, see Remark 2.2. The study of the critical periodic orbits is another issue arising from the famous Hilbert’s 16th Problem and it has strong parallelisms with the research on limit cycles, from both the conceptual and technical point of views. In this regard we can mention for instance that the isochronicity problem (i.e., to decide whether a center has a constant period function) is the counterpart of the center-focus problem. The renowned conjecture claiming that a quadratic differential system can have at most four limit cycles has also an analogue in the context of the period function and it was posed by C. Chicone [5]. More specifically this conjecture asserts that if a quadratic center has some critical periodic orbit then by an affine transformation and a constant rescaling of time it can be brought to Loud normal form  ẋ = −y + Bxy, (1) ẏ = x + Dx 2 + Fy 2 , and that this center has at most two critical periodic orbits for any (B, D, F ) ∈ R3 . In fact there is much analytic evidence that this conjecture is true (see [7,40,41] for instance). The problems that we are interested in take place when the vector field X depends on pa- rameters. To fix notation, let U be an open subset of RN and consider a family of planar vector fields {Xμ , μ ∈ U } such that each Xμ has a center pμ with period annulus Pμ . Let us denote the period function of the center pμ by P ( · ; μ) and observe that, given some μ0 ∈ U , the number of critical periodic orbits of Xμ can vary as we perturb μ ≈ μ0 . Under some regularity assump- tions on the dependence of Pμ with respect to μ it can be proved (see Lemma 2.12) that the emergence/disappearance of critical periodic orbits can only occur from three different places: (a) Bifurcations at the inner boundary of the period annulus (i.e., the center pμ ). (b) Bifurcations at the outer boundary of the period annulus (i.e., the polycycle μ ). (c) Bifurcations at the interior of the period annulus Pμ . 125 D. Marín and J. Villadelprat Journal of Differential Equations 332 (2022) 123–201 Chicone and Jacobs give in their seminal paper [6] a complete description of the bifurcations from the inner boundary for the whole family of quadratic centers. In this case the parameter μ are the coefficients of the vector field and since the center is non-degenerate  P (s; μ) extends analytically to s = 0, so that one can consider its Taylor series P (s; μ) = ∞ i=0 i (μ)s at s = 0, a i whose coefficients ai belong to the polynomial ring R[μ]. On account of this the result about the bifurcations from the isochronous centers (see [6, Theorem 2.2]), which are the most difficult ones to study, follows by analyzing the ideal (a1 , a2 , . . .) of all Taylor coefficients exactly as N. Bautin does in [4] to study the bifurcations of limit cycles from the quadratic centers. In the present paper we resume our study of the bifurcations from the outer boundary that we initiated in [22,23]. Let us recall that the differential system (1) has no critical periodic orbits if B = 0, see [10, Theorem 1]. By means of a rescaling the case B = 0 can be brought to B = 1, i.e., Xν := −y(1 − x)∂x + (x + Dx 2 + Fy 2 )∂y with ν := (D, F ). (2) Here we already adopt the parameter notation that we shall use throughout the paper, which is devoted to the bifurcation of critical periodic orbits from the outer boundary in the family {Xν , ν ∈ R2 }. Since each vector field Xν is polynomial we can consider its Poincaré compact- ification p(Xν ), see [3, §5], which is an analytic vector field on the sphere S2 topologically equivalent to Xν . The outer boundary ν becomes then a polycycle of p(Xν ) that can be studied using local charts of S2 , but even so the period function P (s; ν) cannot be smoothly extended to s = 1. For the family under consideration we show that P (s; ν) has an asymptotic expansion at s = 1 with the remainder being uniformly flat with respect to ν and where the principal part is given in a monomial scale containing a deformation of the logarithm, the so-called Écalle- Roussarie compensator. Our main theorem follows by obtaining the asymptotic expansion to fourth order and computing its coefficients, which are not polynomial in ν but transcendental (more concretely, they are hypergeometric functions). To this end we strongly rely on the tools that we develop in our recent papers [29–31]. The results that we obtain in the present paper can be viewed conceptually as the analogue for the outer boundary of the work carried out by Chicone and Jacobs in [6] on the bifurcation of critical periodic orbits from the inner boundary of the quadratic centers. That being said, the proofs of the results on the outer boundary are tech- nically tougher than the ones on the inner boundary because ν is a polycycle and the period function P (s; ν) cannot be analytically extended there. By way of example, to determine the pa- rameters that vanish simultaneously two coefficients in the asymptotic expansion at s = 1 takes 5 pages of computations dealing with a hypergeometric function (see Appendix C), whereas the same problem for the Taylor series at s = 0 can be solved readily by taking resultants because the coefficients are polynomials. In this paper we use the notion of criticality of the period function at the outer boundary which, roughly speaking, is the number of critical periodic orbits that can emerge or disappear from ν as we perturb ν slightly. It is defined in exactly the same way as the notion of cyclicity of a limit periodic set, which is used to study the bifurcation of limit cycles in the context of Hilbert’s 16th Problem, see [36] for instance. Before giving its precise definition, and the statement of our main contribution, we enumerate the previous results about the bifurcation of critical periodic orbits from the outer boundary ν for the family {Xν , ν ∈ R2 }. In this regard we stress that these results are given according to the dichotomy between local regular value and local bifurcation value (of the period function at the outer boundary) that we introduce in our early paper [23]. This notion (see Definition 2.10) enables to obtain a structure theorem for the bifurcation diagram of the period function in its full domain (see Lemma 2.12), but it has the inconvenience of not being so 126 D. Marín and J. Villadelprat Journal of Differential Equations 332 (2022) 123–201 Fig. 1. The thick (closed) curve B consists of localbifurcation  values  of the period function at the outer boundary according to [23], where the curve that joins − 32 , 32 and − 12 , 1 is the graphic of an analytic function D = G(F ), see Remark 3.4. The dotted lines U correspond to parameters that remained unspecified in that paper and we color the subsequent improvements obtained in [19,24,25,27,28,38,39]. The parameters outside B ∪ U are local regular values by the result in [23]. (For interpretation of the colors in the figure(s), the reader is referred to the web version of this article.) quantitative and geometric as the criticality. In order to simplify the exposition for the moment we can think that ν0 ∈ R2 is a local regular value if and only if the criticality of the period function at ν0 is zero (i.e., no critical periodic orbit bifurcates from ν0 as we perturb ν ≈ ν0 ). That said, let U be the union of dotted straight lines in Fig. 1, whatever its color is. Consider also the thick curve B . (Here the subscripts B and U stand for bifurcation and unspecified respectively.) Then according to [23, Theorem A] the open set R2 \ (B ∪ U ) corresponds to local regular values and B consists of local bifurcation values (of the period function at the outer boundary).  we also conjecture that any parameter in U is regular, except for the segment In that paper {0} × 0, 12 in the vertical axis, that should consist of bifurcation values. Since the formulation of this conjecture there has been some progress in the study of the parameters in U : • From the results in [19,39] it follows that the parameters in blue are regular. In these papers the authors determine a region M in the parameter plane for which the corresponding center has a globally monotonous period function (i.e., it has no critical periodic orbits). The pa- rameters that we draw in blue are inside the interior of M, which prevents the bifurcation of critical periodic orbits. • Along the straight line F = −D there is a breaking of a heteroclinic connection between two hyperbolic saddles at the outer boundary. From the results in [24] it follows that the parameters in red are regular. • Along the two segments in green it occurs a saddle-node bifurcation at the outer boundary of the period annulus. An asymptotic expansion of the Dulac time of this type of unfold- ing is obtained in [25] and as an application it is proved that the parameters in the segment 127 D. Marín and J. Villadelprat Journal of Differential Equations 332 (2022) 123–201 (−1, 0) × {1}, with the exception of (− 12 , 1), are local regular values. A subsequent refine- ment of this approach shows in [27] that the segment (−1, 0) × {0} also consists of local regular values.   • By [28, Theorem B] the parameters in brown, more precisely the segment {0} × 14 , 12 , are local bifurcation values of the period function at the outer boundary. • Along the segment (−1, 0) × { 12 } there is a resonant saddle at ν and the parameters in yellow are local regular values at the outer boundary of the period annulus according to [38, Corollary B]. As we already explained, these results are addressed to solve the dichotomy between local regular value and local bifurcation value (of the period function at the outer boundary). Beyond this dichotomy a challenging problem is the computation of the exact number of critical periodic orbits that can bifurcate from the outer boundary, which constitutes the counterpart of the result by Chicone and Jacobs [6] about the bifurcation from the inner boundary. The following is the precise definition of the number that we aim to compute for the quadratic centers, where dH stands for the Hausdorff distance between compact sets of RP 2 . Definition 1.1. Consider a C ∞ family {Xμ , μ ∈ U } of planar vector fields with a center and fix some μ0 ∈ U . Suppose that the outer boundary of the period annulus varies continuously at μ0 ∈ U , meaning that dH (μ , μ0 ) tends to zero as μ → μ0 . Then, setting  N (δ, ε) = sup # critical periodic orbits γ of Xμ in Pμ with dH (γ , μ0 )  ε and μ − μ0  δ , the criticality of (μ0 , Xμ0 ) w.r.t. Xμ is Crit (μ0 , Xμ0 ), Xμ := infδ,ε N (δ, ε).  We stress that in this definition the vector field Xμ is not required to be polynomial but C ∞ . This is so because in order to define the outer boundary μ of the period annulus Pμ of Xμ we do not compactify the vector field but only the set Pμ and to this end there is no need that Xμ is polynomial. Certainly Crit (μ0 , Xμ0 ), Xμ may be infinite but, if it is not, then it gives the maximal number of critical periodic orbits of Xμ that tend to μ0 in the Hausdorff sense as μ → μ0 . Related with this issue we point out that the contour of the period annulus Pμ0 may change for μ ≈ μ0 . The assumption that the period annulus varies continuously ensures that this change does not occur abruptly. In this regard note that Xμ = −y∂x + (x + μx 3 + x 5 )∂y , with μ ∈ R, is a polynomial family of vector fields with a center at the origin for which the outer boundary does not vary continuously at μ = 2. This is so because the period annulus Pμ is the whole plane for μ < 2, whereas it is bounded for μ = 2 (see [20] for details). In this example Crit (μ0 , Xμ0 ), Xμ , as introduced in Definition 1.1, does not give the number of critical periodic orbits bifurcating from μ as μ → μ0 . Let us mention that this assumption is also required in [17,18], where the authors obtain several results addressed to bound the criticality at the outer boundary of families of vector fields of potential type, i.e., −y∂x + V  (x)∂y . Let us remark at this point that if Chicone’s conjecture about the number of critical periodic orbits of the quadratic centers is true then Crit (ν0 , Xν0 ), Xν  2 for all ν ∈ R2 , see (2). In this paper, by applying our recent results from [29–31], we prove the following (see Fig. 2): 128 D. Marín and J. Villadelprat Journal of Differential Equations 332 (2022) 123–201 Fig. 2. Overview of the criticality results in Theorem A. The blue curves have criticality 1, the green segment has criticality greater or equal than 1 and the two points in red have criticality 2. Any other parameter, except the two dotted segments in black, has criticality 0. Theorem A. Let {Xν , ν ∈ R2 } be the family of quadratic vector fields given in (2) and consider the period function of the center at the origin. Then the following assertions hold: (a) Crit (ν0 , Xν0 ), Xν = 0 for ν0 ∈/ B ∪ D = −1, F ∈ [0, 1] ∪ D = 0, F ∈ [0, 12 ] . (b) Crit (ν0 , Xν0 ), Xν = 1 for ν0 ∈ B \ {D = 0} ∪ (−2, 2), (G( 43 ), 43 ) . (c) Crit (ν0 , Xν0 ), Xν  1 for ν0 ∈ D = 0, F ∈ [ 14 , 2] . (d) Crit (ν0 , Xν0 ), Xν = 2 for ν0 ∈ (−2, 2), (G( 43 ), 43 ) . (e) There is a C 1 curve arriving at ν = (G( 43 ), 43 ) tangent to B and there is a C 0 curve with a exponential flat contact with {F = 2} at ν = (−2, 2), consisting both of local bifurcation values of the period function at the interior. There are some papers containing results related with assertion (b) in Theorem A to be referred. Thus, by [33, Theorem A], Crit (ν0 , Xν0 ), Xν = 1 for any ν0 = (G(F0 ), F0 ) with F0 ∈ ( 43 , 32 ). This is a piece of the curve that joins − 32 , 32 and − 12 , 1 , see Fig. 1, and in this re- gard observe that the criticality is 2 for ν0 = (G( 43 ), 43 ). Furthermore, it is proved in [32, Theorem B] that if ν0 = (D0 , 2) with D0 ∈ (−2, 0) \ {− 12 } then Crit (ν0 , Xν0 ), Xν = 1. The same con- clusion is true for any ν0 = (−F0 , F0 ) with F0 ∈ [ 32 , 2) thanks to [24, Theorem C]. In that paper it is also partially proved the claim about the parameter ν0 = (−2, 2) in assertion (e) of Theo- rem A. Apart from these references to previous results we also want to point out the following issues with regard to the statement and proof of Theorem A: • As expected, the study of the bifurcation of critical periodic orbits, either from the inner or the outer boundary, is much more delicate when we perturb an isochronous center. By the result of W.S. Loud, see [15], we know that there are four nonlinear quadratic isochrones, ν1 = (0, 1), ν2 = (−1/2, 2), ν3 = (−1/2, 1/2) and ν4 = (0, 1/4), (3) which are located in B . Chicone and Jacobs prove, see [6, Theorem 3.1], that the criticality of each isochrone νi at the inner boundary of its period annulus (i.e., the center itself) is 129 D. Marín and J. Villadelprat Journal of Differential Equations 332 (2022) 123–201 one. The proof of this follows by finding a finite set of generators for the ideal of all the coefficients of the Taylor series of P (s; ν) at s = 0. In the present paper we are able to show that ν2 and ν3 have criticality one also at the outer boundary (i.e., the polycycle), see Propositions 4.2 and 4.3 respectively. A crucial point to see this is that, as we prove in [30], the flatness of the remainder in the asymptotic expansion at s = 1 is preserved after the derivation with respect to parameters. This constitutes the cornerstone to obtain Lemma 4.1, which enables us to perform a convenient division in the space of coefficients and proceed then as in the proof of Bautin [4, §3] for the analogous result about the bifurcation of limit cycles from the center. The isochrones ν1 and ν4 cannot be analyzed following this approach because the polycycle at the outer boundary is not hyperbolic. • It is well-known, see [6, Theorem 3.2], that the criticality at the inner boundary of any quadratic center is at most two and that this maximum criticality is achieved at three parame- ter values, the so-called Loud points, which we give in (13). For consistency with Chicone’s conjecture, each one of these three parameters should have a “twin” where the maximum criticality at the outer boundary is attained. In this paper we identify two of these twin pa- rameters, see assertion (d) in Theorem A. We conjecture that each pair of twins is connected by a curve that consists of local bifurcation values at the interior, see Remarks 4.5 and 5.1. • The local bifurcation values of the period function can only occur at the inner boundary (i.e., the center), at the outer boundary (i.e., the polycycle) or at the interior of the period annulus, see Lemma 2.12. (With regard to the latter, its counterpart in the context of Hilbert’s 16th Problem is the bifurcation from a semi-stable limit cycle, which is characterized by the sud- den emergence of a double limit cycle that gives rises to two hyperbolic limit cycles with different stability, see [12, §13.3] for instance). As occurs with limit cycles, the identification of this third type of local bifurcation value is out of reach for the moment and only partial results have been obtained. Thus, in a joint paper with P. Mardešić we prove (see [23, Theo- rem 4.3]) that at each Loud point there exists a germ of analytic curve that consists of local bifurcation values at the interior. Since P (s; ν) extends analytically to s = 0, this follows readily by applying the Weierstrass Preparation Theorem. In the present paper, see assertion (e) in Theorem A, we show the existence of two germs of curve which also consists of local bifurcation values at the interior and that are the mirror image at the outer boundary (i.e., at s = 1) of those previously obtained in [23], see Fig. 6. In another vein it is well-known (see [36, §2.2] for details) that the problem of proving the ex- istence of a uniform bound for the number of limit cycles in a given family, for instance Hilbert’s 16th Problem, can be replaced by a local problem that consists in showing that the cyclicity of each limit periodic set within the family is finite. The proof of this is by a compactness argu- ment and it does not provide an algorithm to compute an explicit upper bound even if we had an explicit bound for the cyclicity of every limit periodic set. In any case this gives a program for solving the existential Hilbert’s 16th Problem that has been posed and implemented for the quadratic vector fields by R. Roussarie and his collaborators (see [9,34]). One can of course transfer this problem to the period function and ask for the existence of a uniform bound for the number of critical periodic orbits in the family of quadratic centers. Similarly as it occurs in the context of limit cycles, an affirmative answer would follow if one can prove that the criticality of the period function at the outer boundary of any quadratic center is finite, cf. Lemma 2.17. On account of Theorem A we are not very far from proving the existence of this uniform bound for the family of reversible quadratic centers. It will follow in particular if one can prove the validity of the following conjecture (see Figs. 2 and 7): 130 D. Marín and J. Villadelprat Journal of Differential Equations 332 (2022) 123–201 Conjecture. Let {Xν , ν ∈ R2 } be the family of quadratic vector fields given in (2) and consider the period function of the center at the origin. Then the following assertions are true: (a) Crit (ν0 , Xν0 ), Xν = 0 for ν0 ∈ {D = −1, F ∈ [0, 1]}. (b) Crit (ν0 , Xν0 ), Xν = 1 for ν0 ∈ {D = 0, F ∈ (0, 2]}. (c) Crit (ν0 , Xν0 ), Xν = 2 for ν0 = (0, 0). (d) There is a curve of local bifurcation values of the period function at the interior arriving to ν = (0, 0) tangent to D = 0. As a matter of fact to show the existence of a uniform bound for the number of critical periodic orbits of the reversible quadratic centers it suffices to verify that Crit (ν0 , Xν0 ), Xν is finite for all ν0 = (D0 , F0 ) inside the segments {−1} × [0, 1] and {0} × [0, 2]. To put this into context let us recall that the differential system (1) has no critical periodic orbits if B = 0 by [10, Theorem 1]. On the other hand, apart from the reversible one, there are essentially three other families of quadratic centers: the Hamiltonian, the codimension four Q4 and the generalized Lotka-Volterra systems QLV 3 . According to Chicone’s conjecture the number of critical periodic orbits should be zero for the centers in these three families. This is known to be true for the Hamiltonian and Q4 families thanks to the results of Coppel and Gavrilov [7] and Zhao [41], respectively. With regard to the family QLV 3 it is proved in [40] that, except for a subset of codimension one in the parameter plane, the criticality at the outer boundary is zero. It is clear then that any contribution to the proof of the above conjecture will constitute a very significant step forward to the existence of a uniform bound for the number of critical periodic orbits in the whole family of quadratic centers. Let us mention in this respect that the singularity at the outer boundary of the period annulus is nilpotent along D = −1 and D = 0. In this situation the results of [30,31] do not apply and new techniques must be developed. The paper is organized in the following way. In Section 2 we recall the definition of local bifurcation value at the outer boundary, that we introduce in our early paper [23] to study the bifurcation diagram of the period function of the family {Xν , ν ∈ R2 }, and we prove several results that relate it with the criticality. We also show how to study the criticality by means of a suitable parametrization of the set of periodic orbits near the outer boundary. Section 3 is devoted to the asymptotic expansion of the period function near the outer boundary, which is the cornerstone in the proof of Theorem A. To this end we prove three results that are addressed to three different parameter subsets according to the contour of the period annulus. As one might expect the proofs of these results are rather long and technical. Furthermore they are based on previous tools developed in [29–31] that need to be introduced appropriately. For these reasons, to ease the paper’s readability we defer some proofs to Appendix A. In Section 4 we study three distinguished parameters. On one hand the two isochrones for which we succeed in proving that the criticality is one (see Propositions 4.2 and 4.3) and, on the other hand, the parameter ν = (G( 43 ), 43 ), which is also rather special because it has criticality two (see Proposition 4.4). Due to the novel approach of its proof we think that each one of these results is of particular interest in the context of Hilbert’s 16th Problem. Section 5 is entirely devoted to the proof of Theorem A. Next, in Appendix A we prove the results stated in Section 3 that we mentioned before and in Appendix B we are concerned with the integral representation of the Beta and hypergeometric functions, which usually appear as coefficients in the asymptotic expansions that we obtain. Finally Appendix C is addressed to prove a technical result that is used to study the vanishing set of two coefficients. 131 D. Marín and J. Villadelprat Journal of Differential Equations 332 (2022) 123–201 2. Criticality vs bifurcation In this section we recap the notion of local bifurcation value of the period function at the outer boundary as we introduced in our early paper [23]. We relate it with the criticality, which is a more quantitative and geometric definition, and prove a general result connecting both notions, see Lemma 2.16. More specifically our aim is to take advantage in the present paper of the results that we obtained in [23] with regard to the period function of Loud’s centers (2) and that are not stated using the notion of criticality. Related with this issue, our goal in this section is also to clarify the usage of a parametrization of the period function near the outer boundary to compute its criticality, see Lemma 2.4. Finally we give a sufficient condition in order that a parameter is a local bifurcation value of the period function at the interior, see Lemma 2.15. Several results in this section are equally valid in the finitely smooth class C k , k ∈ N, the infinitely smooth class C ∞ and the analytic class C ω . For simplicity in the exposition we write C with the wild card ∈ N ∪ {∞, ω}. Our first result is addressed to the regularity properties of the map (p, μ) → P̂ (p; μ) that assigns to each μ ∈ U and p ∈ Pμ the period P̂ of the periodic orbit of Xμ passing through the point p. The result is given under a technical assumption concerning the existence of a continuous parametrization σ (s; μ) of the period annulus Pμ near its outer boundary μ . We point out that from now on, in contrast with the notation used in the introduction, for the sake of convenience s = 0 corresponds to μ and s = 1 to the center. Lemma 2.1. Let us fix ∈ N ∪ {∞, ω} and consider a C family of planar vector fields {Xμ }μ∈U such that, for each μ ∈ U , Xμ has a center pμ ∈ R2 with period annulus Pμ . Suppose that there exists a continuous map σ : (0, δ) × U → R2 verifying, for each fixed μ ∈ U , that (a) the map σ ( · ; μ) : (0, δ) → R2 is C 1 , (b) the vectors ∂s σ (s; μ) and Xμ (σ (s; μ)) are linearly independent for all s ∈ (0, δ), and (c) for each compact set K ⊂ Pμ ∪ {pμ } there exists sK > 0 such that σ (s; μ) ∈ Pμ \ K for all s ∈ (0, sK ). Then the following assertions hold: 1. U = Pμ × {μ} is an open subset of R2 × U , and μ∈U 2. the map (p, μ) → P̂ (p; μ) = {period of the periodic orbit of Xμ passing through p} is C on U . Proof. We consider the family {Xμ }μ∈U as a single C vector field Y on R2 × U whose tra- jectories are contained in the submanifolds μ = constant. Denote the flow of Y by φ(t; p, μ) = (ϕ(t; p, μ), μ). In order to prove the first assertion, for a given (p0 , μ0 ) ∈ U we must show that there is an open subset V of R2 × U such that (p0 , μ0 ) ∈ V ⊂ U . We claim that this is true in the particular case that there exists s0 ∈ (0, δ) such that σ (s0 ; μ0 ) = p0 . Indeed, due to the assumption in (b), note that Y is transverse to ε := {(σ (s; μ), μ); |s − s0 | < ε and μ − μ0 < ε} for all ε > 0 small enough and that (p0 , μ0 ) ∈ ε . Then, since σ : (0, δ) ×U → R2 is continuous, by the flow box theorem (and shrinking ε > 0 if necessary) it follows that 132 D. Marín and J. Villadelprat Journal of Differential Equations 332 (2022) 123–201  V := φ(t; ε ) t∈(−ε,ε) is an open subset of R2 × U . Furthermore, since U is invariant by φ and ε ⊂ U by construc- tion, we have that (p0 , μ0 ) ∈ V ⊂ U and this proves the claim. Let us consider now an arbitrary p0 ∈ Pμ0 . Denote the periodic orbits of Xμ0 passing through q := σ (δ/2; μ0 ) and p0 by γq and γp0 , respectively. For each μ ∈ U we take the orthogonal vector field to Xμ , say Xμ⊥ , pointing inward the periodic orbits in Pμ . We consider the family {Xμ⊥ }μ∈U as a single C vector field Ŷ on R2 × U and denote its flow by φ̂(t; p, μ) = (ϕ̂(t; p, μ), μ). Note that pμ is also a singular point for Xμ⊥ that, by applying the Poincaré-Bendixson Theorem (see for instance [3]), it is easy to show to be asymptotically stable. Observe moreover that φ̂(t; U ) ⊂ U for all t  0. We define  := {ϕ̂(t; q, μ0 ); t  0} ⊂ Pμ0 , which is clearly a transverse section for Xμ0 , and distinguish two cases: • Case 1:  ∩ γp0 = ∅. In this case there exist t1 , t2  0 such that ϕ̂(t2 ; q, μ0 ) = ϕ(t1 ; p0 , μ0 ) ∈ . Since q = σ (δ/2; μ0 ), on account of the claim we can take an open neighborhood V1 of (q, μ0 ) in R2 × U with V1 ⊂ U . Then, by the continuity of solutions with re- spect to initial conditions, there exists an open neighborhood V2 of (p0 , μ0 ) such that φ̂ − t2 ; φ(t1 ; V2 ) ⊂ V1 . Thus V2 ⊂ φ − t1 ; φ̂(t2 ; V1 ) ⊂ U , where the second inclusion follows due to the φ̂(t; U ) ⊂ U for all t  0 and φ(t; U ) = U for all t ∈ R, together with the fact that V1 ⊂ U . • Case 2:  ∩ γp0 = ∅. Note that in this case Int(γq ) ⊂ Int(γp0 ). (Here, given a Jordan curve γ ⊂ R2 , Int(γ ) denotes the bounded connected component of R2 \ {γ }.) Moreover, by the assumption in (c) and taking K = Int(γp0 ), there exists s1 ∈ (0, δ/2) satisfying that σ (s1 ; μ0 ) ∈ Pμ0 \ K. Therefore, since q = σ (δ/2; μ0 ) ∈ Int(γp0 ), by continuity there exists s2 ∈ (s1 , δ/2) such that σ (s2 ; μ0 ) ∈ γp0 . Consequently σ (s2 ; μ0 ) = ϕ(t3 ; p0 , μ0 ) for some t3 ∈ R and on the other hand, again on account of the claim, there exists an open neighbor- hood V3 of (σ (s2 ; μ0 ), μ0 ) in R2 × U with V3 ⊂ U . Thus, exactly as before, by continuity of solutions with respect to initial conditions, there is an open neighborhood V4 of (p0 , μ0 ) such that V4 ⊂ φ(−t3 ; V3 ) ⊂ U . This proves the validity of the first assertion. Let us prove now that the function P̂ : U −→ (0, +∞) defined by (p, μ) → P̂ (p; μ) = {period of the periodic orbit of Xμ passing through p} is C . In what follows we shall use the notation p = (x, y) for the components of a point of R2 . We fix (p̂, μ̂) ∈ U and suppose that the period of the periodic orbit of Xμ̂ passing through p̂ = (x̂, ŷ) ∈ Pμ̂ is τ̂ > 0. Then, due to X(p̂; μ̂) := Xμ̂ (p̂) = (0, 0), there is i ∈ {1, 2} such that ∂t ϕi (τ̂ ; p̂, μ̂) = ∂t ϕi (0; p̂, μ̂) = Xi (p̂; μ̂) = 0. For simplicity in the exposition let us suppose that X1 (p̂; μ̂) > 0. In this case we can apply the Implicit Function Theorem to the equation ϕ1 (t; p, μ) = x at (t, p, μ) = (τ̂ , p̂, μ̂) in order to obtain a C positive function S(p; μ) in a open neighborhood W ⊂ U of (p̂, μ̂) verifying S(p̂; μ̂) = τ̂ and 133 D. Marín and J. Villadelprat Journal of Differential Equations 332 (2022) 123–201 ϕ1 (t; p, μ)|t=S(p;μ) = x for all (p, μ) ∈ W . (4) Clearly we can assume that W is a cube Q(ε1 ) with center (p̂, μ̂) and edge length ε1 > 0. We diminish ε1 if necessary so that X1 (p; μ) > 0 for all (p, μ) ∈ Q(ε1 ). Furthermore, thanks to S(p̂; μ̂) = τ̂ together with the continuity of S and φ, we can take ε2 ∈ (0, ε1 ) such that φ(t; p, μ)|t=S(p;μ) ∈ Q(ε1 ) for all (p, μ) ∈ Q(ε2 ). We claim that P̂ = S on Q(ε2 ). Clearly the claim will follow once we show that ϕ2 (t; p, μ)|t=S(p;μ) = y for all (p, μ) ∈ Q(ε2 ). By contradiction, suppose that there exists (p̄, μ̄) ∈ Q(ε2 ) such that ϕ2 t; p̄, μ̄ |t=S(p̄;μ̄) = ȳ. Due to Q(ε2 ) ⊂ U , the trajectory of Xμ̄ passing through p̄ is a periodic orbit which, for simplic- ity in the exposition, we assume to travel clockwise around the center pμ̄ (the other case follows verbatim). That being said we consider the piece of trajectory  := {ϕ(t; p̄, μ̄); t ∈ [0, S(p̄, μ̄)]} and the vertical segment, recall (4),  := (1 − s)p̄ + sϕ S(p̄, μ̄); p̄, μ̄ ; s ∈ (0, 1) ⊂ {x = x̄}. Arguing on the phase portrait of Xμ̄ , due to X1 (p; μ̄) > 0 for all p ∈ , if ϕ2 S(p̄, μ̄); p̄, μ̄ < ȳ then interior of the Jordan curve  ∪  is a positively but not negatively invariant subset of Pμ̄ . Similarly, if ϕ2 S(p̄, μ̄); p̄, μ̄ > ȳ then we obtain a negatively invariant subset of Pμ̄ which is not positively invariant. In both cases we get a contradiction with the fact that Pμ̄ is foliated by periodic orbits of Xμ̄ and Q(ε2 ) ⊂ U . Consequently ϕ2 S(p, μ); p, μ = y for all (p, μ) ∈ Q(ε2 ) and so the validity of the claim follows. Since Q(ε2 ) is an open neighborhood of an arbitrary point of U and S is C in Q(ε2 ), the claim implies the second assertion in the statement.  The previous result is addressed to a family {Xμ }μ∈U of vector fields and this is the reason why we require the existence of a local transverse section near the outer boundary of the period annulus μ that behaves well with respect to parameters. That being said, Lemma 2.1 can be applied to a single vector field X without this requirement because a trajectory of the orthogonal vector field X ⊥ already provides a transverse section in the whole period annulus. Thus in order to assert that p → P̂ (p; μ) is C on Pμ for each fixed μ ∈ U , it is not necessary to verify the existence of a continuous map σ : (0, δ) × U → R2 satisfying (a), (b) and (c). Remark 2.2. If X is a C vector field, ∈ N ∪ {∞, ω}, with a center then the period function P̂ is a first integral for the flow of X on the period annulus P that, by Lemma 2.1, is C . Consequently the scalar product ∇ P̂ (p) · X(p) is zero for all p ∈ P. This implies that if γ is a critical periodic orbit of X then the gradient ∇ P̂ vanishes on γ . Indeed, if σ : (0, 1) −→ P is a C transverse section to X on P and P (s) := P̂ (σ (s)) then P  (s) = ∇ P̂ (σ (s)) · σ  (s). Thus, since ∇ P̂ (σ (s)) · X(σ (s)) = 0, the transversality of σ implies that P  (s) = 0 if, and only if, ∇ P̂ (σ (s)) = (0, 0). This shows in particular that the condition for γ to be a critical periodic 134 D. Marín and J. Villadelprat Journal of Differential Equations 332 (2022) 123–201 orbit is local and independent of the particular transverse section used to parametrize the set of critical periodic orbits near γ .  We define next the notion that enable us to study the criticality at the outer boundary. Definition 2.3. Let U be an open set of RN and consider a family of functions {h( · ; μ)}μ∈U on (0, ε). Given any μ ∈ U we define Z0 (h( · ; μ), μ ) to be the smallest integer n having the property that there exist δ > 0 and a neighborhood V of μ such that for every μ ∈ V the function h(s; μ) has no more than n isolated zeros on (0, δ) counted with multiplicities.  The hypothesis with regard to the local transverse section in our next result are slightly stronger than in the previous one because we require the continuity at s = 0 and that σ (0; μ) belongs to the outer boundary μ for all μ ∈ U , cf. assumption (c) in Lemma 2.1. We also remark that in the statement P̂ (p; μ) stands for the period of the periodic orbit of Xμ passing through p ∈ Pμ . Lemma 2.4. Let us consider a C ω family {Xμ }μ∈U of planar polynomial vector fields such that, for each μ ∈ U , Xμ has a center pμ ∈ R2 with period annulus Pμ . Let μ ⊂ RP 2 be the outer boundary of Pμ . Suppose there exists a continuous map σ : [0, δ) × U → RP 2 verifying that, for each μ ∈ U , (a) the map σ ( · ; μ) : (0, δ) → R2 is C 1 , (b) the vectors ∂s σ (s; μ) and Xμ (σ (s; μ)) are linearly independent for all s ∈ (0, δ), (c) σ (s; μ) ∈ Pμ for all s ∈ (0, δ) and σ (0; μ) ∈ μ . Then, for each fixed μ ∈ U , the following assertions hold: 1. The Hausdorff distance between the outer boundaries μ and μ tends to zero as μ → μ . 2. If P (s; μ) := P̂ (σ (s; μ); μ) for all (s, μ) ∈ (0, δ) × U , then (2a) Crit (μ , Xμ ), Xμ  Z0 (P  ( · ; μ), μ ). (2b) Crit (μ , Xμ ), Xμ  n if for each open neighborhood V of μ and δ > 0 there exist n different numbers s1 , s2 . . . , sn ∈ (0, δ) and μ̂ ∈ V such that P  (si ; μ̂) = 0 for i = 1, 2, . . . , n. (2c) Crit (μ , Xμ ), Xμ = 0 if, and only if, Z0 (P  ( · ; μ), μ ) = 0. Proof. To show the first assertion note that, since Xμ is polynomial, we can consider its Poincaré compactification p(Xμ ), see [3, §5] for details, which is an analytic vector field on the sphere S2 topologically equivalent to Xμ . The outer boundary μ becomes then a polycycle of p(Xμ ) that can be studied using local charts of S2 . On account of this, the fact that dH (μ , μ ) → 0 as μ → μ follows by the continuity of μ → σ (0; μ) ∈ μ together with the continuity with respect to initial conditions and parameters of the trajectories of p(Xμ). The interested reader is referred to [36, Lemma 22, p. 110] for a related result for limit periodic sets. With regard to the upper bound in (2a) it is clear that if Z0 (P  ( · ; μ), μ ) = +∞ then there is nothing to be proved. So let us assume that Z0 (P  ( · ; μ), μ ) =  ∈ Z≥0 and argue by contradiction. If Crit (μ , Xμ ), Xμ   + 1 then there exist  + 1 sequences {γμki }i∈N , k = 1, 2, . . . ,  + 1, where γμ1i , γμ2i , . . . , γμ+1 i are different critical periodic orbits of Xμi for each 135 D. Marín and J. Villadelprat Journal of Differential Equations 332 (2022) 123–201 i ∈ N, such that μi → μ and dH (γμki , μ ) → 0 as i → +∞. Then, due to dH (μ , μ ) → 0 as μ → μ and dH (γμki , μi )  dH (γμki , μ ) + dH (μi , μ ), we have dH (γμki , μi ) → 0 as i → +∞ for each k = 1, 2, . . . ,  + 1. Since σ (0; μi ) ∈ μi and there is a one-to-one correspondence between zeros of P  (s; μi ) arbitrarily near s = 0 and critical periodic orbits of Xμi arbitrarily close to μi (cf. [36, Lemma 22]), this implies that there exist  + 1 sequences of positive numbers {sik }i∈N , k = 1, 2, . . . ,  + 1, such that P  (sik ; μi ) = 0 and #{si1 , si2 , . . . , si+1 } =  + 1 for each i ∈ N, and limi→+∞ sik = 0 for each k = 1, 2, . . . ,  + 1. This clearly contradicts that Z0 (P  ( · ; μ), μ ) = , see Definition 2.3. The assertion in (2b) follows similarly. Indeed, on account of the assumption and the above mentioned one-to-one correspon- dence between zeros of P  (s; μ) near s = 0 and critical periodic orbits of Xμ close to μi , we can construct n sequences {γμki }i∈N , k = 1, 2, . . . , n, where γμ1i , γμ2i , . . . , γμni are different critical periodic orbits of Xμi for each i ∈ N, such that μi → μ and dH (γμki , μi ) → 0 as i → +∞. Then, using that dH (μ , μ ) → 0 as μ → μ , we can assert that limi→+∞ dH (γμki , μ0 ) = 0 for each k = 1, 2, . . . , n, which implies Crit (μ , Xμ ), Xμ  n, as desired. Finally the as- sertion in (2c) follows easily from the ones in (2a) and (2b). This completes the proof of the result.  Next we introduce the notion of global transverse section for a family of period annuli. Roughly speaking it is a transverse section, joining the center with some point at the outer bound- ary of the period annulus, that behaves well with the parameters. Definition 2.5. Let us fix ∈ N ∪ {∞, ω} and consider a C family {Xμ }μ∈U of planar vector fields such that, for each μ ∈ U , Xμ has a center pμ ∈ R2 with period annulus Pμ . Let μ ⊂ RP 2 be the outer boundary of Pμ . A global transverse section for the family of period annuli {Pμ }μ∈U is a continuous map σ : [0, 1] × U → RP 2 verifying that (a) the map σ ( · ; μ) : [0, 1] → RP 2 is C for each μ ∈ U , (b) the vectors ∂s σ (s; μ) and Xμ (σ (s; μ)) are linearly independent for all (s, μ) ∈ (0, 1) × U and the map ∂s σ : (0, 1) × U → R2 is continuous, (c) σ (s; μ) ∈ Pμ for all s ∈ (0, 1), σ (0; μ) ∈ μ and σ (1; μ) = pμ . When such a global transverse section exists we say that the family of period annuli {Pμ}μ∈U varies continuously.  Remark 2.6. The period annulus of the family of Loud’s quadratic centers given in (2) varies continuously in the sense of Definition 2.5. Indeed, it follows from the proof of [23, Lemma 3.2] that μ = (D, F ) → ξμ := sup{t > 0; (s, 0) ∈ Pμ for all s ∈ (0, t)} is a well-defined continuous function on R2 . Moreover the point (ξμ , 0) belongs to μ and 0 < ξμ  1 for all μ ∈ R2 . Then σ (s; μ) = ((1 − s)ξμ , 0) for (s, μ) ∈ [0, 1] × R2 is clearly a 136 D. Marín and J. Villadelprat Journal of Differential Equations 332 (2022) 123–201 global transverse section. In particular, since the Loud’s system is polynomial, the outer bound- ary of the period annulus varies continuously in the Hausdorff sense by the first assertion in Lemma 2.4.  Note, see (b) in Definition 2.5, that we also require (s, μ) → ∂s σ (s; μ) to be continuous. The reason for this is because if we define P (s; μ) = P̂ (σ (s; μ); μ) then (s, μ) → ∂s P (s; μ) is a continuous function by Lemma 2.1. This continuity is a key point in the forthcoming results. Before that we summarize in the next statement the properties that we get for P (s; μ) as a consequence of Lemma 2.1 and Definition 2.5. Corollary 2.7. Let us fix ∈ N ∪ {∞, ω} and consider a C family of planar vector fields {Xμ }μ∈U such that, for each μ ∈ U , Xμ has a center pμ ∈ R2 with period annulus Pμ . Assume that the family of period annuli varies continuously and let σ : [0, 1] × U → RP 2 be a global transverse section for {Pμ }μ∈U . If P (s; μ) := P̂ (σ (s; μ); μ) for all (s, μ) ∈ (0, 1) × U then the following holds: (a) P ( · ; μ) ∈ C ((0, 1)) for each μ ∈ U , and (b) P and ∂s P are continuous functions on (0, 1) × U . Definition 2.8. Under the assumptions of Corollary 2.7, we say that P (s; μ) = P̂ (σ (s; μ); μ), which is defined for (s, μ) ∈ (0, 1) × U , is a global parametrization of the period function. In contrast, (p, μ) → P̂ (p; μ) = {period of the periodic orbit of Xμ passing through p} is defined on μ∈U Pμ × {μ}, which is not so easy to handle.  One of the main goals in the present section is to relate the concept of local bifurcation value of the period function, as introduced in [23], with the notion of criticality, see Definition 1.1. As we will see the first one concerns with the qualitative properties of the period function, whereas the second is more geometric and quantitative. In doing so we will be able to take advantage of the results about the bifurcation diagram of the period function of the Loud’s centers that we obtained in [23]. For reader’s convenience we next recall the definition of local bifurcation value of the period function. Definition 2.9. Let {Iμ }μ∈U be a continuous family of intervals, i.e., such that Iμ = ((μ), r(μ)) with , r ∈ C 0 (U ), and consider a continuous family of functions {Fμ : Iμ −→ R}μ∈U . We say that μ0 ∈ U is a regular value of the family {Fμ : Iμ −→ R}μ∈U if there exist a neighborhood V of μ0 and an isotopy {hμ : Iμ −→ Iμ0 }μ∈V , with hμ0 = id, such that     sgn Fμ (s) = sgn Fμ0 hμ (s) for all s ∈ Iμ and μ ∈ V , (5) where sgn : R → {−1, 0, 1} is the extended sign function. A parameter μ0 which is not regular is called a bifurcation value.  137 D. Marín and J. Villadelprat Journal of Differential Equations 332 (2022) 123–201 The endpoints of Iμ , the domain of definition of Fμ , depend continuously on μ, so that ∪μ∈U Iμ × {μ} is an open subset of R × U . Thus, by a continuous family of functions {Fμ : Iμ −→ R}μ∈U , we mean that the map (s, μ) → Fμ (s) is continuous on ∪μ∈U Iμ × {μ}. Next we particularize the previous definition to study the period function. To this aim note that, by Corollary 2.7, if {Xμ }μ∈U is a C 1 family of vector fields with a center such that the corre- sponding family of period annuli varies continuously, and we set P (s; μ) = P̂ (σ (s; μ); μ), then {∂s P ( · ; μ)}μ∈U is a continuous family of functions on (0, 1). Definition 2.10. Consider a C 1 family of planar vector fields {Xμ }μ∈U such that, for each μ ∈ U , Xμ has a center pμ ∈ R2 with period annulus Pμ , that we suppose to vary continuously. (a) We say that μ0 ∈ U is a regular (respectively, bifurcation) value of the period function if for some global parametrization of the period function P : (0, 1) × U −→ (0, +∞) we have that μ0 is a regular (respectively, bifurcation) value of the family {P  ( · ; μ) : (0, 1) → R}μ∈U . (b) We say that μ0 ∈ U is a local regular value of the period function at the interior if there is some global parametrization of the period function P : (0, 1) × U −→ (0, +∞) satisfying that for each c ∈ (0, 1) there exists a continuously varying neighborhood Iμ (c) of c in (0, 1) such that μ0 is a regular value of the family {P  ( · ; μ) : Iμ (c) → R}μ∈U . A parameter which is not a local regular value at the interior is called a local bifurcation value at the interior. (c) We say that μ0 ∈ U is a local regular value of the period function at the outer (respectively, inner) boundary if for some global parametrization of the period function P : (0, 1) × U −→ (0, +∞) there exists a continuously varying neighborhood Iμ (c) of c = 0 (respectively, c = 1) such that μ0 is a regular value of the family {P  ( · ; μ) : Iμ (c) ∩ (0, 1) −→ R}μ∈U . A parameter which is not a local regular value at the outer (respectively, inner) boundary is called a local bifurcation value at the outer (respectively, inner) boundary.  Remark 2.11. Let us make the following easy observations with regard to the previous defini- tions: (a) One can replace “some global parametrization” by “any global parametrization”. Indeed, suppose that μ0 ∈ U is a regular value for {P  ( · ; μ)}μ∈U where P (s; μ) = P̂ (σ (s; μ); μ) and consider another global parametrization P̄ (s; μ) = P̂ (σ̄ (s; μ); μ) of the period func- tion, see Definition 2.8. If we denote by τμ (s) the Poincaré map from the transverse section  given by s → σ (s; μ) to the transverse section  ¯ given by s → σ̄ (s; μ) then τμ is an in- creasing diffeomorphism and P (s; μ) = P̄ (τμ (s); μ), so that P  (s; μ) = P̄  (τμ (s); μ)τμ (s). On account of this and following the notation in Definition 2.9, h̄μ := τμ0 ◦ hμ ◦ τμ−1 is a suitable isotopy in order to show that μ0 is a regular value for the family {P̄  ( · ; μ}μ∈U because sgn P̄  (s; μ) = sgn P  (τμ−1 (s); μ) = sgn P  (hμ ◦ τμ−1 )(s); μ0 = sgn P̄  h̄μ (s); μ0 , where we use that τ  (s) > 0. (b) In order to study if a parameter is a local regular value at the outer boundary it is not neces- sary to consider a global transverse section σ : [0, 1] × U −→ RP 2 for the family of period annuli. Indeed, see point (c) in Definition 2.10, it suffices to take a local parametrization 138 D. Marín and J. Villadelprat Journal of Differential Equations 332 (2022) 123–201 σ : [0, δ) × U −→ RP 2 . Similarly, to study the local regular values at the inner boundary it suffices to take a local parametrization σ : (1 − δ, 1] × U −→ R2 .  As expected, μ0 is a bifurcation value of the period function if, and only if, μ0 is either a local bifurcation value at the inner boundary, at the outer boundary or at the interior. This is stated in the following result and the interested reader is referred to [23, Lemma 2.7] for the proof. Lemma 2.12. Let us consider a C 1 family of analytic planar vector fields {Xμ }μ∈U such that, for each μ ∈ U , Xμ has a center pμ ∈ R2 with period annulus Pμ , that we suppose to vary contin- uously. Then the bifurcation diagram of the period function is the union of the local bifurcation diagrams at the inner and outer boundary and in the interior. Under the assumptions and notation in Corollary 2.7, a sufficient condition for μ ∈ U to be a local regular value of the period function at the interior is that P (s; μ ) = 0 for all s ∈ (0, 1). This follows easily by the continuity of (s, μ) → P  (s; μ) on (0, 1) × U and a compactness argument. In case that this function is C 1 then another sufficient condition is that P  ( · ; μ ) has only simple zeros because the application of the Implicit Function Theorem provides the appropriate isotopies. Hence, in this context, the set of local bifurcation values of the period function at the interior is contained in  := {μ ∈ U ; there exists s ∈ (0, 1) such that P  (s; μ) = P  (s; μ) = 0}. If P  (s; μ) was polynomial in s (which is certainly not true) then  would consist of those pa- rameters μ such that the discriminant of P  (s; μ ) is equal to zero. (Recall that the discriminant of q ∈ R[x] is the resultant between q(x) and q  (x), see for instance [8].) One may expect on the other hand that the parameters in  are always local bifurcation values of the period function at the interior. However this is not always the case and the following toy models show that some additional assumptions are needed to this end. Example 2.13. Setting N = 1, we take P  to be F (s; μ) = (s − μ)2 and U = (0, 1). Then it is clear that any μ ∈ U is a local regular value of F at the interior (i.e., there are no local bifurcation values) but we have that  = U . Note that in this case the interior of  is non-empty.  Example 2.14. Setting N = 2, we take P  to be F (s; μ) = (s − μ1 )3 − μ2 and U = (0, 1)2 . Then again it turns out that any μ = (μ1 , μ2 ) ∈ U is a local regular value of F at the interior, whereas  = {μ ∈ U : μ2 = 0}. Observe that in this case the interior of  is empty but F ( · ; μ) has zeroes of multiplicity 3.  The following result provides us with an analytical tool to study the local bifurcation values of the period function at the interior. We emphasize that it has the natural hypothesis in view of the previous discussion. Lemma 2.15. Let {Xμ }μ∈U be an analytic family of planar vector fields such that, for each μ ∈ U , Xμ has a center pμ ∈ R2 with period annulus Pμ . Assume that the family of period annuli varies continuously and let σ : [0, 1] × U → RP 2 be a global transverse section for {Pμ }μ∈U . Setting P (s; μ) = P̂ (σ (s; μ); μ) for all (s, μ) ∈ (0, 1) × U , suppose additionally that 139 D. Marín and J. Villadelprat Journal of Differential Equations 332 (2022) 123–201 (a) the interior of  (as a subset of U ⊂ RN ) is empty, and (b) for each μ ∈ U , the zeros of P  ( · ; μ) have at most multiplicity 2. Then each μ ∈  is a local bifurcation value of the period function at the interior. Proof. Note first that, by Corollary 2.7, the function P ( · ; μ) is analytic on (0, 1) for each μ ∈ U . Let us take any μ0 ∈ . Then there exists s0 ∈ (0, 1) such that P  (s0 ; μ0 ) = P  (s0 ; μ0 ) = 0 and, by the hypothesis in (b), P  (s0 ; μ0 ) = 0. Consequently P  ( · ; μ0 ) has a local extremum at s = s0 and so there exists ε > 0 small enough such that P  ( · ; μ0 ) has the same sign +1 or −1 on (s0 − ε, s0 + ε) \ {s0 }. Assume by contradiction that μ0 is a local regular value of the period function at the interior. Then, taking c = s0 in (b) of Definition 2.10, we can consider a neighborhood V of μ0 , a continuously varying neighborhood Iμ of s0 in (0, 1) and an isotopy hμ : Iμ → Iμ0 for μ ∈ V , with hμ0 = id, verifying the equality in (5). Since  has empty interior we can take μ̂ ∈ V \  and define ŝ := h−1 μ̂ (s0 ) ∈ Iμ̂ . On account of this, particularizing (5)  with μ = μ̂ and s = ŝ we deduce that P (ŝ; μ̂) = 0. Accordingly, due to μ̂ ∈ / , it follows that P  (ŝ; μ̂) = 0. Therefore the function s → P  (s; μ̂) changes sign at s = ŝ. This contradicts (5) taking μ = μ̂ and s ≈ ŝ because P  ( · ; μ0 ) has the same sign on (s0 − ε, s0 + ε) \ {s0 }.  In the statement of our next result p(X) stands for the Poincaré compactification in S2 of a planar polynomial vector field X, see [3, §5] for details. Recall also that any polycycle of an analytic vector field can be desingularized giving a polycycle with only hyperbolic or semi- hyperbolic vertices. By a hyperbolic polycycle we mean that its desingularization does not have semi-hyperbolic vertices (i.e., saddle-nodes). Lemma 2.16. Consider a C ω family of planar polynomial vector fields {Xμ }μ∈U such that, for each μ ∈ U , Xμ has a center pμ ∈ R2 with period annulus Pμ , that we suppose to vary contin- uously. Then the following assertions hold for any given μ ∈ U : (a) If Crit (μ , Xμ ), Xμ = 0 then μ is a local regular value of the period function at the outer boundary. (b) Assuming additionally that the outer boundary μ is a hyperbolic polycycle of p(Xμ ), if μ is a local regular value of the period function at the outer boundary then Crit (μ , Xμ ), Xμ = 0. Proof. Since the family of period annuli varies continuously, see Definition 2.5, we can take a global transverse section σ : [0, 1] × U → RP 2 and consider the global parametrization of the period function given by P (s; μ) := P̂ (σ (s; μ); μ) for (s, μ) ∈ (0, 1) × U , see Corollary 2.7. In order to show (a) note that if Crit (μ , Xμ ), Xμ = 0 then Z0 (P  ( · , μ), μ ) = 0 by assertion (2c) in Lemma 2.4. This implies, see Definition 2.3, the existence of δ > 0 and a neighborhood V of μ such that P  (s; μ) does not vanish on (0, δ) × V . Hence, since (s, μ) → P  (s; μ) is continuous thanks to (b) in Corollary 2.7, the function P  (s; μ) has constant sign on (0, δ) × V . Thus, see Definitions 2.9 and 2.10, taking Iμ = (0, δ) and hμ = id it follows that μ is a regular value of the family {P  ( · ; μ) : Iμ → R}μ∈U as desired. This shows the validity of the assertion in (a). Let us turn next to the assertion in (b). If μ is a local regular value of the period function at the outer boundary then there exist a neighborhood V of μ , a continuous strictly positive 140 D. Marín and J. Villadelprat Journal of Differential Equations 332 (2022) 123–201 function μ → εμ on V and an isotopy {hμ : (0, εμ ) −→ (0, εμ )}μ∈V such that sgn(P  (s; μ)) = sgn(P  (hμ (s)); μ ) for all s ∈ (0, εμ ) and μ ∈ V . From this point we distinguish two cases: 1. If the center of Xμ is not isochronous then, by applying [26, Theorem 1.1], the zeros of P  (s; μ ) do not accumulate to s = 0. Let us remark that to apply this result we take into account that the transverse section σ ( · ; μ ) is analytic at s = 0, see Definition 2.5, and the hypothesis that μ is a hyperbolic polycycle of p(Xμ ). Hence there exists ρ > 0 such that P  (s; μ ) = 0 for all s ∈ (0, ρ). Thus, since we can suppose without loss of generality that εμ ∈ (0, ρ) and δ := inf{εμ : μ ∈ V } > 0, it follows that P  (s; μ) = 0 on (0, δ) × V , which implies (see Definition 2.3) that Z0 (P  ( · , μ), μ ) = 0. Therefore, by assertion (2c) in Lemma 2.4, Crit (μ , Xμ ), Xμ = 0. 2. If the center of Xμ is isochronous then P  ( · ; μ ) ≡ 0. Hence sgn(P  (s; μ)) = sgn(P  (hμ (s)); μ ) = 0 for all s ∈ (0, εμ ) and μ ∈ V . Thus P  ( · ; μ) has not isolated zeros for all μ ∈ V and consequently, see Definition 2.3, Z0 (P  ( · , μ), μ ) = 0. Then Crit (μ , Xμ ), Xμ = 0 by (2c) in Lemma 2.4. This shows (b) and completes the proof of the result.  We conclude this section by showing that, as we explain in the introduction, Theorem A leaves us very close to the proof of the existence of an upper bound for the number of critical periodic orbits in the family {Xν , ν ∈ R2 }. In this respect we note that there are parameter values ν ∈ R2 for which Xν has another center pν apart from the one at the origin (see for instance [23, Figure 4]). The bound also holds for the critical periodic orbits of this second center because one can always find an invertible affine transformation g : R2 −→ R2 with g(pν ) = (0, 0) such that the push-forward of Xν by g verifies g∗ (Xν ) = βXν̂ for some ν̂ ∈ R2 and β = 0. Lemma 2.17. Consider the family of vector fields {Xν , ν ∈ R2 } given in (2). If Crit (ν0 , Xν0 ), Xν is finite for every ν0 ∈ R2 then there exists N ∈ N such that the center at the ori- gin of Xν has at most N critical periodic orbits for all ν ∈ R2 . Proof. By Lemma 2.1, U := ν∈R2 Pν × {ν} is an open subset of R2 × R2 and the map (p, ν) → P̂ (p; ν) = {period of the periodic orbit of Xν passing through p} is analytic on U . We define P (s; ν) := P̂ (1 − s)ξν , 0 ; ν for each (s, ν) ∈ (0, 1) × R2 , see Remark 2.6, which provides us with a suitable global parametrization of the period function. Let us note in particular that ∂sk P (s; ν) is a continuous function on (0, 1) × R2 for each k ∈ N. Moreover, by [19, Theorem A], we know that if ν = (D, F ) ∈ / K := [−7, 2] × [0, 4] then the center at the origin of Xν has no critical periodic orbits. Consequently, if for each fixed ν ∈ R2 we define Nν to be the number of isolated zeros of s → P  (s; ν) on the interval (0, 1) counted without multiplicities, the result will follow once we prove that sup (Nν ) < +∞. ν∈K Let us advance that this will be a consequence of the compactness of [0, 1] × K. With this end in view we fix any (s , ν ) ∈ [0, 1] × K and observe that three different situations may occur: 141 D. Marín and J. Villadelprat Journal of Differential Equations 332 (2022) 123–201 (a) Case s = 1. As a consequence of the result of Chicone and Jacobs, see [6, Theorem 3.1], there exist ε, δ > 0, depending on ν , such if ν ∈ Bε (ν ) := {ν ∈ R2 : ν − ν < ε} then the number of isolated roots of P  (s; ν) = 0 with s ∈ (1 − δ, 1) is at most 2 (counted with multiplicities). (b) Case s = 0. Since  := Crit (ν , Xν ), Xν < +∞ by assumption, (2b) in Lemma 2.4 implies that there exist ε, δ > 0 (depending on ν again) such if ν ∈ Bε (ν ) then the number of isolated roots of P  (s; ν) = 0 with s ∈ (0, δ) is at most  (counted without multiplicities). (c) Case s ∈ (0, 1). (c1) If the center of Xν is not isochronous then there exists k ∈ N, depending on (s , ν ), such that ∂sk P (s ; ν ) = 0. By continuity there is a neighborhood V of (s , ν ) such that ∂sk P (s; ν) = 0 for all (s, ν) ∈ V . Hence the application of Rolle’s Theorem shows that there exist ε, δ > 0 such if ν ∈ Bε (ν ) then the number of roots of P  (s; ν) = 0 with s ∈ (s − δ, s + δ) is at most k (counted with multiplicities). (c2) Let us suppose finally that the center of Xν is isochronous. Since ((1 −s)ξν , 0), ν ∈ U , and by taking for instance the flow of the orthogonal vector field Xν⊥ , there exists a transverse section s̄ → σ (s̄; ν) given by an analytic map σ : (−δ1 , δ1 ) × Bε1 (ν ) −→ U and such that σ (0; ν ) = ((1 − s )ξν , 0), ν . We then define P̄ (s̄; ν) := P̂ (σ (s̄; ν)), which is clearly analytic on (−δ1 , δ1 ) × Bε1 (ν ). We can thus compute its Taylor’s series at s̄ = 0, ∞  P̄ (s̄; ν) = ai (ν)s̄ i , i=0 where each ai is an analytic function on Bε1 (ν ) with ai (ν ) = 0. Working in the local ring R{ν}ν of convergent power series at ν , we consider the ideal B := ai , i ∈ N . The ring is Noetherian and so there exists  ∈ N such that B = (a1 , a2 , . . . , a ). Verba- tim the proof of Chicone and Jacobs for [6, Theorem 2.2] (see also the result of Rous- sarie in [36, §4.3.1] for a similar result for the displacement map), there exist analytic functions hi (s̄; ν) in a neighborhood of (0, ν ) with hi (0; ν) ≡ 1 for i = 1, 2, . . . ,  such that we can write   P̄  (s̄; ν) = ai (ν)s̄ i−1 hi (s̄; ν). i=1 Now, setting ψi (s̄; ν) := s̄ i−1 hi (s̄; ν) and proceeding just like the proof of [6, The- orem 2.2], one can apply the well-known derivation-division algorithm and use re- cursively Rolle’s Theorem to show that there exist δ2 , ε2 > 0 small enough such that if ν ∈ Bε2 (ν ) then the ordered set (ψ1 , ψ2 , . . . , ψ ) is an extended complete Cheby- shev system for s̄ ∈ (−δ2 , δ2 ), see [13] for a definition. Accordingly if ν ∈ Bε2 (ν ) then either P̄  ( · ; ν) ≡ 0 or P  (s̄; ν) = 0 has at most  − 1 roots with s̄ ∈ (−δ2 , δ2 ) counted with multiplicities. Using the original parametrization of the period function, this shows the existence of δ3 , ε3 > 0 small enough such that if ν ∈ Bε3 (ν ) then the 142 D. Marín and J. Villadelprat Journal of Differential Equations 332 (2022) 123–201 number of isolated roots of P  (s; ν) = 0 with s ∈ (s − δ3 , s + δ3 ) is at most  − 1 taking multiplicities into account. Since in each one of the possible cases there is a neighborhood of (s , ν ) where the number of critical periods is finite, the result follows by taking a finite subcover of [0, 1] × K.  3. Asymptotic expansion of the period function From now on we focus on the quadratic family {Xν , ν ∈ R2 } given in (2) and study the period function of the center at the origin. In this section we give its asymptotic expansion near the outer boundary ν for parameters ν inside three specific sets (see Fig. 1): 1 = D = − 12 , F ∈ ( 12 , 1) ∪ F = 12 , D ∈ (−1, 0) , 2 = F = 2, D ∈ (−2, 0) ∪ D = G(F ) : F ∈ (1, 32 ) and 3 = F = 1, D ∈ (−1, 0) . In all the cases the period annulus Pν is unbounded. Since the vector field Xν is polynomial, in order to study the behavior of the trajectories near infinity one can use its Poincaré compact- ification p(Xν ), which is an analytic vector field on the sphere S2 topologically equivalent to Xν , see [3, §5] for details. The outer boundary ν is a polycycle of p(Xν ) that can be studied using local charts of S2 . In doing so one obtains (see [23, Figure 4]) the different phase por- traits in the dehomogeneized Loud’s family {Xν , ν ∈ R2 }. For the parameter values studied in this section it occurs that the polycycle ν of p(Xν ) is hyperbolic if ν ∈ 1 ∪ 2 and has a saddle-node singularity if ν ∈ 3 . With regard to the phase portrait, it happens that the affine part of ν is a straight line for ν ∈ 1 , whereas it is a branch of a hyperbola for ν ∈ 2 . These are the reasons why we split the parameters under consideration in these three subsets, which are studied in the forthcoming subsections. Concerning the behavior of the period function near ν , the dichotomy between local regular value and local bifurcation value (see Definition 2.10) is solved for any ν ∈ 1 ∪ 2 ∪ 3 thanks to the results in [22,23,25,38]. In these papers it is com- puted the asymptotic expansion of the period function to second order, which usually suffices to tackle the regular/bifurcation dichotomy. However in order to study the criticality we need here to go further and compute the third, and even the fourth, order expansion. Let us advance that the asymptotic expansions for ν ∈ 1 are given in Proposition 3.2 and the ones for ν ∈ 2 in Proposition 3.3. Being the proof of both results rather long and technical, for the sake of pa- per’s readability we postpone them to Appendix A, where we also summarize the fundamental results and definitions from [29–31] that we shall use here. Among them we point out the notion of L-flatness satisfied by the remainder and that we advance now for reader’s convenience (see Definition A.2). Given L ∈ R we say that a C ∞ function ψ(s; ν) defined for s > 0 small enough and ν ∈ U ⊂ R2 is L-flat at ν ∈ U if for each  = (0 , 1 , 2 ) ∈ Z3≥0 there exist a neighborhood V of ν and constants C, s0 > 0 such that    ∂ || ψ(s; ν)       Cs L−0 for all s ∈ (0, s0 ) and ν ∈ V .  ∂s 0 ∂ν1 ∂ν2  1 2 143 D. Marín and J. Villadelprat Journal of Differential Equations 332 (2022) 123–201 Fig. 3. Phase portrait in the Poincaré disc of Xν for ν ∈ V , where for convenience the center at (0, 0) is shifted to the left and the vertical invariant line is {x = 1}. In this case we write ψ ∈ FL∞ (ν ). We also consider the Écalle-Roussarie compensator ω(s; α), that is a deformation of the logarithm used in the monomial scale in which the asymptotic ex- pansion is given. Definition 3.1. The function defined for s > 0 and α ∈ R by means of  s −α −1 if α = 0, ω(s; α) := α − log s if α = 0, is called the Écalle-Roussarie compensator. In the sequel we shall also use the notation ωα (s) = ω(s; α).  The asymptotic expansion for ν ∈ 3 is given in Proposition 3.6 and its proof is of a different nature due to the occurrence of a saddle-node bifurcation at the polycycle. 3.1. Study of {D = −1/2, F ∈ (1/2, 1)} and {F = 1/2, D ∈ (−1, 0)} Fig. 3 shows the phase portrait in the Poincaré disc of the vector field Xν in (2) for ν varying inside   V := (D, F ) ∈ R2 : D ∈ (−1, 0), F ∈ (0, 1) . We take transverse sections 1 and 2 parametrized by s → (1 − s, 0) and s → (−1/s, 0) with s > 0, respectively, and define T (s; ν) to be the time that spends the solution of Xν with initial condition at (1 − s, 0) ∈ 1 to arrive at 2 . Thanks to the symmetry of Xν with respect to {y = 0}, it turns out that the period of the periodic orbit passing through (1 − s, 0) ∈ 1 is precisely 2T (s; ν). Consequently the emergence/disappearance of critical periodic orbits from ν corresponds to zeros of T  (s; ν) bifurcating from s = 0, more concretely to the number Z0 (T  ( · ; ν), ν ) as introduced in Definition 2.3. A key point to study these bifurcations is that T (s; ν) is the Dulac time associated to the passage through a hyperbolic saddle, which is at infinity (see Fig. 3 again). Therefore we can apply [30, Theorem A] to obtain the asymptotic expansion of T (s; ν) at s = 0 and use then [31, Theorem A] to compute its first coefficients Tij (ν). Next result gathers all this information, where ( · ) denotes the gamma function. 144 D. Marín and J. Villadelprat Journal of Differential Equations 332 (2022) 123–201 Proposition 3.2. Let T (s; ν) be the Dulac time of the passage from 1 to 2 of the saddle at infinity of the vector field Xν in (2) for ν ∈ V . Then the coefficients T00 , T01 , T10 and T20 in its asymptotic expansion at s = 0 are meromorphic functions on V that can be written as π  − λ T00 (ν) = √ , T01 (ν) = ρ1 (ν) 2 , 2 F (D + 1)  1−λ 2 √  1− 1 π  12 − λ1 T10 (ν) = ρ2 (ν)(2D + 1) 3 2λ1 , T20 (ν) = √ + ρ3 (ν)(2D + 1),  2 − 2λ 2F  1 − λ1 where λ(ν) = F 1−F is the hyperbolicity ratio of the saddle, √   λ+1  λ √ π 2 2 π ρ1 (ν) = 2(1−F ) F D+1 D F −1 and ρ2 (ν) =  , 2 F (D+1)3 and ρ3 is an analytic function on V ∩ 2 3 < F < 1 . In addition the following holds: (a) If ν0 ∈ V ∩ 2 3 < F < 1 then, for all υ > 0 small enough, T (s; ν) = T00 (ν) + T10 (ν)s + T20 (ν)s 2 + FL∞0 −υ (ν0 ) with L0 = min 3, λ(ν0 ) . Moreover T10 (− 12 , F ) = 0 and T20 (− 12 , F ) > 0 for all F ∈ ( 23 , 1). (b) If ν0 ∈ V ∩ 12 < F < 23 then, for all υ > 0 small enough, ∞ T (s; ν) = T00 (ν) + T10 (ν)s + T01 (ν)s λ + F2−υ (ν0 ). Furthermore T10 (− 12 , F ) = 0 and T01 (− 12 , F ) > 0 for all F ∈ ( 12 , 23 ). (c) If ν0 ∈ V ∩ F = 23 then, for all υ > 0 small enough, ∞ T (s; ν) = T00 (ν) + T10 (ν)s + T201 2 (ν)s 2 ω2−λ (s) + T200 2 (ν)s 2 + F3−υ (ν0 ), where T2002 and T 2 are analytic functions in a neighborhood of V ∩ {F = 2 }. Moreover 201 3 T10 (− 2 , 3 ) = 0 and T201 1 2 2 (− 1 , 2 ) = 0. 2 3 (d) If ν0 ∈ V ∩ F = 12 then, for all υ > 0 small enough, ∞ T (s; ν) = T00 (ν) + T101 1 (ν)sω1−λ (s) + T100 1 (ν)s + F2−υ (ν0 ), where 1 T101 (ν) = −ρ4 (ν)(F − 1/2)2 and T100 1 (ν) = ρ5 (ν)(D + 1/2) + ρ6 (ν)(F − 1/2) for some analytic positive functions ρi in a neighborhood of V ∩ {F = 12 } with ρ5 (− 12 , 12 ) = ρ6 (− 12 , 12 ). 145 D. Marín and J. Villadelprat Journal of Differential Equations 332 (2022) 123–201 Fig. 4. Phase portrait in the Poincaré disc of Xν for ν ∈ W with D < −1 (left) and D > −1 (right), where the center at (0, 0) is shifted to the left, the vertical invariant line is {x = 1} and the hyperbola { 12 y 2 − q(x) = 0} appears in boldface type. As we already explained, the proof of this result is postponed to Appendix A. The monomial order in each one of these asymptotic expansions is with respect to the strict partial order ≺ν0 given in [29]. Let us recall in its regard that we write f ≺ν0 g in case that g(s; λ) lim = 0. (s,ν)→(0,ν0 ) f (s; λ) For the monomials under consideration this order is preserved after derivation with respect to s, and so it is the good flatness properties of the remainder. Thus, as it occurs with the Taylor’s series of an analytic function, an upper bound for the number of zeros of T  (s; ν) that can bifurcate from s = 0 follows by identifying the first non-vanishing coefficient in the asymptotic expansion. For the proof and a precise statement of this result, which essentially follows by using the well-known derivation-division algorithm, the reader is referred to [29, Theorem C]. 3.2. Study of {F = 2, D ∈ (−2, 0)} and {D = G(F ) : F ∈ (1, 3/2)} Fig. 4 shows the phase portrait in the Poincaré disc of the vector field Xν in (2) for ν inside W := {(D, F ) ∈ R2 : F + D > 0, D < 0 and F > 1}. In this case the outer boundary of the period annulus of the center at (0, 0) is contained in the union of the line at infinity and an invariant hyperbola C := { 12 y 2 − q(x) = 0}, where q(x) = ax 2 + bx + c with D D−F +1 F −D−1 a := , b := and c := . 2(1 − F ) (1 − F )(1 − 2F ) 2F (1 − F )(1 − 2F ) One can verify that if ν ∈ W then q has two distinct real zeros, that we shall denote by p1 and p2 taking p1 < p2 . That being said, we place two transverse sections 1 and 2 parametrized by s → (p1 − s, 0) and s → (−1/s, 0) with s > 0, respectively, and define T (s; ν) to be the time that takes to the solution of Xν with initial condition at (p1 − s, 0) ∈ 1 to arrive at 2 . Then T (s; ν) is the Dulac time associated to the passage through a hyperbolic saddle at infinity, so that we can apply the results in [30,31] to obtain its asymptotic expansion at s = 0. This is important for the 146 D. Marín and J. Villadelprat Journal of Differential Equations 332 (2022) 123–201 proof of Theorem A because, exactly as in the previous case, the symmetry of Xν with respect to {y = 0} implies that the period of the periodic orbit passing through (1 − s, 0) ∈ 1 is 2T (s; ν). 1−p2 With regard to our next result we remark that 1−p 1 < 1 for all ν ∈ W , which is relevant since the hypergeometric function 2 F1 (a, b; c; · ) is holomorphic on C \ [1, +∞), see Appendix C. Let us also mention that B(·, ·) is the beta function. Proposition 3.3. Let T (s; ν) be the Dulac time of the passage from 1 to 2 of the saddle at infinity of the vector field Xν in (2) for ν ∈ W . Then the coefficients T00 , T01 , T10 and T20 in its asymptotic expansion at s = 0 are meromorphic functions on W that can be written as √ T00 (ν) = √ 2 F a(1−p1 ) 2 1 1, 12 ; 32 ; 1−p2 1−p1 , T01 (ν) = ρ1 (ν)B − λ, 12 , T10 (ν) = ρ2 (ν)B 1 − λ1 , − 12 2 F1 − 1 − λ1 , − 12 ; 12 − λ1 ; 1−p2 1−p1 and T20 (ν) = ρ3 (ν)B 1 − λ2 , − 32 2 F1 − 2 λ − 3, − 32 ; − 12 − λ2 ; 1−p 1−p1 + ρ4 (ν)T10 (ν), 2 where λ(ν) = 2(F1−1) is the hyperbolicity ratio of the saddle and, for i = 1, 2, 3, 4, ρi is an analytic positive function on W . In addition the following holds:   (a) If ν0 ∈ W ∩ 1 < F < 54 then, for all υ > 0 small enough, T (s; ν) = T00 (ν) + T10 (ν)s + T20 (ν)s 2 + FL∞0 −υ (ν0 )   with L0 = min(3, λ(ν0 )). Moreover T20 (ν) = 0 for all ν ∈ W ∩ 1 < F < 54 such that T10 (ν) = 0.   (b) If ν0 ∈ W ∩ 54 < F < 32 then, for all υ > 0 small enough, T (s; ν) = T00 (ν) + T10 (ν)s + T01 (ν)s λ + T20 (ν)s 2 + FL∞0 −υ (ν0 )   with L0 = λ(ν0 ) + 1, and there exists a unique ν ∈ W ∩ 54 < F < 32 such that T10 (ν ) = 0 and T01 (ν ) = 0. Furthermore T20 (ν ) < 0, the gradients of T01 and T10 at ν are linearly independent, and ν = (D , 43 ) with D = G( 43 ) ≈ −1.128.   (c) If ν0 ∈ W ∩ F = 54 then, for all υ > 0 small enough, ∞ T (s; ν) = T00 (ν) + T10 (ν)s + T201 2 (ν)s 2 ω2−λ (s) + T200 2 (ν)s 2 + F3−υ (ν0 ), where T2002 and T 2 are analytic functions in a neighborhood of W ∩ {F = 5 }. Moreover 201 4 T10 (D, 54 ) = 0 if and only if D = −1, and T201 2 (−1, 5 ) = 0. 4 147 D. Marín and J. Villadelprat Journal of Differential Equations 332 (2022) 123–201 (d) If ν0 ∈ W ∩ {F = 2} then, for all υ > 0 small enough, 1 1 ∞ T (s; ν) = T00 (ν) + T01 (ν)s λ + T101 2 (ν)sω1−2λ (s) + T100 2 (ν)s + F3/2−υ (ν0 ), 1 1 2 where T100 2 and T101 are analytic functions in a neighborhood of W ∩ {F = 2}. Moreover 1 T01 (D, 2) = 0 for all D ∈ (−2, 0), T101 2 (D, 2) = 0 if and only if D = − 12 , and the gradients 1 2 of T01 and T101 are linearly independent at (− 12 , 2). The proof of this result is postponed to Appendix A. Remark 3.4. The asymptotic expansions in Proposition 3.3 were already given in [23, Theorem 3.6] but only to second order. In that result it is given, among others, the expression of the coefficient T10 (ν) in terms of a definite improper integral. Furthermore, see [23, Proposition 3.11], it is proved by applying the Implicit Function Theorem that the set of those ν ∈ W1 := W ∩ {F < 3/2} such that T10 (ν) = 0 is the graphic of an analytic function D = G(F ). This is the function that appears in assertion (b) of Proposition 3.3. Thanks to the results in Appendix B we can now identify the improper integral as a hypergeometric function, so that we can write ν ∈ W1 : D = G(F ) = ν ∈ W1 : 2 F1 − 1 − λ1 , − 12 ; 12 − λ1 ; 1−p 1−p1 = 0 , 2 where p1 and p2 with p1 < p2 are the real roots of q(x) = 0 and λ(ν) = 1 2(F −1) .  Remark 3.5. In the statement of Proposition 3.3 we refer to some positive functions ρi ∈ C ω (W ). Let us mention that in the proof we show that 1 1 (p2 − p1 ) 2(F −1) 1 1 ρ1 (ν) = √ ρ2 (ν) = √ 2 2a (p2 − p1 )(1 − p1 ) F 2 2a (F − 1)(1 − p1 ) F −1 3 1 p1 − 1 + 2F (p2 − p1 ) ρ3 (ν) = √ ρ4 (ν) = 8 2a (p2 − p1 )2 (1 − p1 ) (p2 − p1 )(p1 − 1) We do not use the explicit expressions in this paper but they may be relevant for future applications.  3.3. Study of {F = 1, D ∈ (−1, 0)} Our aim in this section is to study the period function of the center at the origin of Xν for ν = (D, F ) with F ≈ 1 and D ∈ (−1, 0). To this end we introduce transverse sections 1 and 2 parametrized, respectively, by means of  (1 − s, 0) if F  1, σ1 (s; ν) := and σ2 (s; ν) := (−1/s, 0), (p1 − s, 0) if F > 1, where recall that q(x) = a(x − p1 )(x − p2 ) with p1 < p2 for F > 1. One can also check that limF →1+ p1 = 1. For each ν = (D, F ) ∈ (−1, 0) × (0, +∞) we define T (s; ν) as the time that 148 D. Marín and J. Villadelprat Journal of Differential Equations 332 (2022) 123–201 Fig. 5. Phase portrait of Xν for ν = (D, F ) ∈ (−1, 0) × (0, +∞) in the Poincaré disc with F  1 (left) and F > 1 (right). In this case, contrary to the previous ones, the singularity at infinity for F = 1 is not a hyperbolic saddle but a saddle-node. spends the solution of Xν starting at σ1 (s; ν) ∈ 1 to arrive at 2 . A key feature of this Dulac time is that the singularity for F = 1 is not a hyperbolic saddle but a saddle-node. Our next result gives the asymptotic expansion of T (s; ν) at s = 0 for F ≈ 1. We point out that this is relevant for the proof of Theorem A because the period of the periodic orbit of Xν passing through σ1 (s; ν) ∈ 1 is precisely 2T (s; ν) due to the symmetry of the vector field. Proposition 3.6. Let T (s; ν) be the Dulac time of the passage from 1 to 2 of the saddle-node unfolding at infinity {Xν }. Then there is an open neighborhood U of (−1, 0) × {1} such that T (s; ν) = T0 (ν) + T1 (ν)s + T2 (ν)s 2 + s 2 h(s; ν), where Ti ∈ C 0 (U ) and, setting  = s∂s , lims→0+ k h(s; ν) = 0 uniformly on compact sets of U for k = 0, 1, 2. Moreover T1 (D, 1) = 0 if, and only if, D = − 12 . Finally T2 (−1/2, 1) = 0. Proof. To study the saddle-node bifurcation that occurs at infinity we work in the projective plane RP 2 and perform the change of coordinates   1−x 1 (u, v) = p(x, y) := , . y y The meromorphic extension of Xν in these coordinates is given by 1  X̃ν := p∗ Xν = uP (u, v; ν)∂u + vQ(u, v; ν)∂v v with P (u, v; ν) = 1 − F − Du2 + (2D + 1)uv − (D + 1)v 2 and Q(u, v; ν) = −F − Du2 + (2D + 1)uv − (D + 1)v 2 . 149 D. Marín and J. Villadelprat Journal of Differential Equations 332 (2022) 123–201 Our first goal is to show that we can bring locally the saddle-node unfolding to a convenient normal form in order that we can apply the tools developed in [25] to study the asymptotic expansion of its Dulac map and Dulac time. With this aim, some long but easy computations show that the local analytic change of coordinates given by   u v (z, w) = (u, v) := √ ,√ , g(u, v) g(u, v) (2D+1) (D+1) 2 where g(u, v) := (2F −1)D uv − 2F D v − 1 2D , brings the vector field X̃ν to 1 X̄ν := z(z2 − 2(F − 1))∂z − w(2F − z2 )∂w , w Ū (z, w; ν)  − 1 (2D+1) (D+1) 2 2 with Ū (z, w; ν) := 2(2F −1) zw − 4F w − D 2 . A technical assumption in order to apply the results from [25] is that for each ν the Taylor’s series of (z, w) → Ū (z, w; ν) at (0, 0) is absolutely convergent for all (z, w) ∈ [−1, 1]2 . This is not fulfilled unless we perform a rescaling which is only well defined provided that ν varies inside a compact subset of (−1, 0) × (0, +∞) and this forces us to work locally. For this reason, as a first step in the proof, we will show a local version of the statement. More concretely, that for each ν = (D , 1) with D ∈ (−1, 0) there exists an open ball Bν = {ν ∈ R2 : ν − ν < δ} such that T (s; ν) = T0ν (ν) + T1ν (ν)s + T2ν (ν)s 2 + s 2 hν (s; ν), with Tiν continuous functions on Bν and lims→0+ k hν (s; ν) = 0 uniformly on Bν for k = 0, 1, 2. To begin with we take δ > 0 small enough so that the closure of Bν is inside (−1, 0) × ( 12 , +∞) and define     (D + 1) 2 2(2D + 1) −1/2 r := inf  w − zw  : |z|  1, |w|  1 and ν − ν  δ , (6) DF D(2F − 1)  which is strictly positive. The pull-back of X̄ν by the rescaling ρ(z, w) := (rz, rw) can now be written as in [25, Eq. 13] because one can easily verify that 1   ρ ∗ X̄ν = z(z2 − ε)∂z − w(2F /r 2 − z2 )∂w with ε := 2(F − 1)/r 2 wU (z, w; ν) and where the Taylor’s series of   − 1 Ū (rz, rw; ν) −D 2 (D + 1) 2 (2D + 1) 2 U (z, w; ν) := = 1+r w − zw r 2r 2F D D(2F − 1) at (z, w) = (0, 0) is absolutely convergent for all (z, w) ∈ [−1, 1]2 and ν ∈ Bν since, on account of (6), 150 D. Marín and J. Villadelprat Journal of Differential Equations 332 (2022) 123–201    + 1) 2 2  (D (2D + 1)  1 r  w − zw   for all (z, w, ν) ∈ [−1, 1]2 × Bν . 2F D D(2F − 1) 2 In these new rescaled coordinates, that we still denote by (z, w) for simplicity, the period annulus is inside the quadrant {w  0, z  ϑε } where √ ε if ε  0, ϑε := 0 if ε < 0. Setting  := ρ −1 ◦ , we take two auxiliary transverse sections n1 := −1  ({w = 1}) and n2 := −1  ({z = 1}) parameterized by σ n (s; ν) := −1 (s + ϑ , 1) and σ n (s; ν) := −1 (1, s), 1  ε 2  respectively (see Fig. 5). We define T (s; ν) and D(s; ν) to be the Dulac time and Dulac map of X̃ν from n1 to n2 , respectively. We remark that, by construction, T (s; ν) is the time that the solution of ρ ∗ X̄ν starting at the point (s + ϑε , 1) spends to arrive at {z = 1} and that the intersec- tion point is precisely (z, w) = (1, D(s; ν)). In this regard, since w = D(z; ν) is a trajectory of the vector field z(z2 − ε)∂z + w(2F /r 2 − z2 )∂w , see [25, p. 6417] for details, the application of (b) in Corollary A of [25] with {μ = 2,  = k = 2, λ = 2F /r 2 } shows that D(s; ν) = s 2 I(Bν ), (7) by shrinking δ > 0 if necessary. Here, and in what follows, I(Bν ) stands for some function h(s; ν) verifying that lims→0+ k h(s; ν) = 0 uniformly on ν ∈ Bν for k = 0, 1, 2. Furthermore, by applying Corollary B in the same paper with {μ = 2,  = k = 2} and shrinking δ > 0 again we can assert that T (s; ν) = b0 (ν) + b1 (ν)s + b2 (ν)s 2 + s 2 I(Bν ) (8) with bi ∈ C 0 (Bν ) for i = 0, 1, 2. Working in the original (x, y) coordinates, we consider next the transition times T1 ( · ; ν) and T2 ( · ; ν) of Xν from 1 to p −1 (n1 ) and from p −1 (n2 ) to 2 , respectively. We define moreover R( · ; ν) to be the transition map from 1 to p −1 (n1 ). Accordingly T (s) = T1 (s) + T ◦ R (s) + T2 ◦ D ◦ R (s), (9) where we omit the dependence on ν for the sake of shortness. By [22, Lemma 3.2], we have that T2 (s; ν) an analytic function at {0} × Bν with T2 (0; ν) = 0. Observe at this point that, setting  0 if F  1, ξν := 1 − p1 if F > 1, we can write the parametrization of 1 as σ1 (s; ν) = (1 − ξν − s, 0). We claim that there exist two functions f (ŝ; ν) and g(ŝ; ν), analytic at {0} × Bν , such that T1 (s; ν) = f (s + ξν ; ν) and R(s; ν) = g(s + ξν ; ν) − ϑε . (10) To show this let us consider two additional transverse sections  ˆ 1 and ˆ n parameterized re- 1 −1 spectively by σ̂1 (ŝ; ν) := (1 − ŝ, 0) and σ̂1 (ŝ; ν) := ( ◦ p) (ŝ, 1), which clearly are analytic n 151 D. Marín and J. Villadelprat Journal of Differential Equations 332 (2022) 123–201 at {0} × Bν . Moreover it is clear that they are related with 1 and n1 through σ1 (s; ν) = σ̂1 (s + ξν ; ν) and σ1n (s; ν) = σ̂1n (s + ϑε ; ν). That being said, the claim follows by noting that if we choose f (ŝ; ν) and g(ŝ; ν) to be, respectively, the transition time and transition map of Xν from  ˆ 1 to  ˆ n , which are clearly analytic at {0} × Bν , then the equalities in (10) hold. 1 Note moreover that g(ξν ; ν) = ϑε since R(0; ν) = 0. On account of the claim, by considering the second order Taylor’s development of f (ŝ; ν) and g(ŝ; ν) at ŝ = ξν , respectively, we get T1 (s; ν) = a0 (ν) + a1 (ν)s + a2 (ν)s 2 + s 2 I(Bν ) and R(s; ν) = c1 (ν)s + c2 (ν)s 2 + s 2 I(Bν ) with ai , ci ∈ C 0 (Bν ) and where we also use that ν → ξν is a continuous function. The combi- nation of the second expression above with (7) and (8) yields D ◦ R (s) = s 2 I(Bν ) and T ◦ R (s) = b̂0 (ν) + b̂1 (ν)s + b̂2 (ν)s 2 + s 2 I(Bν ), respectively, with b̂i ∈ C 0 (Bν ). Summing up, since T2 ◦ D ◦ R (s) = s 2 I(Bν ) due to T2 (0; ν) = 0, from (9) we can assert that T (s; ν) = T0ν (ν) + T1ν (ν)s + T2ν (ν)s 2 + s 2 hν (s; ν) (11) for some functions Tiν that are continuous on Bν and some hν ∈ I(Bν ). This concludes the proof of the local version of the statement, in which we remark that the coefficients Tiν (ν) and the remainder s 2 hν (s; ν) depend by construction on ν . Our next step will be to globalize them and to this end we define  U := Bν ν ∈(−1,0)×{1} which is clearly an open neighborhood of (−1, 0) × {1}. Consider now any ν1 , ν2 ∈ (−1, 0) × {1} such that Bν1 ∩ Bν2 = 0. Then, from (11), we get that T0ν1 (ν) − T0ν2 (ν) + T1ν1 (ν) − T1ν2 (ν) s + T2ν1 (ν) − T2ν2 (ν) s 2 + s 2 hν1 (s; ν) − hν2 (s; ν) = 0 for all s > 0 small enough and ν ∈ Bν1 ∩ Bν2 . Since hν1 − hν2 ∈ I(Bν1 ∩ Bν2 ), taking the limit s → 0+ on both sides of the above equality we deduce that T0ν1 = T0ν2 on Bν1 ∩ Bν2 . Similarly, but taking the first and second derivatives with respect to s, respectively, we get that T1ν1 = T1ν2 and T2ν1 = T2ν2 on Bν1 ∩ Bν2 . Hence, for i = 0, 1, 2, the local functions Tiν ∈ C 0 (Bν ) for ν ∈ (−1, 0) × {1} glue together into a well defined continuous function Ti on U . Exactly the same argument shows that the local functions hν ∈ I(Bν ) for ν ∈ (−1, 0) × {1} glue together into a well defined function h(s; ν) satisfying that lims→0+ k h(s; ν) = 0 uniformly on compact sets of U for k = 0, 1, 2. To show this last assertion it suffices to take a finite subcover Bν1 ∪ . . . ∪ Bνn of the given compact subset of U and use that h|Bνi ∈ I(Bνi ) for i = 1, 2, . . . , n. So far we have proved the first assertion in the statement. Let us turn to the proof of the second one. To this end the key point is that for those ν0 ∈ U ∩ {F < 1} we can also apply (a) in Proposition 3.2 to obtain that ∞ T (s; ν) = T00 (ν) + T10 (ν)s + T20 (ν)s 2 + F3−υ (ν0 ), (12) 152 D. Marín and J. Villadelprat Journal of Differential Equations 332 (2022) 123–201 where, setting λ(ν) = F 1−F ,   √ 1 √ π(2D+1)  1− 2λ  1 1 π  2−λ T10 (ν) =  and T20 (− 12 , F ) = √ . 2 F (1+D)3  32 − 2λ 1 2F  1− λ1 Hence, since Ti ∈ C 0 (U ), from (11) and (12) we can assert that 2D + 1 T1 (D, 1) = lim T10 (D, F ) = , F →1− (1 + D)3/2   1  1− 2λ (1) where we also use that limF →1−   = = √2 . Consequently, as desired, T1 (D, 1) = 0  32 − 2λ 1 ( 32 ) π if and only if D = − 12 . The same argument shows that √ π  12 T2 (−1/2, 1) = lim T20 (−1/2, F ) = √ = 0, F →1− 2 1 and this completes the proof of the result.  4. Distinguished cases This section is devoted to study three specific parameters. Recall that among the quadratic centers there are four nonlinear isochrones, see (3). Chicone and Jacobs show in [6, Theorem 3.1] that the criticality of the period function at the inner boundary (i.e., the center) of P is exactly 1 for each one of the nonlinear isochrones. In this section we prove that for two of them, namely ν = (− 12 , 2) and ν = (− 12 , 12 ), the criticality at the outer boundary (i.e., the polycycle) is also 1, see Propositions 4.2 and 4.3, respectively. In the same vein it is also well-known that the criticality at the inner boundary of any quadratic center is at most two, see [6, Theorem 3.2]. This maximum criticality is achieved in three parameter values, the so-called Loud points, which following the notation in [6] are given by ν = Li with    √ √   √ √  L1 := − 32 , 52 , L2 := −11+20 105 , 15−20 105 and L3 := −11−20 105 , 15+20 105 . (13) As we already explained in the introduction, we conjecture that the criticality at the outer bound- ary of any quadratic center is at most two, and that there are only three parameter values where this maximum criticality is attained. In this paper we identify and prove the validity of the conjec- ture for two of these parameters. We investigate one of them in this section, see Proposition 4.4. The other one was already studied in [24] and we postpone its treatment until the proof of Theo- rem A. The following is a sort of division theorem within the class of flat functions that will be used to study the criticality at the outer boundary for the above-mentioned isochrones. In its statement, and in what follows, we use the notation 0n = (0, 0, . . . , 0) ∈ Rn for the sake of shortness. Lemma 4.1. Let us fix K ∈ N ∪ {∞}, L  0 and n ∈ N. If f (s; μ1 , . . . , μn ) ∈ FLK (0n ) verifies f (s; μ1 , . . . , μk−1 , 0, . . . , 0) ≡ 0, for some k ∈ {1, 2, . . . , n}, 153 D. Marín and J. Villadelprat Journal of Differential Equations 332 (2022) 123–201 n then there exist fk , . . . , fn ∈ FLK−1 (0n ) such that f = i=k μi fi . Proof. We proceed by induction on n ∈ N. For the base case n = 1 we take f (s; μ1 ) ∈ FLK (01 ) 1 with f (s; 0) ≡ 0 and define f1 (s; μ1 ) := 0 ∂2 f (s; μ1 t)dt, so that f = μ1 f1 . To show that f1 ∈ FLK−1 (01 ) we use that, by hypothesis (see Definition A.2), for every ν = (ν0 , ν1 ) ∈ Z2≥0 with |ν| = ν0 + ν1  K − 1 there exist a neighborhood V ⊂ R of 0 and C, s0 > 0 such that |∂s 0 ∂μν11+1 f (s; μ1 )|  Cs L−ν0 for every μ1 ∈ V and s ∈ (0, s0 ). On account of this and applying ν the Dominated Convergence Theorem [37, Theorem 11.30], 1 1  ν  C |∂ f1 (s; μ1 )|  ν ∂ (∂2 f (s; μ1 t)) dt  |∂sν0 ∂2ν1 +1 f (s; μ1 t)|t ν1 dt  s L−ν0 ν1 + 1 0 0 for every μ1 ∈ V and s ∈ (0, s0 ). Hence f1 ∈ FLK−1 (01 ). To prove the inductive step we suppose that n > 1 and consider f (s; μ1 , . . . , μn ) ∈ FLK (0n ) verifying that f (s; μ1 , . . . , μk−1 , 0, . . . , 0) ≡ 0 for some k ∈ {1, 2, . . . , n}. It is clear that we can write f (s; μ1 , . . . , μn−1 , μn ) − f (s; μ1 , . . . , μn−1 , 0) = μn fn (s; μ1 , . . . , μn ) (14) with 1 fn (s; μ1 , . . . , μn ) := ∂n+1 f (s; μ1 , . . . , μn−1 , μn t)dt. 0 Similarly as for the base case, taking f ∈ FLK (0n ) into account, one can easily show that fn ∈ FLK−1 (0n ). Since f (s; μ1 , . . . , μn−1 , 0)|μk =...=μn−1 =0 ≡ 0, by the inductive hypothesis there exist fk (s; μ1 , . . . , μn−1 ), . . . , fn−1 (s; μ1 , . . . , μn−1 ) ∈ FLK−1 (0n−1 ) such that  n−1 f (s; μ1 , . . . , μn−1 , 0) = μi fi (s; μ1 , . . . , μn−1 ). i=k Due to FLK−1 (0n−1 ) ⊂ FLK−1 (0n ), see Definition A.2, the combination of this identity with (14)  shows that f = ni=k μi fi with fk , . . . , fn ∈ FLK−1 (0n ) as desired. This shows the inductive step and concludes the proof of the result.  We state next our first result about the bifurcation of critical periodic orbits from the outer boundary of an isochronous center. With regard to its proof let us advance that, after a conve- nient division in the space of coefficients, we proceed as in the proofs of Bautin [4, §3] and Chicone and Jacobs [6, Theorem 2.2] for the analogous results about limit cycles and critical 154 D. Marín and J. Villadelprat Journal of Differential Equations 332 (2022) 123–201 periods, respectively, bifurcating from the center. Here we tackle the bifurcation from the poly- cycle, which is more challenging because, contrary to the center, the period function cannot be analytically extended there. To overcome this difficulty it is crucial the fact that the flatness of the remainder in the asymptotic expansion is preserved after the derivation with respect to the parameters. Proposition 4.2. If ν0 = (− 12 , 2) then Crit (ν0 , Xν0 ), Xν = 1. Proof. We show first the upper bound Crit (ν0 , Xν0 ), Xν  1, which constitutes the difficult part of the proof. To this end, following the notation introduced in Section 3.2, we define P (s; ν) to be the period of the periodic orbit of Xν passing through the point (p1 − s, 0). Thanks to the reversibility of Xν with respect to {y = 0} it turns out that P (s; ν) = 2T (s; ν) where T ( · ; ν) is the Dulac time that we consider in Proposition 3.3. Thus, by applying (d) in that result and setting λ(ν) = 2(F1−1) , we can assert that 1 1 T (s; ν) = T00 (ν) + T01 (ν)s λ + T101 2 (ν)sω1−2λ (s) + T100 2 (ν)s + r1 (s; ν), ∞ where r1 ∈ F3/2−υ (ν0 ) for all υ > 0 small enough, the coefficients are analytic in a neighborhood 1 of ν0 = (− 12 , 2) and, moreover, the gradients of T01 and T101 2 are linearly independent at ν0 . Since one can verify that ∂s sωα (s) = (1 − α)ωα (s) − 1, the derivation of the above equality yields 1 1 1 s 1−λ T  (s; ν) = λT01 (ν) + 2λT101 2 (ν)s 1−λ ω1−2λ (s) + T100 2 (ν) − T101 2 (ν) s 1−λ + r2 (s; ν) where, by using Lemmas A.3 and A.4 in [29], the remainder r2 := s 1−λ ∂s r1 belongs to F1−υ ∞ (ν ) 0 ∞ because ∂s r1 ∈ F1/2−υ (ν0 ) and, on the other hand, s 1−λ ∞ ∈ F1/2−υ (ν0 ) due to λ(ν0 ) = 1/2. Note 1 furthermore that ν̂ = (ν) := λ(ν)T01 (ν), 2λ(ν)T101 2 (ν) is local analytic change of coordinates at ν0 = (− 2 , 2) such that (ν0 ) = (0, 0). We can thus write 1   R1 (s; ν̂) := s 1−λ(ν) T  (s; ν) = ν̂1 + ν̂2 s 1−λ̂ ω1−2λ̂ (s) + a(ν̂)s 1−λ̂ + h(s; ν̂), (15) ν=−1 (ν̂) where we set λ̂(ν̂) := λ(−1 (ν̂)) for shortness and define 1 1 a(ν̂) := T100 2 − T101 2 (−1 (ν̂)) and h(s; ν̂) := r2 (s; −1 (ν̂)) ∈ F1−υ ∞ (02 ). Recall at this point that the center at the origin of Xν0 is isochronous, so that T  (s; ν0 ) ≡ 0. Consequently, due to (ν0 ) = (0, 0), a(0, 0) = 0 and h(s; 0, 0) ≡ 0. By the Weierstrass Division Theorem (see for instance [11, Theorem 1.8]), the first equality implies that a(ν̂) = ν̂1 a1 (ν̂) + ν̂2 a2 (ν̂) with a1 and a2 analytic functions at (0, 0). On the other hand, by Lemma 4.1, h(s; ν̂) = ν̂1 h1 (s; ν̂) + ν̂2 h2 (s; ν̂) with hi ∈ F1−ν ∞ (0 ). Therefore, from 2 (15), 155 D. Marín and J. Villadelprat Journal of Differential Equations 332 (2022) 123–201     R1 (s; ν̂) = ν̂1 1 + a1 (ν̂)s 1−λ̂ + h1 (s; ν̂) + ν̂2 s 1−λ̂ ω1−2λ̂ (s) + a2 (ν̂)s 1−λ̂ + h2 (s; ν̂) ∞ (0 ), h (s; ν̂) tends to zero uniformly for ν̂ ≈ (0, 0) as s → 0+ (see Defi- Since hi ∈ F1−υ 2 i nition A.2). Due to λ(ν0 ) = 1/2, this is also the case of s 1−λ̂ and s 1−λ̂ ω1−2λ̂ (s) by (c) of Lemma A.4 in [29]. Hence there exists a neighborhood U of (0, 0) such that lims→0+ (1 + a1 (ν̂)s 1−λ̂ + h1 (s; ν̂)) = 1 uniformly on U . Accordingly, the function R1 (s; ν̂) R2 (s; ν̂) := = ν̂1 + ν̂2 (s; ν̂) (16) 1 + a1 (ν̂)s 1−λ̂ + h1 (s; ν̂) with s 1−λ̂ ω1−2λ̂ (s) + a2 (ν̂)s 1−λ̂ + h2 (s; ν̂) (s; ν) := 1 + a1 (ν̂)s 1−λ̂ + h1 (s; ν̂) belongs to the class Cs>0 ∞ (U ), see Definition A.1. We claim that, by shrinking U , there exists s0 > 0 such that R2 (s; ν̂) has at most one zero on (0, s0 ), counted with multiplicities, for all ν̂ = (ν̂1 , ν̂2 ) ∈ U \ {(0, 0)}. Indeed, to show this note first that if ν̂2 = 0 then R2 (s; ν̂) = ν̂1 = 0, so that there is nothing to be proved in this case. Let us study consequently the case ν̂2 = 0. To this end we observe that R2 (s; ν̂) = ν̂2  (s; ν̂) where, using a more compact notation,   ∞ s 1−λ̂ ω1−2λ̂ + a2 s 1−λ̂ + F1−υ   (s; ν̂) = ∂s ∞ 1 + a1 s 1−λ̂ + F1−υ ⎛ ⎞ ω1−2λ̂ λ̂ + a2 −a2 λ̂−1 + F ∞ s 1−λ̂ + a s 1−λ̂ + F ∞ 2  (1 − λ̂)a 1 + F ∞ ⎜ ω1−2λ̂ 1 2 −υ  ω1−2λ̂ 1−υ 2 −υ ⎟ 1  = ⎝ − ⎠. s λ̂ ∞ 1 + a1 s 1−λ̂ + F1−υ ∞ (1 + a1 s 1−λ̂ + F1−υ )2 Here we use the identity ∂s s b ωα (s) = s b−1 ((b − α)ωα (s) − 1) and that, by Lemmas A.3 and A.4 in [29], we have 1/ω1−2λ̂ ∈ F−υ ∞ (0 ) and s −λ̂ ∈ F ∞ 2 −1/2−υ (02 ) for all υ > 0 small enough due to λ̂(0, 0) = 12 and, moreover, that the inclusion FL∞ FL∞ ⊂ FL+L ∞  holds. We also remark that, by (a) of Lemma A.4 in [29], 1 |1 − 2λ̂| − (1 − 2λ̂) lim = uniformly for ν̂ ≈ (0, 0). s→0+ ω1−2λ̂ (s) 2 On account of this, from the above expression of  we obtain that s λ̂  (s; ν̂) lim = b(ν̂) uniformly for ν̂ ≈ (0, 0), s→0+ ω1−2λ̂ (s) where b(ν̂) := λ̂ + 12 (a2 − a2 λ̂ − 1)(|1 − 2λ̂| − (1 − 2λ̂)). Since λ̂(0, 0) = 1/2, it is clear that b(ν̂) is a non-vanishing continuous function in a neighborhood of (0, 0). Accordingly, due to R2 (s; ν̂) = ν̂2  (s; ν̂), we can assert that 156 D. Marín and J. Villadelprat Journal of Differential Equations 332 (2022) 123–201 s λ̂ R2 (s; ν̂) lim = ν̂2 b(ν̂) uniformly for ν̂ ≈ (0, 0). s→0+ ω1−2λ̂ (s) Since ωα (s) only vanishes at s = 1, by shrinking U if necessary, we can assert the existence of some s0 ∈ (0, 1) such that R2 (s; ν̂) = 0 for all s ∈ (0, s0 ) and ν̂ = (ν̂1 , ν̂2 ) ∈ U with ν̂2 = 0. Therefore, by Rolle’s Theorem, R2 (s; ν̂) has at most one zero on (0, s0 ) counted with multiplic- ities. This shows the validity of the claim for the case ν2 = 0. Recall finally that the period function P (s; ν) is twice the Dulac time T (s; ν). Thus, taking the claim into account, from (15) and (16) it turns out that V := −1 (U ) is an open neighborhood of ν0 = (− 12 , 2) verifying that P  (s; ν) has at most one isolated zero on (0, s0 ), counted with mul- tiplicities, for all ν ∈ V . (To be more precise, the claim applies for the punctured neighborhood V \{ν0 } and, on the other hand, P  (s; ν0 ) ≡ 0, so that it has not any isolated zero.) Hence, see Def- inition 2.3, Z0 (P  ( · ; ν), ν0 )  1. Therefore the upper bound Crit (ν0 , Xν0 ), Xν  1 follows from assertion (2a) in Lemma 2.4 since, using the notation in that result, P (s; ν) = P̂ (σ (s; ν); ν) with σ (s; ν) = (p1 − s, 0) for s ∈ [0, δ). Thus it only remains to show that this upper bound is achieved. To this end we recall that, by [23, Theorem A], ν0 = (− 12 , 2) is a local bifurcation value of the period function at the outer boundary, see Definition 2.10. Then, since the period annulus of the centers under consideration varies continuously, see Remark 2.6, by applying (a) in Lemma 2.16 we get that Crit (ν1 , Xν1 ), Xν  1. This completes the proof of the result.  The following is our second result about the criticality of the quadratic isochrones. Proposition 4.3. If ν0 = (− 12 , 12 ) then Crit (ν0 , Xν0 ), Xν = 1. Proof. We prove Crit (ν0 , Xν0 ), Xν  1 first, which is the most complicated part of the proof. To this end, for each s ∈ (0, 1) we denote by P (s; ν) the period of the periodic orbit of Xν passing thought the point (1 − s, 0) ∈ R2 , see Fig. 3. Then, on account of the reversibility of the vector field with respect to {y = 0}, it follows that P (s; ν) = 2T (s; ν), where T ( · ; ν) is the Dulac time introduced before Proposition 3.2. Thanks to that result we have thus the asymptotic expansion of P (s; ν) near the polycycle, which corresponds to s = 0. On the other hand, it is well known that the period function can be analytically extended to the center (which corresponds to s = 1 with this parametrization) because it is non-degenerated. The coefficients of the Taylor’s series of P  (s; ν) at s = 1 belong to the polynomial ring R[D, F ]. Chicone and Jacobs show (see Lemma 3.1 and Theorem 3.9 in [6]) that these coefficients are in the ideal generated by p2 (D, F ) = 10D 2 + 10DF − D + 4F 2 − 5F + 1 and p4 (D, F ) = 1540D 4 + 4040D 3 F + 1180D 3 + 4692D 2 F 2 + 1992D 2 F + 453D 2 + 2768DF 3 + 228DF 2 + 318DF − 2D + 784F 4 − 616F 3 − 63F 2 − 154F + 49 over the local ring R{D, F }νi of convergent power series at νi localized at any of the four quadratic isochrones ν0 := (− 12 , 12 ), ν1 := (0, 1) ν2 := (0, 14 ) and ν3 = (− 12 , 2). With regard to the first one, we claim that the ideal B := (p2 , p4 ) is equal to D + F, (2F − 1)2 over the local ring R{D, F }ν0 . Indeed, to prove this we use that 157 D. Marín and J. Villadelprat Journal of Differential Equations 332 (2022) 123–201      p2 q11 q12 (2F − 1)2 = (17) p4 q21 q22 D+F with q11 = 1, q12 = 10D − 1, q21 = 52F 2 + 44F + 49 and q22 = 576F 3 +(2192D −584)F 2 +(2500D 2 +812D −135)F +1540D 3 +1180D 2 +453D −2. (The idea to obtain this is that the zero of p2i |D=−F at F = 1/2 has multiplicity two.) From (17) we get that p2i ∈ D + F, (2F − 1)2 over the polynomial ring R[D, F ]. Conversely, since one can verify that the determinant q11 q22 − q21 q12 is different from zero at ν0 = (− 12 , 12 ), by in- verting the matrix in (17) it follows that (2F − 1)2 ∈ B and D + F ∈ B over the local ring R{D, F }ν0 . This proves the validity of the claim. Consequently, thanks to the result of Chicone and Jacobs mentioned above, we have the following equality between ideals over the local ring R{D, F }ν0 : B = D + F, (2F − 1)2 = P (i) (1; ν), i ∈ N . Now the crucial point is that the ideal P (i) (s0 ; ν), i ∈ N does not depend on the point s0 ∈ (0, 1]. Indeed, this follows verbatim the argument that R. Roussarie gives in [35, pp. 76–78] or [36, §4.3.1] to justify the same property about the ideal of the displacement map, the so-called Bautin ideal. Here we also use that, such as the displacement map, the period function P (s; ν) extends analytically to the non-degenerate center (i.e., s = 1). Accordingly, B = D + F, (2F − 1)2 = P (i) (s0 ; ν), i ∈ N for all s0 ∈ (0, 1]. (18) We turn next to the study of the period function near the polycycle (i.e., s = 0). In this regard by applying (d) in Proposition 3.2 we can assert that, for all υ > 0 small enough, ∞ P (s; ν) = 2T00 (ν) + 2T101 1 (ν)sω1−λ (s) + 2T100 1 (ν)s + F2−υ (ν0 ), where λ(ν) = F 1−F and 1 T101 (ν) = −ρ4 (ν)(F − 1/2)2 and T100 1 (ν) = ρ5 (ν)(D + 1/2) + ρ6 (ν)(F − 1/2) (19) for some analytic positive functions ρ4 , ρ5 and ρ6 in a neighborhood of ν0 = (− 12 , 12 ). Conse- quently, on account of the identity ∂s sωα (s) = (1 − α)ωα (s) − 1 and assertion (f ) of Lemma A.3 in [29], P  (s; ν) = 2λT101 1 (ν)ω1−λ (s) + 2 T100 1 − T101 1 ∞ (ν) + F1−υ (ν0 ). Furthermore, from (19) it follows that    ν̂ = (ν) := (F − 1/2) 2λρ4 (ν), 2 T100 1 − T101 1 (ν) is an analytic local change of coordinates in a neighborhood √ of ν = ν0 because one can verify that its Jacobian at ν0 = (− 12 , 12 ) is equal to −2ρ5 (ν0 ) 2ρ4 (ν0 ) = 0. Setting ν̂ = (ν̂1 , ν̂2 ), observe that then 158 D. Marín and J. Villadelprat Journal of Differential Equations 332 (2022) 123–201 P  (s; −1 (ν̂)) = −ν̂12 ω1−λ̂ (s) + ν̂2 + f (s; ν̂), (20) ∞ (0 ) and we denote λ̂ := λ(−1 (ν̂)) for shortness. where f ∈ F1−υ 2 We claim that B = (ν̂12 , ν̂2 ) over the local ring R{D, F }ν0 . To show this we note that  ν̂2 D=−F = (2F − 1) ρ6 − ρ5 (−F, F ) + 2(F − 1/2)2 ρ4 (−F, F ). Since ρ5 (ν0 ) = ρ6 (ν0 ) by (d) in Proposition 3.2, it follows that ρ6 − ρ5 (−F, F ) = (F − 1/2)r1 (F ) for some analytic function r1 at F = 1/2. Consequently ν̂2 |D=−F = (F − 1/2)2 r2 (F ) with r2 analytic at F = 1/2. Taking this into account, the Weierstrass Division Theorem (see [11, Theorem 1.8]) shows that ν̂2 = (D + F )q(ν̂) + (F − 1/2)2 r2 (F ) for some analytic function q at ν̂ = (0, 0) which, from (19), verifies q(0, 0) = 2ρ5 (ν0 ) = 0. Hence we can write  2    ν̂1 0 2λρ4 (ν) D+F = , ν̂2 q((ν)) r2 (F ) (F − 1/2)2 where the matrix has an analytic inverse at ν = ν0 . Taking (18) into account this shows that B = (ν̂12 , ν̂2 ) over the local ring R{D, F }ν0 , as desired. Recall at this point that the center of Xν0 is isochronous. Hence P  (s; ν0 ) ≡ 0. Thus, taking (ν0 ) = (0, 0) into account, from (20) we get that f (s; 0, 0) ≡ 0. Having this in mind we write the remainder in (20) as f (s; ν̂) = f1 (s; ν̂) + f2 (s; ν̂1 ) with f1 (s; ν̂) := f (s; ν̂1 , ν̂2 ) − f (s; ν̂1 , 0) and f2 (s; ν̂1 ) := f (s; ν̂1 , 0). Since f1 (s; ν̂1 , 0) ≡ 0, the application of Lemma 4.1 shows the existence of g1 ∈ F1−υ ∞ (0 ) such that f (s; ν̂) = 2 1 ν̂2 g1 (s; ν̂). Due to f2 (s; 0) ≡ 0 and again by Lemma 4.1, f2 (s; ν̂1 ) = ν̂1 g2 (s; ν̂1 ) with g2 ∈ ∞ (0 ). We also have g (s; 0) ≡ 0 because, otherwise, it would exist s > 0 such that F1−υ 2 2 0 g2 (s0 ; ν̂1 ) = 0 for all ν̂1 ≈ 0. In this case, taking the claim into account together with (18) and (20), P  (s0 ; −1 (ν̂)) = −ν̂12 ω1−λ̂ (s0 ) + ν̂2 + ν̂2 g1 (s0 ; ν̂) + ν̂1 g2 (s0 ; ν̂1 ) ∈ B = (ν̂12 , ν̂2 ). From here, since each gi (s0 ; ν̂) is analytic at ν̂ = (0, 0) and g2 (s0 ; ν̂1 ) = 0 for ν̂1 ≈ 0, we would get that ν̂1 ∈ (ν̂12 , ν̂2 ) over the local ring R{D, F }ν0 , which is clearly a contradiction. Concerning the analyticity of gi (s0 ; ν̂), let us remark that it follows by applying the Weierstrass Division Theorem thanks to the analyticity of f (s0 ; ν̂) at ν̂ = (0, 0), which in its turn follows from (20) noting that: • P  (s0 ; ν) is analytic at ν = ν0 because {Xν }ν∈R2 is an analytic family of the vector fields and hence, by Lemma 2.1, (s, ν) → P (s; ν) = P̂ ((1 − s, 0); ν) is analytic on (0, 1) × R2 , • the change of coordinates ν̂ = (ν) is analytic at ν = ν0 , and 159 D. Marín and J. Villadelprat Journal of Differential Equations 332 (2022) 123–201 e−x −1 • ωα (s0 ) is analytic at α = 0 because ωα (s0 ) = F (α ln s0 )α with F (x) = x . ∞ (0 ). Hence g2 (s; 0) ≡ 0 and, by Lemma 4.1 once again, f2 (s; ν̂1 ) = ν̂12 g3 (s; ν̂1 ) with g3 ∈ F1−υ 2 Summing-up all this information about the remainder, from (20) we get that P  (s; −1 (ν̂)) = −ν̂12 ω1−λ̂ (s) + F1−υ ∞ ∞ (02 ) + ν̂2 1 + F1−υ (02 ) . We are now in position to complete the proof by showing that there exist s0 > 0 and an open neighborhood U of ν̂ = (0, 0) such that ∞ (0 ) P  (s; −1 (ν̂)) 1 + F1−υ 2 G(s; ν̂) := ∞ = −ν̂12 + ν̂2 ∞ (0 ) ω1−λ̂ (s) + F1−υ (02 ) ω1−λ̂ (s) + F1−υ 2 has at most one zero on (0, s0 ), counted with multiplicities, for all ν̂ = (ν̂1 , ν̂2 ) ∈ U \ {(0, 0)}. This is clear in case that ν̂2 = 0. To tackle the case ν̂2 = 0 we compute the derivative with respect to s to obtain that     1 + F1−υ∞ ∞ 1 + F1−υ  G (s; ν̂) = ν̂2 ∂s ∞ = ν̂2 ∂s ∞ ) ω1−λ̂ (s) + F1−υ ω1−λ̂ (s)(1 + F1−2υ   ∞ 1 + F1−2υ ν̂2 ∞ ν̂2 = ν̂2 ∂s = (1 + F1−2υ )+ F∞ ω1−λ̂ (s) 2−λ̂ s ω (s) 2 ω1−λ̂ (s) −2υ 1−λ̂ ν̂2 ∞ ∞ ν̂2 ∞ = 1 + F1−2υ + s 2−λ̂ ω1−λ̂ (s)F−2υ = 1 + F1−3υ . s 2−λ̂ ω2 (s) s 2−λ̂ ω2 (s) 1−λ̂ 1−λ̂ Here, in the second equality we apply first assertion (c) of Lemma A.4 in [29] to get that ∞ for all υ > 0 small enough, due to λ̂(0, 0) = 1, and use next that F ∞ F ∞ ⊂ 1/ω1−λ̂ (s) ∈ F−υ −υ 1−υ ∞ F1−2υ 1 from (g) of Lemma A.3 in [29]. In the third equality, on account of 1+s − 1 ∈ F1∞ and by (h) of Lemma A.3 in [29], we use first the inclusion 1+F1∞ ⊂ 1 + F1−2υ ∞ . Then, by using (d) 1−2υ ∞ )(1 + F ∞ ) ⊂ and (g) of Lemma A.3 in [29], we expand the numerator to get that (1 + F1−υ 1−2υ ∞ . Next, in the fourth equality we use that ∂ ω (s) = s −α−1 and assertion (f ) of 1 + F1−2υ s α ∞ Lemma A.3 in [29] to deduce that ∂s F1−2υ ∞ . Finally in the last equality we apply (c) of ⊂ F−2υ ∞ and we use again that F ∞ F ∞ ⊂ F ∞ . Lemma A.4 in [29] to get that s 2−λ̂ ω1−λ̂ (s) ∈ F1−υ 1−υ −2υ 1−3υ On account of Definition A.2 we can assert the existence of some s0 ∈ (0, 1) and a neighbor- hood U of (0, 0) such that G (s; ν̂) = 0 for all s ∈ (0, s0 ) and ν̂ ∈ U with ν̂2 = 0. Consequently P  (s; −1 (ν̂)) has at most one isolated zero on (0, s0 ), counted with multiplicities, for all ν̂ ∈ U \ {(0, 0)}. Thus, on account of Definition 2.3 and the fact that (ν0 ) = (0, 0), we get Z0 (P  ( · ; ν), ν0 )  1. Finally the upper bound Crit (ν0 , Xν0 ), Xν  1 follows from asser- tion (2a) in Lemma 2.4 since, using the notation in that result, P (s; ν) = P̂ (σ (s; ν); ν) with σ (s; ν) = (1 − s, 0) for s ∈ [0, δ). Therefore it only remains to show that this upper bound is attained. To this end we recall that, by [23, Theorem A], ν0 = (− 12 , 12 ) is a local bifurcation value of the period function at the outer boundary, see Definition 2.10. Then, since the period annulus of the centers under consideration varies continuously, see Remark 2.6, by applying (a) in Lemma 2.16 we get that Crit (ν1 , Xν1 ), Xν  1. This finishes the proof of the result.  160 D. Marín and J. Villadelprat Journal of Differential Equations 332 (2022) 123–201 As we explain at the beginning of this section, the maximum criticality of the period function at the inner boundary is 2 and it is achieved at the three Loud points ν = Li , see (13). We refer the interested reader to the paper of Chicone and Jacobs [6] for a proof of this result. In a joint paper with P. Mardešić, see [23, Theorem 4.3], we prove that at each ν = Li there exists a germ of analytic curve that consists of local bifurcation values of the period function at the interior, see Definition 2.10. Since the period function extends analytically to the center, this follows readily by applying the Weierstrass Preparation Theorem. In our next result we identify a parameter ν = ν for which the criticality at the outer boundary is 2. Furthermore we prove that at ν = ν there exists a C 1 germ of curve of local bifurcation values of the period function at the interior. Hence, roughly speaking, this parameter is the mirror image at the outer boundary of one of the Loud points, see Fig. 6 and Remark 4.5. In the statement, following the notation introduced at the beginning of Section 3.2, for each s ∈ (0, p1 ) and ν ≈ ν we denote by P (s; ν) the period of the periodic orbit of Xν passing through the point (p1 − s, 0) ∈ R2 . We also remark that T10 and T01 are the coefficients given in Proposition 3.3, which vanish at ν = (G(4/3), 4/3). Proposition 4.4. Let us consider ν = (G(4/3), 4/3). Then the following holds: (a) Crit (ν , Xν ), Xν = 2. (b) There exist an open neighborhood U of ν and s0 > 0 such that the set  := {ν ∈ U ; there exists s ∈ (0, s0 ) such that P  (s; ν) = P  (s; ν) = 0} satisfies the following conditions: (b1) Each ν ∈  is a local bifurcation value of the period function at the interior, (b2) there are ε > 0 and a C 1 injective curve δ : (−ε, ε) → U with δ(0) = ν , δ((0, ε)) =  and such that δ  (0) = (0, 0) is tangent to {ν ∈ U ; T10 (ν) = 0}, (b3) for each ν ∈  there exists a unique sν ∈ (0, s0 ) such that P  (sν ; ν) = P  (sν ; ν) = 0 and, moreover, limν→ν sν = 0+ , (b4)  ⊂ {ν ∈ U ; T10 (ν) < 0 and T01 (ν) > 0}, and (b5) for any ν0 ∈  and any neighborhood V of ν0 there exist ν̄ ∈ V and different s1 , s2 ∈ (0, s0 ) such that P  (s1 ; ν̄) = P  (s2 ; ν̄) = 0. Proof. We observe first of all that σ (s; ν) := (p1 − s, 0) is a parametrization of the outer bound- ary of the period annulus verifying the hypothesis in Lemma 2.4. This will enable us to relate Crit (ν , Xν ), Xν with Z0 (P  ( · ; ν), ν ). That being said, thanks to the reversibility of the vector field with respect to {y = 0}, we note that P (s; ν) = 2T (s; ν), where T ( · ; ν) is the Dulac time considered in Proposition 3.3. From point (b) in that result we can assert that, for all υ > 0 small enough, ∞ P (s; ν) = 2T00 (ν) + 2T10 (ν)s + 2T01 (ν)s λ + 2T20 (ν)s 2 + F5/2−υ (ν ), where λ(ν) = 2(F1−1) , T10 (ν ) = T01 (ν ) = 0, T20 (ν ) < 0 and the gradients ∇T10 (ν ) and ∇T01 (ν ) are linearly independent. Due to T20 (ν ) = 0, by applying [30, Theorem C] we get that Z0 (P  ( · ; ν), ν )  2. (For readers convenience, let us explain that [30, Theorem C] is a general result addressed to the Dulac time which, by using the well-known derivation-division algorithm, gives a bound for Z0 (T  ( · ; ν), ν0 ) in terms of the position of the first non-vanishing 161 D. Marín and J. Villadelprat Journal of Differential Equations 332 (2022) 123–201 coefficient in the asymptotic expansion of T (s; ν) at s = 0.) Consequently, by assertion (2a) in Lemma 2.4, Crit (ν , Xν ), Xν  2. In addition, since F1 (s; ν) := P  (s; ν) = 2T10 (ν) + 2λT01 (ν)s λ−1 + 4T20 (ν)s + F3/2−υ ∞ (ν ) (21) and the gradients ∇T10 (ν ) and ∇T01 (ν ) are linearly independent, by [30, Proposition 4.2] it turns out that Z0 (P  ( · ; ν), ν )  2. As a matter of fact, from the proof of that result, this lower bound is achieved by means of two different sequences of zeros of P  ( · ; ν) and, therefore, by assertion (2b) in Lemma 2.4, Crit (ν , Xν ), Xν  2. Accordingly Crit (ν , Xν ), Xν = 2 and this proves (a). Let us turn next to the proof of the assertions in (b). For this purpose, from (21) and by applying Lemmas A.3 and A.4 in [29], we get F2 (s; ν) := s 2−λ P  (s; ν) = 2λ(λ − 1)T01 (ν) + 4T20 (ν)s 2−λ + F1−υ ∞ (ν ). (22) Setting Uε := {ν ∈ R2 : ν − ν < ε}, the map F := (F1 , F2 ) is well-defined for (s, ν) ∈ (0, ε) × Uε taking ε > 0 small enough. Since T10 (ν ) = T01 (ν ) = 0, T20 (ν ) = 0 and the gra- dients ∇T10 (ν ) and ∇T01 (ν ) are linearly independent, we can assume by reducing ε > 0 if necessary that ν̂ = (ν), defined by means of   T10 (ν) λ(ν)T01 (ν) (ν) := , , (23) 2T20 (ν) 2T20 (ν) is an analytic change of coordinates from Uε to the neighborhood Ûε̂ := (−ε̂, ε̂)2 of (0, 0) = (ν ). Recall that our aim is to study the solutions of the system of equations {P  = 0, P  = 0} which, on account of (21) and (22), is equivalent to {F1 = 0, F2 = 0}. In order to study the latter we first lift  to an analytic change of variables  given by (ŝ, ν̂) = (s, ν) := (s 2−λ(ν) , (ν)), which (diminishing ε and ε̂ if necessary) is defined from Uε := (0, ε) × Uε to Ûε̂ := (0, ε̂) × Ûε̂ , and then we consider the map F̂ : Ûε̂ → R2 defined by F̂ (ŝ, ν̂) = (F̂1 (ŝ, ν̂), F̂2 (ŝ, ν̂)) with F1 (−1 (ŝ, ν̂)) F2 (−1 (ŝ, ν̂)) F̂1 (ŝ, ν̂) := and F̂ 2 (ŝ, ν̂) := . 4T20 (−1 (ν̂)) 4T20 (−1 (ν̂)) By assertions (h) and (c) of Lemmas A.3 and A.4 in [29], respectively, it follows that λ̂−1 F̂1 (ŝ, ν̂) = ν̂1 + ν̂2 ŝ 2−λ̂ + f1 (ŝ; ν̂) and F̂2 (ŝ, ν̂) = (λ̂ − 1)ν̂2 + ŝ + f2 (ŝ; ν̂), ∞ (0 ) for some υ > 0 small enough and we set λ̂(ν̂) := λ(−1 (ν̂)) for where f1 , f2 ∈ F2−υ 2 shortness. Here we also use that F3−υ ∞ (0 ) ⊂ F ∞ (0 ) and s = ŝ 1/(2−λ̂) ∈ F ∞ (0 ) due to 2 2−υ 2 2−υ 2 λ(ν ) = 3/2. Observe on the other hand that, via the diffeomorphism , the system {P  (s; ν) = 0, P  (s; ν) = 0} on Uε is equivalent to the system {F̂1 (ŝ, ν̂) = 0, F̂2 (ŝ, ν̂) = 0} on Ûε̂ . With re- gard to the latter note that, by [29, Lemma A.1], the remainders f1 and f2 extend to C 1 functions in a neighborhood of (0, 0, 0) satisfying that ∇f1 (0, 0, 0) = ∇f2 (0, 0, 0) = (0, 0, 0). Observe in 162 D. Marín and J. Villadelprat Journal of Differential Equations 332 (2022) 123–201 particular that F̂2 (ŝ, ν̂) extends to a C 1 function in a neighborhood of (0, 0, 0). Hence, taking λ̂(0, 0) = 3/2 into account, by the Implicit Function Theorem there exists a C 1 function h(ŝ, ν̂1 ) in a neighborhood of (0, 0) such that, by shrinking ε̂ > 0 if necessary, F̂2 (ŝ, ν̂) = 0 with (ŝ, ν̂) ∈ Ûε̂ ⇔ ν̂2 = h(ŝ, ν̂1 ). Furthermore h satisfies h(0, ν̂1 ) ≡ 0 and ∇h(0, 0) = (−2, 0). Our next task is to substitute ν̂2 = h(ŝ, ν̂1 ) in F̂1 (ŝ, ν̂1 , ν̂2 ) = 0 and analyze the resulting equation. To this end we extend F̂1 (ŝ, ν̂1 , ν̂2 )|ν̂2 =h(ŝ,ν̂1 ) on a neighborhood of (ŝ, ν̂1 ) = (0, 0) by means of (ŝ, ν̂1 ) → ν̂1 + h(ŝ, ν̂1 )|ŝ|e(ŝ,ν̂1 ) + fˆ1 (ŝ, ν̂1 ),  λ̂−1  where e(ŝ, ν̂1 ) := and fˆ1 (ŝ, ν̂1 ) = f1 (ŝ; ν̂1 , h(ŝ, ν̂1 )) are clearly C 1 in a neigh- 2−λ̂ λ̂=λ̂(ν̂1 ,h(ŝ,ν̂1 )) borhood of (0, 0). We claim that the function g(ŝ, ν̂1 ) := h(ŝ, ν̂1 )|ŝ|e(ŝ,ν̂1 ) is C 1 in a neigh- borhood of (0, 0) as well and that its gradient vanishes at (0, 0). To show this notice first that g(0, ν̂1 ) = 0 and, consequently, ∂ν̂1 g(0, ν̂1 ) = 0. Moreover, using that h(0, ν̂1 ) = 0, we get h(ŝ, ν̂1 )|ŝ|e(ŝ,ν̂1 ) h(ŝ, ν̂1 ) − h(0, ν̂1 ) ∂ŝ g(0, ν̂1 ) = lim = lim lim |ŝ|e(ŝ,ν̂1 ) = ∂ŝ h(0, ν̂1 ) · 0 = 0 ŝ→0 ŝ ŝ→0 ŝ ŝ→0 because h is C 1 and e(0, 0) = 1 implies e(0, ν̂1 ) > 0 for ν̂1 ≈ 0. Similarly, if ŝ = 0 then   e(ŝ, ν̂1 ) ∂ŝ g(ŝ, ν̂1 ) = (∂ŝ h(ŝ, ν̂1 ))|ŝ|e(ŝ,ν̂1 ) + h(ŝ, ν̂1 )|ŝ|e(ŝ,ν̂1 ) log |ŝ|∂ŝ e(ŝ, ν̂1 ) + ŝ h(ŝ, ν̂1 ) − h(0, ν̂1 ) e(ŝ,ν̂1 )   = (∂ŝ h(ŝ, ν̂1 ))|ŝ|e(ŝ,ν̂1 ) + |ŝ| ŝ log |ŝ|∂ŝ e(ŝ, ν̂1 ) + e(ŝ, ν̂1 ) ŝ and ∂ν̂1 g(ŝ, ν̂1 ) = (∂ν̂1 h(ŝ, ν̂1 ))|ŝ|e(ŝ,ν̂1 ) + h(ŝ, ν̂1 )|ŝ|e(ŝ,ν̂1 ) log |ŝ|∂ν̂1 e(ŝ, ν̂1 ) tend to zero as ŝ → 0 uniformly on ν̂1 ≈ 0. This clearly implies that g is C 1 in a neighborhood of (0, 0) and ∇g(0, 0) = (0, 0), so that the claim is true. Thus, by applying the Implicit Function Theorem to the “extended equation” ν̂1 + h(ŝ, ν̂1 )|ŝ|e(ŝ,ν̂1 ) + fˆ1 (ŝ, ν̂1 ) = 0 and reducing ε̂ > 0 once again, we obtain a C 1 function (ŝ) on (−ε̂, ε̂) such that F̂1 (ŝ, ν̂1 , h(ŝ, ν̂1 )) = 0 with (ŝ, ν̂1 ) ∈ (0, ε̂) × (−ε̂, ε̂) if, and only if, ν̂1 = (ŝ). Moreover (0) =  (0) = 0. Accordingly, after shrinking ε̂ > 0 once again if necessary, we can assert that F̂1 (ŝ, ν̂) = F̂2 (ŝ, ν̂) = 0 with (ŝ, ν̂) ∈ Ûε̂ ⇔ ν̂ = (ν̂1 , ν̂2 ) = (ŝ), h(ŝ, (ŝ) . At this point, since we reduced the original ε̂ > 0, we also diminish ε > 0 so that (s, ν) = (s 2−λ(ν) , (ν)) is still a diffeomorphism from Uε into Ûε̂ . Then, from (21) and (22), the follow- 163 D. Marín and J. Villadelprat Journal of Differential Equations 332 (2022) 123–201 Fig. 6. Arrangement of the three types of local bifurcation curve (inner boundary, interior and outer boundary) near the parameters ν = L3 and ν = ν , see Remark 4.5, and ν = L1 and ν = (2, 2), see Remark 5.1. ing assertions are equivalent: (1) ν0 ∈  := {ν ∈ Uε ; there exists s ∈ (0, ε) such that P  (s; ν) = P  (s; ν) = 0}, (2) (ν0 ) = (ŝ0 ), h(ŝ0 , (ŝ0 )) for some ŝ0 ∈ (0, ε̂), (3) ν0 = δ(t0 ) for some t0 ∈ (0, ε̂), where δ(t) := −1 (t), h(t, (t)) . It is clear from these equivalences that δ : (−ε̂, ε̂) −→ Uε ⊂ R2 is a C 1 parametrized curve with δ(0) = ν satisfying that δ (0, ε̂) = . One can easily verify, taking  (0) = 0 and ∂1 h(0, 0) = 0 into account, together with the definition of  in (23), that δ  (0) is a non-zero vector tangent to {ν ∈ Uε ; T10 (ν) = 0}. In particular, on account of δ  (0) = (0, 0) and by reducing ε̂ > 0, we have that δ is one-to-one. This proves the assertion (b2) in the statement. Due to Z0 (P  ( · ; ν), ν )  2, and after shrinking ε > 0 if necessary, note also that the zeros of P  ( · ; ν) on (0, ε) can have at most multiplicity two. Therefore, since the interior of  = δ (0, ε̂) is empty (as a subset of R2 ), by applying Lemma 2.15 we can assert that each ν0 ∈  is a local bifurcation value of the period function at the interior, which shows the validity of (b1) in the statement. With regard to the assertions in (b3), we note that the uniqueness of sν and limν→ν sν = 0+ follow from the point (2) above using that ŝ → h(ŝ, (ŝ)) is invertible at ŝ = 0 and that, by definition, ŝ = s 2−λ(ν) . On the other hand, since P  (sν ; ν) = 0 for all ν ∈ , from (22) we get that T01 (ν)T20 (ν) < 0 for all ν ∈ . Here we also use that limν→ν sν = 0+ to take advantage of the properties of the remainder and the fact that λ(ν ) = 3/2. By arguing similarly, on account of P  (sν ; ν) = 0 for all ν ∈ , from (21) it follows that T01 (ν)T10 (ν) < 0 for all ν ∈ . Taking this into account the assertion in (b4) is a consequence of T20 (ν ) < 0, see (b) in Proposition 3.3. Finally, in order to prove (b5), let us consider ν ∈  and note that from (21) we obtain  lim ∂ν1 P  (s; ν), ∂ν2 P  (s; ν) s=s = 2∇T10 (ν ), ν→ν ν 164 D. Marín and J. Villadelprat Journal of Differential Equations 332 (2022) 123–201 ∞ where we use that the flatness of the remainder F3/2−υ (ν ) is preserved after derivation with respect to parameters, see Definition A.2. Similarly, in this case from (22) and using also P  (sν ; ν) ≡ 0, we get   3 lim s 2−λ(ν) ∂ν1 P  (s; ν), ∂ν2 P  (s; ν)  = ∇T01 (ν ). ν→ν s=sν 2 Thus, since the vectors ∇T10 (ν ) and ∇T01 (ν ) are linearly independent, so they are ∇P  (s; ν)|s=sν and ∇P  (s; ν)|s=sν for all ν ∈ Uε (after shrinking ε > 0 if necessary). That being said, we fix any ν0 ∈  and compute the second order Taylor’s expansion of P (s; ν) at s = sν0 , P  (s; ν) = P  (sν0 ; ν) + P  (sν0 ; ν)(s − sν0 ) + o(s − sν0 ). Then, due to P  (s; ν0 ) ≡ 0, P  (sν0 ; ν0 ) = P  (sν0 ; ν0 ) = 0 and the fact that the gradients ∇P  (sν0 ; ν) and ∇P  (sν0 ; ν) are linearly independent at ν = ν0 , the application of [30, Proposi- tion 4.2] shows that for each open neighborhood V of ν0 there exist ν̄ ∈ V and two s1 , s2 ∈ (0, ε) such that P  (s1 ; ν̄) = P  (s2 ; ν̄) = 0. This proves the validity of the assertion in (b5) and com- pletes the proof of the result.  Remark 4.5. Let us finish this section contextualizing the results in Proposition 4.4. In Fig. 6 we display the ellipse C that consists of local bifurcation values of the period function at the inner boundary (i.e., the center) of P. It corresponds, see [6, Lemma 3.1], to the vanishing of the first period constant p2 (ν) = 10D 2 + 10DF − D + 4F 2 − 5F + 1. Moreover the curve B consists of local bifurcation values at the outer boundary (i.e., the poly- cycle) of P, see [23, Theorem A]. It is made of the arc {D = G(F ) : F ∈ (1, 32 )} joining (− 32 , 32 ) and (− 12 , 1) together with several straight segments. According to Proposition 4.4, the germ of curve  at ν = ν (that we depict as 3 in Fig. 6 for consistency) is inside the set of local bi- furcation values of the period function at the interior of P. Inside this set there is also the germ of a curve δ3 at ν = L3 by [23, Theorem 4.3]. At this moment we do not have any analytical tool to fully characterize this set of interior bifurcations. We conjecture that 3 and δ3 connect with each other to delimit a region of parameters for which the corresponding center has exactly two critical periodic orbits. With regard to this conjecture it is proved in [23, Theorem 5.2] that the center of any parameter inside one of the two light gray regions in Fig. 6 has at least two critical periodic orbits, cf. Fig. 7. The boundary of these regions is inside C , B and 0 . For completeness let us explain that the curve 0 consists of those parameters such that the period function tends to 2π as the periodic orbits tend to the outer boundary.  5. Proof of Theorem A Proof of Theorem A. The statement covers all the parameters ν0 ∈ R2 outside the vertical seg- ments 0 := {D = −1, F ∈ [0, 1]} ∪ {D = 0, F ∈ [0, 14 )}. For simplicity in the exposition, instead of proving the five assertions in the statement separately, we split R2 \ 0 depending on the result and tool applied to study the corresponding criticality. For reader’s convenience we enumerate the different cases that we obtain in this way. 165 D. Marín and J. Villadelprat Journal of Differential Equations 332 (2022) 123–201 1. Let us consider first of all the set 1 := R2 \ (B ∪ U ), where recall (see Fig. 1) that U is the union of the dotted straight lines, whatever its color is, and B is the Jordan curve in boldface type. Then, by [23, Theorem A], we know that any ν0 ∈ 1 is a local regular value of the period function at the outer boundary, see (c) in Definition 2.10. On account of this, by (b) in Lemma 2.16 we get that Crit (ν0 , Xν0 ), Xν = 0. Here we also use that the period annuli of the Loud’s centers vary continuously, see Remark 2.6, and that the outer boundary of Pν for ν ∈ / B ∪ U is a hyperbolic polycycle, see for instance [23, §3.1]. 2. The criticality at 2 := {D = − 12 , F ∈ ( 12 , 1)} ∪ {F = 12 , D ∈ (− 12 , 0)} and 3 := {F = 12 , D ∈ (−1, − 12 )} follows from the results in Section 3.1. In this case σ (s; ν) = (1 − s, 0) is a parametrization of the outer boundary of the period annulus verifying the hypothesis (a), (b) and (c) in Lemma 2.4. Moreover denoting by P (s; ν) the period of the periodic orbit of Xν passing through σ (s; ν), we have that P (s; ν) = 2T (s; ν), where T is the Dulac map considered in Proposition 3.2. By applying this result we know that the first non-vanishing coefficient in the asymptotic expansion of P (s; ν) at s = 0 is the third one for all ν ∈ 2 . Therefore [30, Theorem C] implies that Z0 (P  ( · ; ν), ν0 )  1 for all ν0 ∈ 2 . On account of this, by assertion (2a) in Lemma 2.4 it follows that Crit (ν0 , Xν0 ), Xν  1 for all ν0 ∈ 2 . On the other hand, due to 2 ⊂ B , we know by [23, Theorem A] that these parameters are local bifurcation values of the period function at the outer boundary. Thus, since the period an- nuli of the Loud’s centers vary continuously (see Remark 2.6), by applying (a) in Lemma 2.16 we get that Crit (ν0 , Xν0 ), Xν  1 for all ν0 ∈ 2 . Hence Crit (ν0 , Xν0 ), Xν = 1 for all ν0 ∈ 2 . We turn now to the criticality in the segment 3 . So let us fix any ν0 = (D0 , 12 ) with D0 ∈ (−1, − 12 ) and note that then, by (d) in Proposition 3.2, T  (s; ν) = −ρ4 (ν)(F − 1/2)2 λω1−λ (s) − 1 + ρ5 (ν)(D + 1/2) + ρ6 (ν)(F − 1/2) + R(s; ν), where R ∈ F1−υ ∞ (ν ) for all υ > 0 small enough. To obtain the derivative of the Dulac 0 time, we use that ∂s (sωα (s)) = (1 − α)ωα (s) − 1 and that, by (f ) in Lemma A.3 in [29], ∞ (ν ) ⊂ F ∞ (ν ). From this equality, since λω ∂s F2−υ 0 1−υ 0 1−λ (s) − 1 tends to +∞ as (s, ν) → (0, ν0 ) due to λ(ν0 ) = 1 (see Definition 3.1), R(s; ν) tends to 0 as (s, ν) → (0, ν0 ), ρi (ν0 ) > 0 and D0 + 12 < 0, we can assert the existence of an open neighborhood V of ν0 and ε > 0 such that P  (s; ν) = 2T  (s; ν) < 0 for all ν ∈ V and s ∈ (0, ε). Consequently Z0 (P  ( · ; ν), ν0 ) = 0 and so, by applying (2c) in Lemma 2.4, we conclude that Crit (ν0 , Xν0 ), Xν = 0. 3. We turn next to study the horizontal segments 4 := F = 2, D ∈ (−2, 0) \ {− 12 } and the curve 5 := D = G(F ) : F ∈ (1, 32 ) . Here we set ν := (G(4/3), 4/3) because this parameter yields to a distinguished case. We begin by noting (see the first paragraph in Section 3.2) that σ (s; ν) = (p1 − s, 0) is a parametrization of the outer boundary of the period annulus verifying the assumptions in Lemma 2.4 and that if we denote the period of the periodic orbit of Xν passing through σ (s; ν) by P (s; ν) then P (s; ν) = 2T (s; ν), where T is the Dulac map considered in Proposition 3.3. Thus, by applying first that result and then [30, Theorem C] we obtain that Z0 (P  ( · ; ν), ν0 )  1 for all ν0 ∈ 4 ∪ 5 \ {ν }. Moreover [23, Theorem A] shows that these parameters are local bifurcation values of the period function at the outer boundary because 4 ∪ 5 ⊂ B . Since the period annuli of the Loud’s centers vary continuously (see Remark 2.6), by applying (a) in Lemma 2.16 we have Crit (ν0 , Xν0 ), Xν  1 for all ν0 ∈ 4 ∪ 5 \ {ν }. Therefore Crit (ν0 , Xν0 ), Xν = 1 for all ν0 ∈ 4 ∪ 5 \ {ν }. 166 D. Marín and J. Villadelprat Journal of Differential Equations 332 (2022) 123–201 On the other hand we have that Crit (ν , Xν ), Xν = 2 by assertion (a) in Proposi- tion 4.4. Finally the fact that there is a curve of local bifurcation values of the period function at the interior of P arriving at ν = ν tangent to B follows from assertion (b) in the same result. 4. Next we analyze the parameters in the segment 6 := {F = 1, D ∈ (−1, 0)}, that corresponds to a case in which there is a saddle-node singularity at the outer boundary of the period annulus. This is treated in Section 3.3, where we introduce the map  (1 − s, 0) if F  1, σ (s; ν) := (p1 − s, 0) if F > 1, that provides a parametrization of the outer boundary of the period annulus verifying the assumptions in Lemma 2.4. In addition if we denote by P (s; ν) the period of the periodic orbit of Xν passing through σ (s; ν) then P (s; ν) = 2T (s; ν), where T is the Dulac map considered in Proposition 3.6. From that result we get the existence of an open neighborhood U of 6 = (−1, 0) × {1} such that P  (s; ν) = 2T1 (ν) + 4T2 (ν)s + s ĥ(s; ν) with T1 , T2 ∈ C 0 (U ) and where ĥ(s; ν) and s∂s ĥ(s; ν) tend to zero as s → 0+ uniformly on compact subsets of U . We know moreover that T1 (ν) = 0 if, and only if, ν = ν := (− 12 , 1) and that T2 (ν ) = 0. If we take any ν0 ∈ 6 \{ν } then, thanks to the good properties of the remainder, we get that lim(s,ν)→(0,ν0 ) P  (s; ν) = 2T1 (ν0 ) = 0 and this easily implies Z0 (P  ( · ; ν), ν0 ) = 0. Hence, by applying assertion (2c) in Lemma 2.4, Crit (ν0 , Xν0 ), Xν = 0 for all ν0 ∈ 6 \ {ν }. In order to study the criticality of ν we observe that lim(s,ν)→(0,ν ) P  (s; ν) = 4T2 (ν ) = 0 and, consequently, Z0 (P  ( · ; ν), ν0 )  1 by Rolle’s Theorem. Therefore, by assertion (2a) in Lemma 2.4, Crit (ν , Xν ), Xν  1. On the other hand, the application of [23, Theorem A] together with (a) in Lemma 2.16 shows that Crit (ν , Xν ), Xν  1 due to ν ∈ B . Hence Crit (ν , Xν ), Xν = 1. 5. We proceed with the study of the segment 7 := {F = 0, D ∈ (−1, 0)} which, as in the previous case, corresponds to period annuli having a saddle-node singularity at the outer boundary. In order to compute the criticality of any ν0 ∈ 7 we apply the results obtained in [27]. In that paper it is proved that for each ν0 ∈ 7 there exist δ > 0, an open neighbor- hood V of ν0 and a continuous function σ : [0, δ) × V −→ RP 2 verifying the hypothesis in Lemma 2.4. Moreover, denoting the period of the periodic orbit of Xν passing through σ (s; ν) by P (s; ν), the proof of [27, Theorem B] shows that P  (s; ν) tends to −∞ as (s, ν) → (0, ν0 ). Consequently Z0 (P  ( · ; ν), ν0 ) = 0 and hence, by applying (c) in Lemma 2.4, we get that Crit (ν0 , Xν0 ), Xν = 0. 6. We analyze next the parameters inside the segment 8 := D = 0, F ∈ [ 14 , 2] . So let us fix any ν0 = (0, F0 ) with F0 ∈ [ 14 , 2]. By [23, Theorem A] we can assert that if F0 ∈ [ 12 , 2] then ν0 is a local bifurcation value of the period function at the outer boundary. On the other hand, if F0 ∈ [ 14 , 12 ] then we can conclude the same by applying [28, Theorem B]. Hence, since the period annuli of the Loud’s centers vary continuously, see Remark 2.6, assertion (a) in Lemma 2.16 shows that Crit (ν0 , Xν0 ), Xν  1 for all ν0 ∈ 8 . 7. From the results in [19,39] it follows that Crit (ν0 , Xν0 ), Xν = 0 for any parameter ν0 in- side the set 9 := {D = 0, F ∈ / [0, 2]} ∪ {D = −1, F < 0} ∪ {D + F = 0, F < 0}. Indeed, 167 D. Marín and J. Villadelprat Journal of Differential Equations 332 (2022) 123–201 in those papers the authors determine a region M in the parameter plane for which the cor- responding centers have a globally monotonic period function. Taking this into account the assertion follows easily from the fact that 9 is contained in the interior of M, see Defini- tion 1.1. 8. We consider now the half-line 10 := {D + F = 0, F > 1}, so let us take a parameter ν0 = (−F0 , F0 ) with F0 > 1. In this case the assertions with regard to its criticality follow from the results in [24]. It is proved there that there exists a function ξ = ξ(ν) in a neighborhood U of ν0 such that σ (s; ν) = 0, ξ(ν)(1 − s) is a C 0 map on [0, δ) × U verifying the hypothesis (a), (b) and (c) in Lemma 2.4. Moreover if we denote by P (s; ν) the period of the periodic orbit of Xν passing through σ (s; ν) then [24, Theorem B] shows that • Z0 (P  ( · ; ν), ν0 ) = 0 if F0 ∈/ [3/2, 2], • Z0 (P  ( · ; ν), ν0 ) = 1 if F0 ∈ [3/2, 2) and • Z0 (P  ( · ; ν), ν0 ) = 2 if F0 = 2. In the first case Crit (ν0 , Xν0 ), Xν = 0 by (2c) in Lemma 2.4, whereas in the second case the combination of (2a) and (2b) implies Crit (ν0 , Xν0 ), Xν = 1. In the third case, by applying (2a) we get Crit (ν0 , Xν0 ), Xν  2. To show that this upper bound is attained we also apply (2b) in Lemma 2.4 but to this end we must check the assumption that for each open neighborhood V of ν0 = (−2, 2) and δ > 0 there exist distinct s1 , s2 ∈ (0, δ) and ν̂ ∈ V such that P  (si ; ν̂) = 0 for i = 1, 2. To verify this we note first, see [24, §4], that we can write P  (s; ν) = δ1 (ν)f1 (s; ν) + δ2 (ν)f2 (s; ν) + f3 (s; ν) where the coefficients δ1 and δ2 are independent at ν0 in the sense of [30, Definition 4.1] and, for i = 1, 2, lims→0+ ffi+1 (s;ν) i (s;ν) = 0. On account of this and P  (s; ν0 ) ≡ 0, the fact that the mentioned assumption is verified follows from the proof of [30, Proposition 4.2]. Related with this let us also mention, see again [24, §4], that the ordered set (f1 , f2 , f3 ) is an extended complete Chebyshev system on (0, ε) for ε > 0 sufficiently small (see [13] for a definition). On the other hand, by assertion (a) in [24, Theorem C], there exist a neighborhood U of ν0 = (−2, 2), s0 > 0 and a injective C 0 curve ρ : (−ε, ε) −→ U satisfying ρ(0) = (−2, 2) and ρ((0, ε)) =  := {ν ∈ U ; there exists s ∈ (0, s0 ) such that P  (s; ν) = P  (s; ν) = 0}. Furthermore, assertion (b) in that result shows that the curve  has an exponentially flat contact with the straight line {F = 2} at ν0 = (−2, 2), see Fig. 6 where we denote  by 1 for consistency. Since the interior of  is clearly empty and the Chebyshev property explained above prevents the zeros of P  ( · ; ν) to have multiplicity greater than 2, the application of Lemma 2.15 shows that  consists of local bifurcation values of the period function at the interior. 9. Finally the fact that the criticality at the outer boundary of the isochrones ν1 := (− 12 , 2) and ν2 := (− 12 , 12 ) is 1 follows from Propositions 4.2 and 4.3, respectively. i=1 i ∪ {ν1 } ∪ {ν2 }, this concludes the proof of the result.  Since R2 \ 0 = ∪10 Remark 5.1. By assertion (e) in Theorem A there exists a germ of curve 1 at ν = (−2, 2), see Fig. 6, which is inside the set of local bifurcation values of the period function at the interior 168 D. Marín and J. Villadelprat Journal of Differential Equations 332 (2022) 123–201 Fig. 7. In red, blue and black, respectively, the bifurcation curves at the inner boundary, the outer boundary and the interior. Rendered with a red (respectively, blue) circle, the three parameters with criticality 2 at the inner (respectively, outer) boundary. Depicted with a violet square, the four isochrones. We conjecture (see Remark 5.2 for details) that the parameters in gray, orange and uncolored, respectively, correspond to centers with exactly 2, 1 and 0 critical periodic orbits. of P. Inside this set there is also the germ of a curve δ1 at ν = L1 by [23, Theorem 4.3]. Similarly as we explain in Remark 4.5 for 3 and δ3 , we conjecture that 1 and δ1 connect each other to delimit a region of parameters for which the period function has exactly two critical periodic orbits. In this regard [23, Theorem 5.2] shows that the center of any parameter inside the light gray sector has at least two critical periodic orbits. We know now, see point 8 in the proof of Theorem A, that ν = (−2, 2) is at the boundary of this region with exactly two critical periodic orbits. The numerical visualization of this fact is a challenging problem because 1 has a exponential flat contact with {F = 2} at ν = (−2, 2).  Remark 5.2. Following the observations made in Remarks 4.5 and 5.1 with regard to the local bifurcation curves displayed in Fig. 6, we conclude this section with the conjecture about the behavior of the period function in its whole domain. This is done in Fig. 7, which in turn is an update of the conjecture stated in our earliest paper on the issue, see [23, Figure 3]. Of course, as it is widely accepted, we conjecture that the center at the origin of (2) has at most two critical periodic orbits for any ν = (D, F ) ∈ R2 . More specifically, see Fig. 7, we conjecture that the parameters inside the gray, orange and uncolored regions, respectively, have exactly 2, 1 and 0 critical periodic orbits. The main difference with respect to our original conjecture is about the size of the gray region in the middle, which we know now to be much smaller and not arriving to the diagonal F = −D. This conjecture has been partially proved. Indeed, besides the two subsets inside the gray region with at least 2 critical periodic orbits explained in Remark 5.1, we can quote [23, Theorem 5.1], which shows that all the parameters in the orange region have at least 169 D. Marín and J. Villadelprat Journal of Differential Equations 332 (2022) 123–201 one critical periodic orbit. Furthermore the authors in [19,39] determine an unbounded and rather large subset in the parameter plane for which the corresponding center has no critical periodic orbits.  Appendix A. Coefficient formulas A.1. Previous results about the Dulac time This appendix is entirely devoted to the proof of Propositions 3.2 and 3.3 in Section 3. For the parameter values under consideration in both results, and thanks to the symmetry of the vector field Xν in (2), it turns out that the period function is twice the Dulac time associated to the pas- sage through a hyperbolic saddle at infinity. The asymptotic expansion of this type of passage is the subject of our recent papers [29–31] and in order to prove the results in Section 3 we strongly rely on the tools developed there. For this reason we first summarize for reader’s convenience the definitions and results from those papers that are indispensable here. We recap the results in three theorems. In short, Theorem A.3 will provide us with the monomial scale needed in each asymptotic expansion, which only depends on the hyperbolicity ratio of the saddle, whereas Theorem A.4 will give the explicit expression of their coefficients in terms of a sort of Mellin transform that is introduced in Theorem A.5. In order to facilitate the application of the above-mentioned results we particularize them to fit in the context needed to prove Propositions 3.2 and 3.3. Thus, following the notation that we use in [30], let us consider the parameter μ̂ := (λ, μ) ∈ Ŵ := (0, +∞) × W , where W is an open set of RN , and the family of vector fields {Xμ̂ }μ̂∈Ŵ with 1  Xμ̂ (x1 , x2 ) := x1 P1 (x1 , x2 ; μ̂)∂x1 + x2 P2 (x1 , x2 ; μ̂)∂x2 , (24) x2 where • P1 and P2 belong to C ω (U × Ŵ ) for some open set U of R2 containing the origin, • P1 (x1 , 0; μ̂) > 0 and P2 (0, x2 ; μ̂) < 0 for all (x1 , 0), (0, x2 ) ∈ U and μ̂ ∈ Ŵ , • λ = − PP2 (0,0; μ̂) (0,0;μ̂) . 1 Moreover, for i = 1, 2, let σi : (−ε, ε) × Ŵ −→ i be a C ω transverse section to Xμ̂ at xi = 0 defined by σi (s; μ̂) = σi1 (s; μ̂), σi2 (s; μ̂) such that σ1 (0, μ̂) ∈ {(0, x2 ); x2 > 0} and σ2 (0, μ̂) ∈ {(x1 , 0); x1 > 0} for all μ̂ ∈ Ŵ . Then The- orem A.3 is concerned with the time T (s; μ̂) that spends the solution of Xμ̂ passing through the point σ1 (s; μ̂) ∈ 1 to arrive at 2 , see Fig. 8. More concretely it shows that T (s; μ̂) has an asymptotic expansion at s = 0 with the remainder having good flatness properties with respect to the parameters. We specify these properties in the following two definitions. Definition A.1. Given an open subset U ⊂ Ŵ ⊂ RN +1 , we say that a function ψ(s; μ̂) belongs ∞ (U ) if there exists an open neighborhood  of to the class Cs>0 170 D. Marín and J. Villadelprat Journal of Differential Equations 332 (2022) 123–201 Fig. 8. Definition of T ( · ; μ̂), where ϕ(t, p; μ̂) is the solution of Xμ̂ passing through the point p ∈ U at time t = 0. {(s, μ̂) ∈ RN +2 ; s = 0, μ̂ ∈ U } = {0} × U in RN +2 such that (s, μ̂) → ψ(s; μ̂) is C ∞ on  ∩ (0, +∞) × U .  Definition A.2. Consider an open subset U ⊂ Ŵ ⊂ RN +1 . Given any L ∈ R and μ̂0 ∈ U , we say that a function ψ(s; μ̂) ∈ Cs>0 ∞ (U ) is L-flat with respect to s at μ̂ if for each ν = 0 N +2 (ν0 , . . . , νN +1 ) ∈ Z≥0 there exist a neighborhood V of μ̂0 and C, s0 > 0 such that    ∂ |ν| ψ(s; μ̂)     ν   Cs L−ν0 for all s ∈ (0, s0 ) and μ̂ ∈ V ,  ∂s 0 ∂ μ̂11 · · · ∂ μ̂N +1  ν ν N+1 where |ν| = ν0 + · · · + νN +1 . In this case!we write ψ ∈ FL∞ (μ̂0 ). If W is a (not necessarily open) subset of U then we define FL∞ (W ) := μ̂0 ∈W FL∞ (μ̂0 ).  Next result merges the statements of Theorem 1.6, Theorem 4.3 and Corollary B in [31]. Following the notation in that paper, we particularize them to the case (n1 , n2 ) = (0, 1) for sim- plicity. Moreover, for the sake of shortness, we only include those items that will be used in the present paper. Theorem A.3. Let T (s; μ̂) be the Dulac time of the hyperbolic saddle (24) from 1 and 2 . Then, setting D00 = ∅, D10 = N 1 , D01 = N, D20 = N 2 and D02 = N 2 , for each (i, j ) ∈ {(0, 0), (1, 0), (0, 1), (2, 0), (0, 2)} there exists a meromorphic function Tij (μ̂) on Ŵ = (0, +∞) × W , having poles only along Dij × W , such that the following assertions hold: (1) If λ0 ∈ (1, 2) then T (s; μ̂) = T00 (μ̂) + T10 (μ̂)s + T01 (μ̂)s λ + T20 (μ̂)s 2 + FL∞ ({λ0 } × W ) for any L ∈ 2, λ0 + 1 . ∞  λ0 > 2 then T (s; μ̂) = T00 (μ̂) + T10 (μ̂)s + T20 (μ̂)s + FL ({λ0 } × W ) for any L ∈ (2) If 2 2, min(3, λ0 ) . (3) If λ0 = 12 then T (s; μ̂) = T00 (μ̂) + T01 (μ̂)s λ + sT λ100 (ω; μ̂) + FL∞ ({λ0 } × W ) for any L ∈ [1, 32 ), where ω = ω(s; α), α = 1 − 2λ and T λ100 (w; μ̂) ∈ C ∞ (Û )[w] for some open neighborhood Û of {λ0 } × W . Moreover 171 D. Marín and J. Villadelprat Journal of Differential Equations 332 (2022) 123–201 λ T 100 (ω; μ̂) = T10 (μ̂) + T02 (μ̂)(1 + αω) for λ = λ0 . (4) If λ0 = 1 then T (s; μ̂) = T00 (μ̂) + sT 100 (ω; μ̂) + FL∞ ({λ0 } × W ) for any L ∈ (1, 2), where λ ω = ω(s; α), α = 1 − λ and T 100 (w; μ̂) ∈ C ∞ (Û )[w] for some open neighborhood Û of λ {λ0 } × W . Moreover T λ100 (ω; μ̂) = T10 (μ̂) + T01 (μ̂)(1 + αω) for λ = λ0 . (5) If λ0 = 2 then T (s; μ̂) = T00 (μ̂) + T10 (μ̂)s + s 2 T 200 (ω; μ̂) + FL∞ ({λ0 } × W ) for any λ  L ∈ 2, 3 , where ω = ω(s; α), α = 2 − λ and T 200 (w; μ̂) ∈ C ∞ (Û )[w] for some open λ neighborhood Û of {λ0 } × W . Moreover λ T 200 (ω; μ̂) = T20 (μ̂) + T01 (μ̂) (1 + αω) for λ = λ0 . We focus next on the expression of the coefficients Tij and the result that we state below in this regard follows from assertion (c) in [31, Theorem A] particularized to (n1 , n2 ) = (0, 1). In its statement we use the following functions: u   u   P1 (0, z) 1 dz P2 (z, 0) dz L1 (u) := exp + L2 (u) := exp +λ P2 (0, z) λ z P1 (z, 0) z 0 0 1 L2 (u) A1 (u) := A2 (u) := (25) P2 (0, u) P1 (u, 0)   P1 M1 (u) := L1 (u)∂1 (0, u) B1 (u) := L1 (u)∂1 P2−1 (0, u) P2 C1 (u) := L21 (u)∂12 P2−1 (0, u) + 2L1 (u)M̂1 (1/λ, u)∂1 P2−1 (0, u) Here, given α ∈ R \Z≥0 and a real valued function f (x) that is C ∞ in an open interval containing x = 0, fˆ(α, x) is a sort of incomplete Mellin transform (see Theorem A.5 below). Moreover, for the sake of shortness, in the following statement we use the compact notation σij k for the kth derivative at s = 0 of the j th component of σi (s; μ̂), i.e., σij k (μ̂) := ∂sk σij (0; μ̂). Also with regard to the statement, note that Dij refers to the discrete sets in Theorem A.3. Theorem A.4. For each (i, j ) ∈ {(0, 0), (1, 0), (0, 1), (2, 0)}, the following expression of Tij (μ̂) is valid provided that λ ∈ / Dij : T00 (μ̂) = − σ120 Â1 (−1, σ120 ), λ σ120 σ111 T01 (μ̂) = λ Lλ (σ Â2 (λ, σ210 ), σ210 1 120 ) 172 D. Marín and J. Villadelprat Journal of Differential Equations 332 (2022) 123–201 σ121 σ120 σ111 T10 (μ̂) = − − B̂1 1/λ − 1, σ120 , P2 (0, σ120 ) L1 (σ120 ) and σ120 σ122 1 2 T20 (μ̂) = − − σ121 ∂2 P2−1 (0, σ120 ) − σ121 σ111 ∂1 P2−1 (0, σ120 ) 2σ120 P2 (0, σ120 ) 2 2 σ120 σ111 σ120 σ111 − Ĉ1 (2/λ − 1, σ120 ) − S1 B̂1 (1/λ − 1, σ120 ), 2L21 (σ120 ) L1 (σ120 ) where   σ112 σ121 P1 σ111 S1 = − (0, σ120 ) − M̂1 (1/λ, σ120 ). 2σ111 σ120 P2 L1 (σ120 ) As we already explained, the following result (that merges Theorem B.1 and Corollary B.3 in [31]) is the third ingredient needed in the proof of Propositions 3.2 and 3.3. Theorem A.5. Consider an open interval I of R containing x = 0 and an open subset U of RN . (a) Given f (x; ν) ∈ C ∞ (I × U ), there exists a unique fˆ(α, x; ν) ∈ C ∞ ((R \ Z≥0 ) × I × U ) such that x∂x fˆ(α, x; ν) − α fˆ(α, x; ν) = f (x; ν). −α (b) If x ∈ I \ {0} then ∂x (fˆ(α, x; ν)|x|−α ) = f (x; ν) |x|x and, taking any k ∈ Z≥0 with k > α,  k−1 i x   ∂ f (0; ν) ds fˆ(α, x; ν) = x x + |x| i f (s; ν) − T0k−1 f (s; ν) |s|−α , α i!(i − α) s i=0 0  where T0k f (x; ν) = ki=0 i!1 ∂xi f (0; ν)x i is the k-th degree Taylor polynomial of f (x; ν) at x = 0. (c) If f (x; ν) is analytic on I × U then fˆ(α, x; ν) is analytic on (R \ Z≥0 ) × I × U . Finally, for each (α0 , x0 , ν0 ) ∈ Z≥0 × I × U the function (α, x, ν) → (α0 − α)fˆ(α, x; ν) extends analytically to (α0 , x0 , ν0 ). (d) If f (x; ν) = x n g(x; ν) with g ∈ C ∞ (I × U ) and n ∈ N then fˆ(α, x; ν) = x n ĝ(α − n, x; ν). The following simple observation will be useful in order to study the coefficients of the asymp- totic expansions that we shall deal with.  Remark A.6. If m i=1 ai x + ψ(x) = 0 for all x ∈ (0, ε), where λi ∈ R with λ1 < λ2 < · · · < λi λm , a1 , a2 , . . . , am ∈ R and ψ(x) = o(x λm ) then a1 = a2 = · · · = am = 0.  We are now in position to begin the proof of the two first results in Section 3. 173 D. Marín and J. Villadelprat Journal of Differential Equations 332 (2022) 123–201 Fig. 9. Projective coordinate change in the proof of Proposition 3.2. A.2. Proof of Proposition 3.2 Proof of Proposition 3.2. We follow the approach in [22, §5] to take advantage of the general setting developed in [30]. To this end we will work on an extended parameter space ν̄ ∈ V̄ that we specify as follows. Firstly, introducing an auxiliary parameter η ≈ 0, we consider two local η η transverse sections 1 and 2 parametrized respectively by s → (1 − s, η) and s → (− 1s , ηs ), for s > 0, cf. Fig. 3. Secondly, taking any α, β ∈ R such the straight line y = αx + β does not η η intersect any solution of Xν while traveling from 1 to 2 . One can readily see that a sufficient condition for this to hold is that α + β < η and η > −α. Then, setting ν̄ := (D, F, α, β, η), we will work on the extended parameter space   V̄ := ν̄ ∈ R5 : D ∈ (−1, 0), F ∈ (0, 1), α + β < η, −α < η . Taking this into account, we consider the projective change of coordinates, see Fig. 9,   1−x 1 (x1 , x2 ) = , . y − αx − β y − αx − β η η One can verify that in these coordinates the parametrizations of 1 and 2 become     s 1 1+s s σ1 (s) := , and σ2 (s) := , , η + αs − α − β η + αs − α − β α + η − βs α + η − βs (26) respectively, whereas the vector field (2) is brought to 1 X̄ν̄ := x1 P1 (x1 , x2 ; ν̄)∂x1 + x2 P2 (x1 , x2 ; ν̄)∂x2 (27) x2 with 174 D. Marín and J. Villadelprat Journal of Differential Equations 332 (2022) 123–201 P1 (x1 , x2 ; ν̄) = (1 − αx1 )2 + (α + β)x2 − x22 + (1 − α 2 − αβ)x1 x2 − F (1 + α(x2 − x1 ) + βx2 )2 − D(x1 − x2 )2 and P2 (x1 , x2 ; ν̄) = α 2 x12 − αx1 − x22 + (1 − α 2 − αβ)x1 x2 − F (1 + α(x2 − x1 ) + βx2 )2 − D(x1 − x2 )2 . The reason why we introduce the auxiliary parameters α, β and η is because the computations are much easier taking the projective change of coordinates that sends y = 0 to infinity (i.e., with α = β = 0), which is not compatible with the placement of the original transverse sections (i.e., with η = 0). Since the parameters ν̄ with α = β = η = 0 are in the boundary of the admissible set V̄ , we will work in the interior and then make a limit argument. By introducing these auxiliary parameters we end up in a setting where the assumptions to apply the results in Section A.1 are fulfilled. Observe in particular that P1 and P2 are analytic on R2 × V̄ . Following the notation introduced there, note that the hyperbolicity ratio of the saddle P2 (0, 0) F λ=− = P1 (0, 0) 1 − F η η depends on μ = ν̄. We denote the Dulac time of X̄ν̄ from 1 to 2 by T̄ (s; ν̄). Note that, by construction, it does not depend on α and β as long as α + β < η and η > −α holds. Moreover, and this is the key point for our purposes, the Dulac time T (s; ν) in the statement is precisely T̄ (s; ν̄) for η = 0. Let us fix any ν̄0 = (D0 , F0 , α0 , β0 , η0 ) ∈ V̄ with F0 ∈ [ 12 , 1). Observe that λ(ν̄0 ) > 2 if F0 ∈ ( 3 , 1), λ(ν̄0 ) ∈ (1, 2) if F0 ∈ ( 12 , 23 ), λ(ν̄0 ) = 2 if F0 = 23 and λ(ν̄0 ) = 1 if F0 = 12 . Consequently, 2 by applying (2), (1), (5) and (4) in Theorem A.3, respectively, we get that (a  ) T̄ (s; ν̄) = T̄00 (ν̄) + T̄10 (ν̄)s + T̄20 (ν̄)s 2 + FL∞0 −υ (ν̄0 ) if F0 ∈ ( 23 , 1), where L0 = min(3, λ(ν̄0 )), ∞ (ν̄ ) if F ∈ ( 1 , 2 ), (b ) T̄ (s; ν̄) = T̄00 (ν̄) + T̄10 (ν̄)s + T̄01 (ν̄)s λ + F2−υ 0 0 2 3  (c ) T̄ (s; ν̄) = T̄00 (ν̄) + T̄10 (ν̄)s + T̄201 (ν̄)s ω2−λ (s) + T̄200 (ν̄)s 2 + F3−υ 2 2 2 ∞ (ν̄ ) if F = 3 , where 0 0 2 T̄201 (ν̄) and T̄200 (ν̄) are smooth in a neighborhood of {ν̄ ∈ V̄ : λ(ν̄) = 2} and, moreover, 2 2 2 T̄201 (ν̄) = (2 − λ)T̄01 (ν̄) and T̄200 2 (ν̄) = T̄20 (ν̄) + T̄01 (ν̄) for λ(ν̄) = 2, (d  ) T̄ (s; ν̄) = T̄00 (ν̄) + T̄101 ∞ 1−λ (s) + T̄100 (ν̄)s + F2−υ (ν̄0 ) if F0 = 2 , where T̄101 (ν̄) and 1 (ν̄)sω 1 1 1 T̄100 (ν̄) are smooth in a neighborhood of {ν̄ ∈ V̄ : λ(ν̄) = 1} and, moreover, 1 1 T̄101 (ν̄) = (1 − λ)T̄01 (ν̄) and T̄100 1 (ν̄) = T̄10 (ν̄) + T̄01 (ν̄) for λ(ν̄) = 1. Here υ is a small enough positive number depending on ν̄0 . Furthermore the coefficients T̄ij (ν̄) are meromorphic functions on V̄ with poles only at those F ∈ (0, 1) such that λ(ν̄) = 1−F F ∈ Dij , where D00 = ∅, D10 = N , D01 = N and D20 = N . 1 2 175 D. Marín and J. Villadelprat Journal of Differential Equations 332 (2022) 123–201 We claim that the coefficients T̄ij (ν̄) do not depend on α and β. Indeed, to see this recall that the Dulac time T̄ (s; ν̄) does not depend on α and β provided that α + β < η and η > −α, which is verified for ν̄ ∈ V̄ . Hence ∂α T̄ (s; ν̄) ≡ 0 and ∂β T̄ (s; ν̄) ≡ 0. Thus from (b ) we get that, for each fixed ν̄ ∈ V̄ ∩ { 12 < F < 23 }, ∂α T̄00 (ν̄ ) + ∂α T̄10 (ν̄ )s + ∂α T̄01 (ν̄ )s λ(ν̄ ) + o(s 2−2υ ) = 0 for all s > 0, where we use that the flatness order of the remainder is preserved when derived with respect to parameters (see Definition A.2). Then, since 1 < λ < 2 for F ∈ ( 12 , 23 ) and we can choose υ > 0 arbitrary small (depending on ν̄ ), by taking Remark A.6 into account we can assert that ∂α T̄00 (ν̄ ) = 0, ∂α T̄10 (ν̄ ) = 0 and ∂α T̄01 (ν̄ ) = 0. Since V̄ ∩ { 12 < F < 23 } is open and the coefficients are meromorphic on V̄ , by Lemma B.1 it fol- lows that ∂α T̄00 , ∂α T̄10 and ∂α T̄01 are identically zero. The claim for ∂β and the other coefficients follows verbatim. On account of the claim and the fact that T̄ (s; ν̄)|η=0 = T (s; ν), the assertions in (a)–(d) with regard to the asymptotic expansion of T (s; ν) follow from (a  )–(d  ), respectively, by setting  Tij (ν) := T̄ij (ν̄)η=0 . Next we proceed with the computation of the expression of each coefficient. To this end observe that  Tij (ν) = T̄ij (D, F, α, β, η)η=0 = lim T̄ij (D, F, α, β, η) = lim T̄ij (D, F, 0, 0, η), (28) η→0+ η→0+ where the second follows by the continuity of ν̄ → T̄ij (ν̄) on V̄ and the last one on account of the previous claim. In view of this the plan is to compute T̄ij (D, F, 0, 0, η) with η > 0 by applying Theorem A.4 and then make η → 0. With this aim it is first necessary to obtain the functions in (25). In doing so, and setting 1 D+1 1 D δ1 = , κ1 = , δ2 = and κ2 = , 2F F 2(1 − F ) F −1 for shortness, one can verify that Li (u) = h(u; δi , κi ) for i = 1, 2, where h(u; δ, κ) := (1 + κu2 )δ . We stress that L1 and L2 are defined in (25) in terms of the functions P1 (x1 , x2 ; ν̄) and P1 (x1 , x2 ; ν̄) given in (27) and that, as we already explained, we take α = β = 0 here and in what follows. Similarly, setting 2D + 1 γ1 = − F2 for the sake of shortness again, some computations show that M1 (u) = γ1 uh(u; δ1 − 2, κ1 ), 176 D. Marín and J. Villadelprat Journal of Differential Equations 332 (2022) 123–201 A1 (u) = −2δ1 h(u; −1, κ1 ), B1 (u) = M1 (u) and C1 (u)|D=− 1 = −4δ12 h(u; 2δ1 − 2, δ1 ). (29) 2 Moreover A2 (u) = 2δ2 h(u; δ2 − 1, κ2 ). With regard to the parametrization of the transverse sections in the expression of the coefficients, see (26), using the compact notation σij k (ν̄) := ∂sk σij (0; ν̄) we get that σ112 = σ121 = σ122 = 0 and σ111 = σ120 = σ210 = σ221 = 1/η. (30) We are now in position to apply Theorem A.4 to obtain the coefficients T̄ij (ν̄). (We omit the computation leading to T00 (ν) because it is given in [22, Proposition 5.2].) In doing so we obtain that λ σ120 σ111 ηλ T̄01 (ν, 0, 0, η) = Â2 (λ, σ210 ) = 2δ2 ĥ(λ, 1/η; δ2 − 1, κ2 ). λ Lλ (σ σ210 1 120 ) (η2 + κ1 )δ2 Therefore λ   2δ2 κ22 λ λ T01 (ν) = lim T̄01 (ν, 0, 0, η) = δ B − , −δ2 + 1 + η→0+ κ 2 2 2 2 1  δ2   √   λ+1  λ δ2 κ2 λ 1 π F 2 D 2 (− λ2 ) =√ B − , = , κ2 κ1 2 2 2(1 − F ) D+1 F −1 ( 1−λ 2 ) where the first equality follows from (28), the second one by (a) in Proposition √ B.3 (pro- vided that λ ∈/ N), and the last one by using (46) and that ( 12 ) = π . Since ρ1 (ν) = √ λ+1 λ π F 2 D 2 2(1−F ) D+1 F −1 is an analytic positive function on V , this proves the equality in the statement because we know that T01 (ν) is meromorphic on V with poles only at those F ∈ (0, 1) such that λ(ν) ∈ N. Let us turn next to the computation of T10 . With this aim we note that   σ121 σ111 T̄10 (ν, 0, 0, η) = −σ120 + B̂1 1/λ − 1, σ120 σ120 P2 (0, σ120 ) L1 (σ120 ) η−2 −1 η−3+2δ1 =− B̂1 (1/λ − 1, η ) = −γ 1 ĥ(1/λ − 2, η−1 ; δ1 − 2, κ1 ). L1 (η−1 ) (η2 + κ1 )δ1 Here the first equality follows by Theorem A.4, the second one from (30) and the last one by applying (d) in Theorem A.5 to the function B1 (u) = γ1 uh(u; δ1 − 2, κ1 ), see (29). Since −3 + 2δ1 = λ1 − 2, by applying (a) in Proposition B.3 we get 1 −1   √ γ1 κ12λ 1 1 π(2D + 1) (1 − 2λ 1 ) T10 (ν) = lim T10 (ν, 0, 0, η) = − B 1− , =  η→0+ 2κ1δ1 2λ 2 2 F (1 + D) ( 2 − 2λ ) 3 3 1 √ π and, due to ρ2 (ν) =  , this proves the validity of the expression for T10 given in the 2 F (1+D)3 statement. Let us finally compute the coefficient T20 . In this case, on account of (30), by Theo- rem A.4 we get 177 D. Marín and J. Villadelprat Journal of Differential Equations 332 (2022) 123–201   −1 η−2 −1 η−1 S1 −1 T̄20 (ν, 0, 0, η) = −η Ĉ1 (2/λ − 1, η ) + B̂1 (1/λ − 1, η ) . 2L21 (η−1 ) L1 (η−1 ) Thus, since γ1 = 0 for D = − 12 , from (29) it turns out that 2δ12 η−3+4δ1  −1  T̄20 (−1/2, F, 0, 0, η) = ĥ(2/λ − 1, η ; 2δ1 − 2, δ1  ) (η2 + κ1 )2δ1 D=− 12 Hence, due to −3 + 4δ1 = 2 λ − 1, by applying (a) in Proposition B.3 once again, 1 − 12    κ1λ  1 1 1 T20 (−1/2, F ) = lim T̄20 (−1/2, F, 0, 0, η) = 2δ1  B − , η→0+ 4F 2 κ1 1 D=− 2 2 λ 2 √ π ( 12 − λ1 ) =√ . 2F (1 − λ1 ) Since T20 (ν) is a meromorphic function having poles only at those ν0 ∈ V such that λ(ν0 ) ∈ D20 = N 2 and, on the other hand, λ(ν) > 2 for all ν ∈ V ∩ { 23 < F < 1}, by applying the Weier- strass Division Theorem (see for instance [11, Theorem 1.8]), we can assert the existence of an analytic function ρ3 on V ∩ { 23 < F < 1} such that √ π ( 12 − λ1 ) T20 (ν) = √ + ρ3 (ν)(2D + 1). 2F (1 − λ1 ) It only remains to prove the assertions with regard to the properties of the coefficients in the respective asymptotic expansions. Being the ones in (a) and (b) an easy consequence of well- known properties of the gamma function (see for instance [1]), we proceed with the other two: (c) Let us take any ν0 ∈ V ∩ {F = 23} and note that, from (c ), the functions T201 2 (ν) := T̄201 (ν̄)η=0 and T200 (ν) := T̄200 (ν̄)η=0 are smooth in a neighborhood of {ν ∈ V : λ(ν) = 2 2 2 2} = V ∩ {F = 23 } and 2 T201 (ν) = 2 − λ(ν) T01 (ν) and T200 2 (ν) = T20 (ν) + T01 (ν) for λ(ν) = 2. Recall that T01 (ν) and T20 (ν) are meromorphic with a pole at those ν such that λ(ν) = 2 ∈ D01 ∩ D20 . What is more, by Propositions 3.2 and 3.6 in [31], respectively, we know that in both cases the pole is simple. Consequently by the Weierstrass Division Theorem 2 (ν) and T 2 (ν) are analytic in (or, more directly, by [31, Lemma 2.8]) it follows that T201 200 a neighborhood of V ∩ {F = 3 }. On the other hand, from the already proved part of the 2 statement, if ν = (− 12 , 23 ) then T10 (ν ) = 0 and ρ1 (ν ) ρ1 (ν ) 2 T201 (ν ) = lim (2 − λ)T01 (ν) = lim (−λ/2)(2 − λ) = −2 =0 ν→ν (− 12 ) ν→ν (− 12 ) because limx→−1 (x + 1)(x) = −1 (see [1, §6] for instance). 178 D. Marín and J. Villadelprat Journal of Differential Equations 332 (2022) 123–201  1 (ν) := T̄ 1 (ν̄) (d) Consider finally any ν0 ∈ V ∩ {F = 12 }. Then, from assertion (d  ), T101  101 η=0 and T 1 (ν) := T̄ 1 (ν̄) 100 100 are smooth functions in a neighborhood of {ν ∈ V : λ(ν) = 1} = η=0 V ∩ {F = 12 } and, in addition, 1 T101 (ν) = 1 − λ(ν) T01 (ν) and T100 1 (ν) = T10 (ν) + T01 (ν) for λ(ν) = 1. Since T10 (ν) and T01 (ν) are meromorphic with a pole at those ν such that λ(ν) = 1 ∈ D10 ∩ 1 (ν) and T 1 (ν) are analytic in a D01 , the above equality shows exactly as before that T101 100 neighborhood of V ∩ {F = 2 }. Moreover, from the expression for T01 in the statement that 1 we already proved and using that 1 − λ = 2 FF−1/2 −1 , we can write ρ1 (ν) 4(− λ2 ) 1 T101 (ν) = (1 − λ)T01 (ν) = −ρ4 (ν)(F − 1/2)2 with ρ4 (ν) := . (F − 1)2 ( 1−λ 2 )(λ − 1) The function ρ4 is analytic at ν0 = (D0 , 12 ) because λ(ν0 ) = 1 and (z) has simple pole √ z = 0. In addition, since limz→0 z(z) = 1 and (− 12 ) = −2 π , we get that ρ4 (ν0 ) = at √ 16 πρ1 (ν0 ) > 0. From the expressions in the statement as well, we get 1 T100 (ν) = ρ5 (ν)(D + 1/2) + ρ6 (ν)(F − 1/2) 1 (1− 2λ ) with ρ5 (ν) := 2ρ2 (ν) and ρ6 (ν) := 1−F 2 ρ4 (ν), that are analytic and positive at ν0 = ( 32 − 2λ 1 ) (D0 , 12 ) due to λ(ν0 ) = 1. Finally a computation shows that ρ5 (− 12 , 12 ) = ρ6 (− 12 , 12 ). This concludes the proof of the result.  A.3. Proof of Proposition 3.3 Proof of Proposition 3.3. We will adapt the arguments in [23, §3.2.1] to take advantage of the general setting developed in [31]. To this end, as we did in the proof of the previous result, we will work in an extended parameter space W̄ to be specified. In this case the computations are a little bit more involved because we also need to straighten the separatrices of the saddle, see Fig. 4. With this aim in view we first take ε ∈ R and consider the local change of coordinates given by   q(x) − 12 y 2 p2 − p1 (x1 , x2 ) = φε (x, y) := , , a(p2 − x + εy)2 p2 − x + εy where recall that q(x) = a(x − p1 )(x − p2 ) with a = 2(1−FD ) > 0 and p1 , p2 ∈ R, p1 < p2 , for all ν ∈ W . In what follows, for the sake of shortness we set √ κ1 := p2 − p1 and κ2 := 1/ 2a. One can check that the Jacobian determinant of φε vanishes at (x, y) if and only if y −εq  (x) = 0, where 179 D. Marín and J. Villadelprat Journal of Differential Equations 332 (2022) 123–201 Fig. 10. Auxiliary transverse sections in the proof of Proposition 3.3. q  (x) = 2ax − a(p1 + p2 )x, and that this straight line is mapped by φε to Dε (x1 , x2 ) := 2a(1 − x1 − x2 ) + a 2 ε 2 (4x1 + x22 ) = 0. We claim that φε is an analytic map from ε := {(x, y) ∈ R2 : p2 − x + εy > 0, y − εq  (x) > 0} to Uε := {(x1 , x2 ) ∈ R2 : x2 > 0, Dε (x1 , x2 ) > 0} with a well defined analytic inverse given by  1 κ1 (εDε (x1 , x2 ) 2 − 1) + (p2 − ε 2 a(p1 + p2 ))x2 ψε (x1 , x2 ) := , (1 − 2aε 2 )x2 1  κ1 (Dε (x1 , x2 ) 2 + aε(x2 − 2)) . (1 − 2aε 2 )x2 Indeed, the claim follows by checking that φε ◦ ψε = Id on {Dε (x1 , x2 ) > 0, x2 = 0} and that  (x) ψε ◦ φε = Id on { py−εq 2 −x+εy > 0}. To show the second identity we use that (Dε ◦ φε )(x, y) =   2 y−εq (x) p2 −x+εy . Consequently (x1 , x2 ) = φε (x, y) is an analytic global change of variables from ε to Uε for all ε. In what follows we require |ε| < √1 in order that the straight line {εy + p2 − x = 2a 0} does not intersect the left branch of the hyperbola C = { 12 y 2 − q(x) = 0}, see Fig. 10. Next we introduce a second auxiliary parameter η ∈ R and consider two additional transverse η η sections 1 and 2 laying on the straight line η := {y = η(p2 − x)}, see Fig. 10. Observe √ that η intersects the right branch of the hyperbola C at (p2 , 0) for all η. We require |η| < 2a additionally in order that η intersects the hyperbola at a point (xη , yη ) in the left branch. Then η η we parametrize 1 and 2 , respectively, by s → xη − s, η(p2 − xη + s) and s → − 1/s, η(p2 + 1/s) , (31) 180 D. Marín and J. Villadelprat Journal of Differential Equations 332 (2022) 123–201 for s > 0 small enough. We also require ε(η2 + 2a) + 2η > 0 so that (xη , yη ) ∈ ε . Summing up, the admissible conditions 1 √ |ε| < √ , |η| < 2a and ε(η2 + 2a) + 2η > 0 2a η η guarantee that any solution of Xν going from 1 to 2 is inside the domain ε of the coordinate change (x1 , x2 ) = φε (x, y). Thus, setting ν̄ := (ν, ε, η), we will work on the extended parameter space √ W̄ := ν̄ ∈ R4 : F + D > 0, D < 0, F > 1, |ε| < √1 , |η| < 2a and ε(η2 + 2a) + 2η > 0 . 2a √ Clearly the sets {(ν, ε, 0) : ν ∈ W and ε ∈ (0, √1 )} and {(ν, 0, η) : ν ∈ W and η ∈ (0, 2a)} are 2a inside W̄ , that will be crucial in the forthcoming steps. At this point we define σ1 ( · ; ν̄) and σ2 ( · ; ν̄) to be, respectively, the composition with φε of η η the parametrization of 1 and 2 given in (31). In its regard one can check that   s(1 − κ22 η2 )2 κ1 (1 − κ22 η2 ) σ1 (s; ν̄)|ε=0 = , (32) κ1 + s(1 − κ22 η2 ) κ1 + s(1 − κ22 η2 ) and   1 + p1 s κ1 s σ2 (s; ν̄)|ε=0 = − κ22 η2 , . (33) 1 + p2 s 1 + p2 s One can also verify that the coordinate change (x1 , x2 ) = φε (x, y) brings the vector field Xν in (2) to 1 X̄ν̄ (x1 , x2 ) = x1 P1 (x1 , x2 ; ν̄)∂x1 + x2 P2 (x1 , x2 ; ν̄)∂x2 , x2 where P1 and P2 analytic functions on {(x1 , x2 , ν̄) ∈ R2 × W̄ : Dε (x1 , x2 ) > 0}. The hyperbolic- ity ratio of the saddle at the origin is P2 (0, 0) 1 λ=− = . P1 (0, 0) 2(F − 1) 1√ Moreover P1 |ε=0 = R P̄1 and P2 |ε=0 = R P̄2 where R(x1 , x2 ) = κ2 1 − x 1 − x2 , P̄1 (x1 , x2 ) = 2κ1 (F − 1) + 2(p2 − 1)x2 and P̄2 (x1 , x2 ) = −κ1 + (p2 − 1)x2 . (34) (It will be clear in a moment the reason why it suffices to restrict to ε = 0.) For each ν̄ ∈ W̄ , we η η define T̄ (s; ν̄) to be the Dulac time of X̄ν̄ between the transverse sections φε (1 ) and φε (2 ) parametrized by σ1 ( · ; ν̄) and σ2 ( · ; ν̄), respectively. We point out that, by construction, T̄ (s; ν̄) does not depend on ε and that, furthermore, T̄ (s; ν̄)η=0 = T (s; ν). 181 D. Marín and J. Villadelprat Journal of Differential Equations 332 (2022) 123–201 Next we will apply Theorem A.3 to obtain the asymptotic expansion of T̄ (s; ν̄) at s = 0. Note to this end that, by construction, given any ν̄0 ∈ W̄ there exists a relatively compact neighbor- hood V0 of (x1 , 0) : x1 ∈ [0, σ̄21 (0; ν̄0 )] ∪ (0, x2 ) : x2 ∈ [0, σ̄12 (0; ν̄0 )] η η in R2 and a neighborhood W̄0 of ν̄0 in W̄ such that φε 1 ∪ 2 ⊂ V0 for all ν̄ ∈ W̄0 and V0 × W̄0 ⊂ {(x1 , x2 , ν̄) ∈ R2 × W̄ : Dε (x1 , x2 ) > 0}. Here we use (see also Fig. 10) that φε maps the straight line {y − εq  (x) = 0} to {Dε (x1 , x2 ) = 0}. The above inclusion guarantees that P1 and P2 are analytic on V0 × W̄0 , so that we can apply Theorem A.3 to study the Dulac time of X̄ν̄ for ν̄ ≈ ν̄0 . Accordingly, with this aim, let us fix any ν̄0 = (D0 , F0 , ε0 , η0 ) ∈ W̄ with F0 ∈ (1, 32 ) ∪ {2}. Observe that λ(ν̄0 ) > 2 if F0 ∈ (1, 54 ), λ(ν̄0 ) ∈ (1, 2) if F0 ∈ ( 54 , 32 ), λ(ν̄0 ) = 2 if F0 = 54 and λ(ν̄0 ) = 12 if F0 = 2. Then, by applying (2), (1), (5) and (3) in Theorem A.3, respectively, we can assert that (a  ) T̄ (s; ν̄) = T̄00 (ν̄) + T̄10 (ν̄)s + T̄20 (ν̄)s 2 + FL∞0 −υ (ν̄0 ) if F0 ∈ (1, 54 ), where L0 = min(3, λ(ν̄0 )), (b ) T̄ (s; ν̄) = T̄00 (ν̄) + T̄10 (ν̄)s + T̄01 (ν̄)s λ + T̄20 (ν̄)s 2 + FL∞0 −υ (ν̄0 ) if F0 ∈ ( 54 , 32 ), where L0 = λ(ν̄0 ) + 1, (c ) T̄ (s; ν̄) = T̄00 (ν̄) + T̄10 (ν̄)s + T̄201 ∞ 2−λ (s) + T̄200 (ν̄)s + F3−υ (ν̄0 ) if F0 = 4 , where 2 (ν̄)s 2 ω 2 2 5 T̄201 (ν̄) and T̄200 (ν̄) are smooth in a neighborhood of {ν̄ ∈ W̄ : λ(ν̄) = 2} and, moreover, 2 2 2 T̄201 (ν̄) = (2 − λ)T̄01 (ν̄) and T̄200 2 (ν̄) = T̄20 (ν̄) + T̄01 (ν̄) for λ(ν̄) = 2, 1 1 (d  ) T̄ (s; ν̄) = T̄00 (ν̄) + T̄01 (ν̄)s λ + T̄101 2 (ν̄)sω1−2λ (s) + T̄100 2 ∞ (ν̄)s + F3/2−υ (ν̄0 ) if F0 = 2, where T̄101 (ν̄) and T̄100 (ν̄) are smooth in a neighborhood of {ν̄ ∈ W̄ : λ(ν̄) = 1/2} and, moreover, 1 1 1 1 2 T̄101 (ν̄) = (1 − 2λ)T̄02 (ν̄) and T̄100 2 (ν̄) = T̄10 (ν̄) + T̄02 (ν̄) for λ(ν̄) = 1/2. Here υ is a small enough positive number depending on ν̄0 . Furthermore by applying locally Theorem A.3 we know that the coefficients T̄ij (ν̄) are meromorphic functions on W̄ with poles only at those F > 1 such that λ(ν̄) = 2(F1−1) ∈ Dij , where D00 = ∅, D10 = N 1 , D01 = N, D20 = 2 N and D02 = N 2 . We claim that the coefficients T̄ij (ν̄) do not depend on ε. To prove this observes that the Dulac time T̄ (s; ν̄) does not depend on ε as long as ν̄ ∈ W̄ . Accordingly ∂ε T̄ (s; ν̄) ≡ 0. Thus from (b ) we get that, for each fixed ν̄ ∈ W̄ ∩ { 54 < F < 32 }, ∂ε T̄00 (ν̄ ) + ∂ε T̄10 (ν̄ )s + ∂ε T̄01 (ν̄ )s λ(ν̄ ) + ∂ε T̄20 (ν̄ )s 2 + o(s L0 −2υ ) = 0 for all s > 0, where L = λ(ν̄ ) + 1 and we use that the flatness order of the remainder is preserved when derived with respect to parameters (see Definition A.2). Then, since 1 < λ < 2 for F ∈ ( 54 , 32 ) and we can choose υ > 0 arbitrary small (depending on ν̄ ), the application of Remark A.6 shows that 182 D. Marín and J. Villadelprat Journal of Differential Equations 332 (2022) 123–201 ∂ε T̄00 (ν̄ ) = 0, ∂ε T̄10 (ν̄ ) = 0, ∂ε T̄01 (ν̄ ) = 0 and ∂ε T̄20 (ν̄ ) = 0. Since W̄ ∩ { 54 < F < 32 } is open and the coefficients are meromorphic on W̄ , by Lemma B.1 it follows that ∂ε T̄00 , ∂ε T̄10 , ∂ε T̄01 and ∂ε T̄20 are identically zero. The claim for ∂ε T̄02 follows verbatim.  Thanks to the claim and the fact that T̄ (s; ν̄)η=0 = T (s; ν) by construction, the assertions in (a)–(d) concerning the asymptotic expansion of T (s; ν) at s = 0 follow from (a  )–(d  ), respec- tively, by setting  Tij (ν) := T̄ij (ν̄)η=0 . We proceed next with the computation of these coefficients and for this purpose the idea is that if ν ∈ W and ε > 0 then Tij (ν) = T̄ij (ν, ε, 0) = lim T̄ij (ν, ε, η) = lim T̄ij (ν, 0, η), (35) η→0+ η→0+ where in the second equality we use the continuity of T̄ij (ν̄) at any ν̄0 ∈ W̄ with λ(ν̄0 ) = 2(F0 −1) ∈ 1 / Dij and in the last one the fact that T̄ij (ν̄) does not depend on ε. Hence our first goal is to obtain T̄ij (ν̄) for ε = 0 and to this end we shall apply Theorem A.4. (We point out that from now on all the computations are performed taking ε = 0 and η > 0.) In doing so, and setting p2 − 1 κ0 := p2 − p 1 for shortness, from (25) we obtain that L1 (u) = (1 − κ0 u)2F , L2 (u) ≡ 1, M1 (u) ≡ 0 and κ2 1 κ2 1 A1 (u) = − (1 − κ0 u)−1 (1 − u)− 2 A2 (u) = (1 − u)− 2 κ1 2κ1 (F − 1) (36) κ2 3 3κ2 5 B1 (u) = − (1 − κ0 u)2F −1 (1 − u)− 2 C1 (u) = − (1 − κ0 u)4F −1 (1 − u)− 2 . 2κ1 4κ1 From (32) and (33), the necessary information with regard to the transverse sections is the fol- lowing: (1−κ22 η2 )2 2(1−κ22 η2 )3 σ120 = σ210 = 1 − κ22 η2 , σ111 = −σ121 = and σ122 = −σ112 = . (37) κ1 κ12 Taking this into account we obtain that T00 (ν) = lim T̄00 (ν, 0, η) = − lim (1 − κ22 η2 )Â1 (−1, 1 − κ22 η2 ) η→0+ η→0+ κ2 2κ2 κ0 = κ1 B(1, 12 ) 2 F1 (1, 1; 32 ; κ0 ) = κ1 (1−κ0 ) 2 F1 (1, 2 ; 2 ; κ0 −1 ) 1 3 where in the second equality we use Theorem A.4, in the third one we apply (b) in Proposi- tion B.3 taking {α = −1, γ = 1, δ = 12 , x = κ0 } and in the last one [1, §15.3]. Since κ0κ−1 0 = 1−p2 1−p1 , 183 D. Marín and J. Villadelprat Journal of Differential Equations 332 (2022) 123–201 this shows the validity of the expression for T00 given in the statement. Similarly, from (36) and (37) again, T01 (ν) = lim T̄01 (ν, 0, η) η→0+ κ1−λ −2λF = lim  (λ, 1 − k22 η2 ) = κ2 (1−κ (1−κ0 )2λF η→0+ 2 0) B(−λ, 12 ). 2κ1λ+1 (F −1) Here the last equality follows by applying (b) in Proposition B.3 taking {α = λ, δ = 12 , x = 0}, so that α = λ(ν) = 2(F1−1) ∈/ Z≥0 , and noting that 2 F1 (a, b; c; 0) = 1, see (47). Since ρ1 (ν) := κ2 (1−κ0 )−2λF is an analytic positive function on W , this proves the expression for T01 given in the 2κ1λ+1 (F −1) statement. Let us study next the coefficient T10 . For this purpose we apply Theorem A.4, which on / D10 = N account of (37) shows that if η > 0 and λ(ν) ∈ 1 then  1 −1 + κ22 η2 T̄10 (ν, 0, η) = κ1 η (κ1 + (p2 − 1)(1 − κ22 η2 )) (38) (1 − κ22 η2 )3   − B̂1 1/λ − 1, 1 − κ2 η 2 2 . (1 − κ0 (1 − κ22 η2 ))2F κ2 Following the notation in Proposition B.3, see (36), we can write B1 (1 − κ22 η2 ) = − 2κ 1 g(y; δ, γ ; x) with {y = 1 − κ2 η , δ = − 2 , γ = 2F, x = κ0 } but we cannot apply it to get the limit of 2 2 1 B̂1 1/λ − 1, 1 − κ22 η2 as η → 0+ because the condition δ > 0 is not satisfied. As a matter of fact this is coherent because, since the first summand in (38) is divergent as η → 0+ , it happens that limη→0+ B̂1 1/λ − 1, 1 − κ22 η2 diverges as well. Hence the approach to compute this coefficient hast to be different. The idea is to take advantage of [23, Theorem 3.6], which shows that if F ∈ (1, 32 ) then T10 (ν) = ρ2 (ν)ρ̄2 (ν) where 1   1 ds κ2 1 1 ρ2 (ν) := and ρ̄2 (ν) := −2 + (1 − s)− λ (1 − κ̄s)1+ λ − 1 s − 2 2κ1 (1 − p1 ) s 0 1 with κ̄ := 1−p2 By applying assertion (b) in Theorem A.5 taking f (s; ν) = (1 − s)− λ (1 − 1−p1 . κ̄s)1+ λ , k = 1 and α = 12 we can assert that ρ̄2 (ν) = lims→1− fˆ( 12 , s; ν). Observe on the other 1 hand that, following the notation in Proposition B.3, we can write f (s; ν) = g(y; δ, γ ; x) with {y = s, δ = 1 − λ1 , γ = −1 − λ1 , x = κ̄}. Thus, since one can verify that δ = 1 − λ1 > 0 and x = κ̄ < 1 for all ν ∈ W ∩ {1 < F < 32 }, the application of assertion (b) in that result gives ρ̄2 (ν) = limy→1− fˆ( 12 , y; ν) = B − 12 , 1 − 1 λ 2 F1 −1 − λ1 , − 12 ; 12 − λ1 ; κ̄ . Due to B(− 12 , 1 − λ1 ) = B(1 − λ1 , − 12 ), see (46), this proves the validity of the expression for T10 (ν) in the statement for all ν ∈ W ∩ {1 < F < 32 }. Accordingly, since T10 is meromorphic on W , this is also the case of ρ̄2 thanks to Lemma B.2, and ρ2 is analytic on W , the application of 184 D. Marín and J. Villadelprat Journal of Differential Equations 332 (2022) 123–201 the real version of Lemma B.1 implies the validity of the equality on W . Observe moreover that ρ2 is positive on W . We proceed with the computation of the coefficient T20 . In first instance, for the sake of conve- nience we shall work with ν ∈ W ∩ {1 < F < 54 }, so that λ(ν) > 2. Due to M1 ≡ 0, Theorem A.4 shows that if λ(ν) ∈/ D20 = N 2 then σ120 σ122 1 2 T̄20 (ν, 0, η) = − − σ121 ∂2 P2−1 (0, σ120 ) − σ121 σ111 ∂1 P2−1 (0, σ120 ) (39) 2σ120 P2 (0, σ120 ) 2 2 σ120 σ111 − Ĉ1 (2/λ − 1, σ120 ) 2L21 (σ120 )     σ112 σ121 P1 σ120 σ111 − − (0, σ120 ) B̂1 (1/λ − 1, σ120 ). 2σ111 σ120 P2 L1 (σ120 ) √  Since P2 |ε=0 = R P̄2 , R(0, σ120 ) = κ12 1 − x2 x =1−κ 2 η2 = η and P̄2 (0, 1) = p1 − 1 = 0, 2 2 see (34) and (37), it follows that we can write σ120 σ122 1 2 φ1 (η2 ) − − σ121 ∂2 P2−1 (0, σ120 ) − σ121 σ111 ∂1 P2−1 (0, σ120 ) = (40) 2σ120 P2 (0, σ120 ) 2 η3 and   σ112 σ121 P1 − + (0, σ120 ) = φ2 (η2 ), (41) 2σ111 σ120 P2 where here (and in what follows) φi (x) stands for an analytic function at x = 0. (In fact φi depends also on ν and this dependence is analytic on W . We omit this dependence for brevity when there is no risk of confusion.) Following this notation, from (38) and using that limη→0+ L1 (1 − κ22 η2 ) = (1 − κ0 )2F = 0, we can assert that σ120 σ111 φ4 (η2 ) B̂1 (1/λ − 1, σ120 ) = φ3 (η2 )T̄10 (ν, 0, η) + (42) L1 (σ120 ) η as long as λ(ν) ∈ / D10 = N 1 . On the other hand, since λ(ν) > 2 for all ν ∈ W ∩ {1 < F < 54 }, by applying assertion (b) in Theorem A.5 with k = 0 and α = λ2 − 1, we get 2 2 σ120  2η 1−κ 1− 2 2 du 2 σ120 λ Ĉ1 (2/λ − 1, σ120 ) = C1 (u)u1− λ = C1 (u)u− λ du u 0 0 1 3κ2 5 2 =− (1 − κ0 )4F −1 (1 − κ̄x)4F −1 x − 2 (1 − x)− λ dx , 4κ1 κ22 η2 " #$ % I 185 D. Marín and J. Villadelprat Journal of Differential Equations 332 (2022) 123–201 κ0 where in the second equality we perform the change of variable u = 1 − x and use that κ0 −1 = 1−p2 1−p1 = κ̄. Next we split the above integral as I = I1 + I2 with 1   3 dx 1 3 dx 4F −1 − λ2 −2 I1 := (1 − κ̄x) (1 − x) − (1 + βx) x and I2 := (1 + βx)x − 2 , x x κ̄22 η2 κ̄22 η2 where we take β := λ2 − (4F − 1)κ̄ so that I1 converges as η → 0. Due to I2 = − 23 − 2β + 2β 2β 3 (κ η)3 + κ2 η , we can write I = J + 3 (κ η)3 + κ2 η with 2 1 2 1 2 2 1   3 dx 2 2 J (ν, η) := (1 − κ̄x)4F −1 (1 − x)− λ − (1 + βx) x − 2 − − 2β. x 3 κ22 η2 Consequently we obtain that   3κ2 (1 − κ0 )4F −1 2 2β Ĉ1 (2/λ − 1, σ120 ) = − J (ν, η) + + . (43) 4κ1 (1 − κ 2 η2 )1− λ2 3(κ2 η)3 κ2 η 2 2 On the other hand, setting g(x; ν) := (1 − κ̄x)4F −1 (1 − x)− λ we get that     2 3 3 1 2 lim J (ν, η) = lim ĝ(3/2, x; ν) = B 1 − , − 2 F1 1 − 4F, − ; − − ; κ̄ =: J0 (ν). η→0 x→1 − λ 2 2 2 λ Here the first equality follows by (b) in Theorem A.5 taking {α = 32 , k = 2} and the second one by (b) in Proposition B.3 taking {α = 32 , γ = 1 − 4F, δ = 1 − λ2 , x = κ̄} and using that δ > 0 thanks to λ(ν) > 2 for all ν ∈ W ∩ {1 < F < 54 }. That being said, substituting (40), (41), (42) and (43) into (39) and gathering next the analytic functions at η = 0 we obtain that T̄20 (ν, 0, η) = φ5 (ν, η2 )J (ν, η) + φ6 (ν, η2 )T̄10 (ν, 0, η) + φ7 (ν, η2 )/η3 , with φi (ν, x) analytic at x = 0. (Here we specify again the dependence on ν for the sake of consistency in the exposition.) Consequently, from (35), T20 (ν) = lim T̄20 (ν, 0, η) = φ5 (ν, 0)J0 (ν) + φ6 (ν, 0)T10 (ν), η→0+ where we use that φ7 (ν, η2 ) = O(η4 ) because the limit must be finite. One can easily check that 3κ2 p1 − 1 + 2F κ1 ρ3 (ν) := φ5 (ν, 0) = and ρ4 (ν) := φ6 (ν, 0) = . 8κ1 (1 − κ0 ) 3 κ1 (p1 − 1) This proves the validity of the expression for T20 (ν) for all ν ∈ W ∩ {1 < F < 54 }. Similarly as before, by applying Lemma B.1 this equality extends to W since ρ3 and ρ4 are analytic on W 186 D. Marín and J. Villadelprat Journal of Differential Equations 332 (2022) 123–201 and, on the other hand, J0 is meromorphic on W by Lemma B.2 and so are T̄10 and T̄20 . Finally an easy computation shows that ρ3 and ρ4 are positive on W . So far we have proved the validity of the expression of the coefficients Tij (ν) that we give in the first part of the statement. Moreover, since T (s; ν) = T̄ (s; ν̄)|η=0 and Tij (ν) = T̄ij (ν̄)|η=0 , the assertions with regard to the asymptotic expansion of the Dulac map at s = 0 in (a)–(d) follow, respectively, from (a  )–(d  ). The only remaining point concerns the behavior of the coefficients in each one of these cases. This is our final task, that we carry out case by case: (a) Let us consider any ν0 = (D0 , F0 ) ∈ W ∩ {1 < F < 54 } such that T10 (ν0 ) = 0. We claim that then D0 ∈ (−1, − 12 ). Indeed, to prove this we first use Proposition 3.11 in [23], which shows that the set ν ∈ W : T10 (ν) = 0 with 1 < F < 32 is the graphic of an analytic function D = G(F ) verifying −F < G(F ) < − 12 and limF →1+ G(F ) = − 12 . Therefore it is clear that the claim will follow once we prove that T10 (−1, F ) = 0 for all F ∈ (1, 54 ). In order to show this we note that p2 |D=−1 = 1 and, consequently,  2  T10 (−1, F ) = ρ2 (ν)B 1 − λ1 , − 12 2 F1 − 1 − λ1 , − 12 ; 12 − λ1 ; 1−p 1−p1  =0 D=−1 because 2 F1 (a, b; c; 0) = 1 by definition and, on the other hand, λ(ν) = 1 2(1−F ) > 2 for all (1− λ1 )(− 12 ) F ∈ (1, 54 ) and one can check that B 1 − λ1 , − 12 = = 0 for all λ > 2. This ( 12 − λ1 ) proves the claim. Recall at this point that T20 (ν) = ρ3 (ν)B 1 − λ2 , − 32 2 F1 − 2 λ − 3, − 32 ; − 12 − λ2 ; 1−p 1−p1 + ρ4 (ν)T10 (ν). 2 (44) Accordingly, since T10 (ν0 ) = 0 with ν0 ∈ Q := W ∩ {1 < F < 54 } ∩ {−1 < D < − 12 } and ρ2 and ρ3 are positive functions, in order to prove that T20 (ν0 ) = 0 it suffices to show that the linear combination B 1 − λ2 , − 32 2 F1 − 2 λ − 3, − 32 ; − 12 − λ2 ; 1−p2 1−p1 − 4B 1 − λ1 , − 12 2 F1 − 1 − λ1 , − 12 ; 12 − λ1 ; 1−p2 1−p1 1−p2 does not vanish on Q. Since one can easily verify that 2 λ ∈ (0, 1) and 1−p1 ∈ (−1, 0) for all ν ∈ Q, this follows directly by applying Proposition C.1 with {a = 2 λ,b = − 1−p 1−p1 }. 2 (b) We already proved that T01 (ν) = ρ1 (ν)B − λ, 12 with ρ1 an analytic positive function (−λ)( 12 ) on W . The function B(−λ, 12 ) = vanishes only when 1 − λ(ν) ∈ Z≤0 and, for ( 12 −λ) 2 ν ∈ W ∩ { 54 < F < 32 }, this occurs if and only if λ(ν) = 32 , i.e., F = 43 . Recall moreover that, by Proposition 3.11 in [23], the set ν ∈ W : T10 (ν) = 0 with 1 < F < 32 is the graphic of an analytic function D = G(F ) on (1, 32 ). Consequently there exists a unique ν = (D , 43 ) inside W ∩ { 54 < F < 32 } such that T10 (ν ) = 0 and one can prove that D ∈ (−1.15, −1.10). The gradients of T10 and T01 are linearly independent at ν because ∂D T10 (ν) = 0 for all ν ∈ W ∩ {1 < F < 32 } by (a) of Lemma 3.13 in [23] and, on the other hand, ∂D T01 (ν ) = 187 D. Marín and J. Villadelprat Journal of Differential Equations 332 (2022) 123–201 B − λ, 12 |ν=ν ∂D ρ1 (ν ) = 0 and ∂F T10 (ν ) = 0 since the gamma function has simple poles at Z≤0 with non-zero residue. Finally the fact that T20 (ν ) < 0 follows noting that, from (44) and λ(ν ) = 32 , we get  11 1−p2  T20 (ν ) = ρ3 (ν ) B − 13 , − 32 2 F1 − 3 , − 2 ; − 6 ; 1−p1 13 3  , ν=ν 11 1−p2 which is negative because one can easily check that D → 2 F1 − 3 , − 2 ; − 6 ; 1−p1 13 3 |F = 4 3 is positive on (−1.15, −1.10) and B − ≈ −2.6. 3,−2 1 3 (c) Let us fix any ν0 ∈ W ∩ {F = 54 }, so that λ(ν0 ) = 2. Then, from (c ), T201 2 (ν) := T̄ 2 (ν̄)| 201 η=0 and T200 (ν) := T̄200 (ν̄)|η=0 are smooth functions in a neighborhood of {ν ∈ W : λ(ν) = 2} 2 2 and, in addition, 2 T201 (ν) = 2 − λ(ν) T01 (ν) and T200 2 (ν) = T20 (ν) + T01 (ν) for λ(ν) = 2. The functions T01 (ν) and T20 (ν) are meromorphic with a pole at those ν such that λ(ν) = 2 ∈ D01 ∩ D20 , and the pole is simple in both cases by Propositions 3.2 and 3.6 in [31], re- spectively. Therefore by the Weierstrass Division Theorem (or, more directly, by [31, Lemma 2 (ν) and T 2 (ν) are analytic in a neighborhood of W ∩ {F = 5 }. Fur- 2.8]) it follows that T201 200 4 thermore, by [23, Lemma 3.13] once again, T10 (D, 54 ) = 0 if and only if D = −1. Finally, (−λ)( 12 ) since λ(−1, 54 ) = 2, T01 (ν) = ρ1 (ν)B(−λ, 12 ) and B(−λ, 12 ) = , we have that ( 12 −λ) ( 12 ) 2 T201 (−1, 5/4) = lim 2 − λ(ν) T01 (ν) = ρ1 (−1, 5/4) lim (2 − λ)(−λ) = 0, ν→(−1, 54 ) (− 32 ) λ→2 because the gamma function has simple poles with non-zero residues at Z≤0 . (d) Consider finally any ν0 ∈ W ∩ {F = 2}, so that λ(ν0 ) = 12 . Since T01 (ν) = ρ1 (ν)B − λ, 12 (−λ)( 12 ) with ρ1 = 0, λ(ν) = 1 2(F −1) and B − λ, 12 = , there exists an analytic non- ( 12 −λ) vanishing function 1 in a neighborhood of W ∩ {F = 2} such that T01 (ν) = (F − 2)1 (ν). 1 On the other hand, from (d  ) and arguing as in the previous case, the functions T101 2 (ν) := 1 1 1 2 T̄101 2 (ν̄)|η=0 and T100 (ν) := T̄100 2 (ν̄)|η=0 are analytic in a neighborhood of {ν ∈ W : λ(ν) = 1 2 } and 1 1 2 T101 (ν) = (1 − 2λ)T02 (ν) and T100 2 (ν) = T10 (ν) + T02 (ν) for λ(ν) = 1/2. In particular we have that the sum of residues of T10 and T02 along {ν ∈ W : λ(ν) = 12 } = W ∩ {F = 2} is equal to zero, which saves us from computing the explicit value of T02 . (1− λ1 )(− 12 ) Indeed, since λ(ν0 ) = 1 and B(1 − λ1 , − 12 ) = , we obtain that 2 ( 12 − λ1 ) 1 2 T101 (ν0 ) = lim (1 − 2λ(ν))T02 (ν) = − lim (1 − 2λ(ν))T10 (ν) ν→ν0 ν→ν0 188 D. Marín and J. Villadelprat Journal of Differential Equations 332 (2022) 123–201  (− 12 ) 2 = −ρ2 (ν0 )2 F1 − 3, − 12 ; − 32 ; 1−p limλ→ 1 (1 − 2λ)(1 − λ1 ) 1−p1 ν=ν 0 (− 32 ) 2  = 34 ρ2 (ν0 )2 F1 − 3, − 1 ; − 3 ; 1−p2  2 2 1−p1 ν=ν0 , where in the second equality we use the expression of T10 already proved and in the last (− 12 ) one that = − 32 and limx→−1 (x + 1)(x) = −1, see for instance [1, §15]. From the (− 32 ) same reference we get that 2 F1 − 3, − 12 ; − 32 ; x) = (x + 1)(x − 1)2 . Moreover one can  2 verify that D → 1−p 1−p1 F =2 maps diffeomorphically (−2, 0) to (−∞, 1) and that it is equal 1 to −1 at D = − 12 . Accordingly we can assert that T101 2 (D, 2) = (D + 12 )2 (D) where 2 is 1 a non-vanishing analytic function on (−2, 0) and, consequently, ∂D T101 2 (− 12 , 2) = 0. Since ∂D T01 (− 12 , 2) = 0 and ∂F T01 (− 12 , 2) = 1 (− 12 , 2) = 0, this proves that the gradients of T01 1 2 and T101 are linearly independent at ν = (− 12 , 2) as desired. This finishes the proof of the result.  Appendix B. Beta and hypergeometric functions In this appendix we are concerned with the integral representation of the Beta and hypergeo- metric functions (see [2]). The Beta integral is defined for Re(z) > 0 and Re(w) > 0 by 1 +∞ B(z, w) := t z−1 (1 − t)w−1 dt = t z−1 (1 + t)−z−w dt. (45) 0 0 This function can be analytically extended for z, w ∈ C \ Z≤0 thanks to the identity (z)(w) B(z, w) = , (46) (z + w) where  is the gamma function. Recall in this regard that 1/(z) is an entire function with simple zeros at z ∈ Z≤0 . On the other hand, if we consider a, b, c ∈ C with c ∈ / Z≤0 and z inside the complex open unit disc D := {z ∈ C : |z| < 1}, then Gauss hypergeometric function is defined by the power series ∞  (a)n (b)n zn 2 F1 (a, b; c; z) := , (47) (c)n n! n=0 where for a given x ∈ C we use the Pochhammer symbol (x)n := x(x + 1) · · · (x + n − 1) = (x+n) (x) . In this section by a meromorphic function of several complex variables we mean a function that locally writes as a quotient of two holomorphic functions. Recall that a function f :  −→ C, where  is a connected open set of C n , is holomorphic if for each z0 ∈  there exists an open polydisc Dr (z0 ) such that f can be written as an absolutely and uniformly convergent power 189 D. Marín and J. Villadelprat Journal of Differential Equations 332 (2022) 123–201  series at z0 , i.e., f (z) = α aα (z − z0 )α for all z ∈ Dr (z0 ). On account of this one can readily obtain the following result about uniqueness of meromorphic continuation. Lemma B.1. Consider two functions φ and ϕ that are meromorphic on a connected open set  ⊂ C n . If there exists an open subset V of  such that φ|V = ϕ|V then φ = ϕ. Let us remark that the previous result is also true in the real setting, i.e., for functions in Rn that locally write as a quotient of real analytic functions. The following result is well-known but since we did not find its statement in its fullness we give it here for the sake of completeness. 2 F1 (a,b;c;z) Lemma B.2. The function (a, b, c, z) → (c) extends holomorphically to C 3 × (C \ [1, +∞)). Proof. Following [2, p. 65], we show first that the function extends holomorphically to C 3 × D. To prove this claim we write ∞  2 F1 (a, b; c; z) (a + n)(b + n)zn = fn (a, b, c, z) with fn (a, b, c, z) := . (c) (a)(b)(c + n)(1 + n) n=0 √ 1 Stirling’s asymptotic formula (z) ∼ 2πzz− 2 e−z as Re(z) → +∞ (see [2, Theorem 1.4.1]) shows that    (a + n)(b + n)     (c + n)(1 + n)  ∼ n as n → +∞. Re(a+b+c−1) Fix any compact set K ⊂ C 3 ×D and suppose that Re(a +b +c −1)  m ∈ N and |z|  r < 1 for all (a, b, c, z) ∈ K. Then, on account of the above asymptotic estimate and the fact that 1/(z) is an entire function, there exists C > 0 such that |fn (a, b, c, z)|  Cnm r n for all (a, b, c, z) ∈ K. By applying the Weierstrass M-test this proves that the series ∞ n=0 fn converges uniformly on compact sets of C 3 ×D. So the claim follows because the uniform limit of holomorphic functions is holomorphic (see [16, Proposition 2]). Finally the result follows by Pfaff and Kummer’s formulas (see [1, §15] or [2, Theorem 2.3.2]) relating the values of 2 F1 (a, b; c; · ) at z, z−1 z and 1z , which enable to extend holomorphically to C × (C \ [1, +∞)) the map (a, b, c, z) → 2 F1 (a,b;c;z) 3 (c) . (These formulas are usually proved under some restrictions on the parameters a b and c but they are always satisfied thanks to the claim and Lemma B.1.) This concludes the proof of the result.  It is worth to mention that by Hartogs’s theorem (see [14, §1.2]), a function of several complex variables is holomorphic if, and only if, it is holomorphic (in the classical one-variable sense) in each variable separately. The main concern in this section is Euler’s integral representation of 2 F1 , see for instance [2, Theorem 2.2.1], that is given by 1 1 2 F1 (a, b; c; z) = t b−1 (1 − t)c−b−1 (1 − zt)−a dt (48) B(b, c − b) 0 190 D. Marín and J. Villadelprat Journal of Differential Equations 332 (2022) 123–201 provided that Re(c) > Re(b) > 0 and z ∈ C \ [1, +∞). Our goal is to use this formula to com- pute fˆ(α, x) (see Theorem A.5) for some specific functions f (x). Next result is addressed to this problem. Proposition B.3. The following holds: (a) Consider h(y; δ; κ) = (1 + κy 2 )δ with κ > 0. Then, for any δ ∈ R and α ∈ R \ Z≥0 such that 2δ < α, κ2  α α α lim y −α ĥ(α, y; δ; κ) = B − , −δ + . y→+∞ 2 2 2 (b) Consider g(y; δ, γ ; x) = (1 − y)δ−1 (1 − xy)−γ with y ∈ (0, 1) and x < 1. Then, for any δ > 0, γ ∈ R and α ∈ R \ Z≥0 , lim ĝ(α, y; δ, γ ; x) = B(−α, δ)2 F1 (γ , −α; δ − α; x). y→1− Proof. In order to prove (a) we define  := {(α, δ, κ) ∈ R3 : 2δ < α and κ > 0}, which is con- nected. Note then that we must show the validity of the identity on  ∩ {α ∈ / Z≥0 }. We will show first the identity on an open set of V ⊂  and then extend it by using the real version of Lemma B.1. With this aim observe that if we work on V :=  ∩ {α < 0} then the application of assertion (b) of Theorem A.5 with k = 0 yields +∞ α  +∞ −α 2 δ −α−1 κ2 α lim y ĥ(α, y; δ, κ) = (1 + κu ) u du = (1 + v)δ v − 2 −1 dv y→+∞ 2 0 0 κ  α α 2 α = B − , −δ + , 2 2 2 where in the second equality we perform the change of variable v = κu2 and in the third one we use (45). Note at this point that the function of the right hand side is meromorphic on  because 1/ is entire and B(z, w) = (z)(w) (z+w) . We claim that the function on the left hand side is also meromorphic on . To show this we work first on  ∩ {α ∈ / Z≥0 } because in doing so we can apply assertion (b) of Theorem A.5 with any k > α to obtain  k−1  y −α h(i) (0) i−α du y ĥ(α, y) = y + h(u) − T0k−1 h(u) i!(i − α) uα+1 i=0 0  k−1  1 y h(i) (0) i−α du du = y + h(u) − T0k−1 h(u) + h(u) i!(i − α) uα+1 uα+1 i=0 0 1 y du − T0k−1 h(u) uα+1 1 191 D. Marín and J. Villadelprat Journal of Differential Equations 332 (2022) 123–201  k−1  1  y h(i) (0) du du = + h(u) − T0k−1 h(u) α+1 + h(u) α+1 i!(i − α) u u i=0 0 1 y = ĥ(α, 1; δ, κ) + (1 + κu2 )δ u−α−1 du. 1 Here we denote ∂yi h(y; δ, κ) = h(i) (y) for shortness. Consequently, since 2δ < α, +∞ −α lim y ĥ(α, y; δ, κ) = ĥ(α, 1; δ, κ) + (1 + κu2 )δ u−α−1 du y→+∞ 1 1 = ĥ(α, 1; δ, κ) + κ δ (1 + κ −1 v 2 )δ v α−2δ−1 dv 0 = ĥ(α, 1; δ, κ) + κ δ ĥ(2δ − α, 1; δ, κ −1 ), where in the second equality we make the change of variable v = 1/u and in the last one we apply (b) in Theorem A.5 with k = 0. By (c) in Theorem A.5 the second summand is analytic on , whereas the first one is meromorphic on . This shows the validity of the claim and so the result follows by applying the real version of Lemma B.1. In order to prove (b) we fix δ, γ and x and apply (b) in Theorem A.5 to the function y → g(y; δ, γ ; x). Then, taking any k ∈ N with k > α and setting ∂yi g(y; δ, γ ; x) = g (i) (y) for shortness, we get  1   k−1 g (r) (0)  k−1 (r) g (0) du lim ĝ(α, y) = + g(u) − ur y→1− r!(r − α) r! uα+1 r=0 0 r=0 ⎛ ⎞ 2   1 ⎜  k−1 k−1 (r) g (r) (0) g (0) du ⎟ = 2α ⎝ + 2−α g(u) − ur ⎠ r!(r − α)2r r! uα+1 r=0 0 r=0 1  k−1 (r)  1 −γ −α−1 g (0) du + (1 − u) δ−1 (1 − xu) u du − ur−α r! u 1 r=0 1 2 2  k−1 g (r) (0) + (1 − 2α−r ) r!(r − α) r=0 1 2 = 2α ĝ (α, 1/2; δ, γ ; x) + s δ−1 (1 − x(1 − s))−γ (1 − s)−α−1 ds 0 192 D. Marín and J. Villadelprat Journal of Differential Equations 332 (2022) 123–201   1 x = 2α ĝ (α, 1/2; δ, γ ; x) + 2−δ (1 − x)−γ ĝ −δ, ; −α, γ ; 2 x−1 where in the last equality we use (b) in Theorem A.5 with k = 0 and also take δ > 0 into account. Thus, by applying (c) in Theorem A.5 to each summand in the last expression, this shows that the function (γ , α, δ, x) → lim ĝ(α, y; δ, γ ; x) y→1− is meromorphic on the open connected set  ˆ := R2 × (0, +∞) × (−∞, 1). Note also that if we ˆ consider parameter values in V̂ :=  ∩ {α < 0} ∩ {δ > 0} then 1 lim ĝ(α, y; δ, γ ; x) = (1 − u)δ−1 (1 − xu)−γ u−α−1 du = B(−α, δ)2 F1 (γ , −α; δ − α; x), y→1− 0 where in the first equality we apply (b) in Theorem A.5 with k = 0 and in the second one we use Euler’s integral representation (48). We have just proved that the left hand side expression is a meromorphic function on . ˆ Furthermore, by applying Lemma B.2 and taking B(−α, δ) = (−α)(δ) (δ−α) into account, we can assert that the right hand side is also a meromorphic function on ˆ In view of this the identity in (b) for the parameters under consideration follows by applying . the real version of Lemma B.1.  It is worth to point out that the application of (b) in Proposition B.3 provides integral rep- resentations of the hypergeometric function in a range of parameters not covered by Euler’s formula (48). Indeed, by applying also (b) in Theorem A.5 with any k > α we get  k−1 g (r) (0) B(−α, δ)2 F1 (γ , −α; δ − α; x) = r!(r − α) r=0 1    k−1 (r) g (0) −α−1 −γ + t (1 − t)δ−1 (1 − xt) − t r dt, r! 0 r=0 which holds for any δ > 0, γ ∈ R and α ∈ R \ Z≥0 , where g(t) = (1 − t)δ−1 (1 − xt)−γ . We stress that (48) gives an integral representation for 2 F1 (γ , −α; δ − α; x) only in case that δ > 0 and α < 0. Appendix C. A technical result for the proof of Proposition 3.3 Proposition C.1. The function (a, b) = B(1 − a, − 32 )2 F1 (−3 − a, − 32 ; − 12 − a; −b) − 4B(1 − a2 , − 12 )2 F1 (−1 − a2 , − 12 ; 12 − a2 ; −b) is strictly positive for all a, b ∈ (0, 1). 193 D. Marín and J. Villadelprat Journal of Differential Equations 332 (2022) 123–201 Proof. In what follows given a smooth function of several variables ψ(u) with u = (u1 , u2 , . . . , un ) ∈ Rn , for each fixed i ∈ {1, 2, . . . , n} and m ∈ N we denote by Tumi =0 ψ(u) the m-th or-   1 ∂kψ   der Taylor polynomial of ui → ψ(u) at ui = 0, i.e., Tumi =0 ψ(u) = m k=0 k! ∂uk  uki . Setting i ui =0 P (a, b) := Tb=0 3 (a, b) we claim that the following inequalities hold: (i) (a, b)  P (a, b) for all a, b ∈ (0, 1), and (ii) P (a, b) > 0 for all a, b ∈ (0, 1). It is clear that the result will follow once we prove this. For this purpose our first task will be to express the function  in terms of a definite integral and to this end we define 1 (a, b) :=B(− 32 , 1 − a)2 F1 (−3 − a, − 32 ; − 12 − a; −b) and 2 (a, b) :=B(− 12 , 1 − a2 )2 F1 (−1 − a2 , − 12 ; 12 − a2 ; −b), so that, taking B(x, y) = B(y, x) into account,  = 1 − 42 . Then by applying (b) in Proposition B.3 with {α = 32 , δ = 1 − a, γ = −3 − a, x = −b} we can assert that 1 (a, b) = limy→1− ĝ1 ( 32 , y; a, b) where g1 (y; a, b) := (1 − y)−a (1 + by)3+a . Next we apply (b) in Theo- rem A.5 taking {f = g1 , α = 32 , k = 2, x = 1} to obtain that 1 5 1 (a, b) = lim ĝ1 (3/2, y; a, b) = 2κ1 + g1 (s) − r1 (s) s − 2 ds y→1− 0 1 5 = g1 (s) − r1 (s) + κ1 s 2 s − 2 ds, 0 where κ1 = − 13 − κ̄ and r1 (s; a, b) = 1 + κ̄s with κ̄ := (3 + a)b + a. Similarly by (b) in Proposition B.3 with {α = 12 , δ = 1 − a2 , γ = −1 − a2 , x = −b} we obtain that 2 (a, b) = a a limy→1− ĝ2 ( 12 , y; a, b) where g2 (y; a, b) := (1 − y)− 2 (1 + by)1+ 2 . Next we apply (b) in Theo- rem A.5 with {f = g2 , α = 12 , k = 2, x = 1} to get that 1 3 2 (a, b) = lim ĝ2 (1/2, y; a, b) = 2κ2 + g2 (s) − r2 (s) s − 2 ds y→1− 0 1 5 = sg2 (s) − sr2 (s) + κ2 s 2 s − 2 ds, 0 where κ2 = −1 + κ̄ and r2 (s; a, b) = 1 + κ̄s. Accordingly an easy computation yields 194 D. Marín and J. Villadelprat Journal of Differential Equations 332 (2022) 123–201 1 5 (a, b) = 1 (a, b) − 42 (a, b) = h(s; a, b)s − 2 ds, 0 where a a h(s; a, b) := (1 + bs)3+a (1 − s)−a − 4s(1 + bs)1+ 2 (1 − s)− 2 − 1 + (4 − κ̄)s + (11/3 − κ̄)s 2 and consequently 1 5 P (a, b) = Tb=0 3 (a, b) = h0 (s; a, b)s − 2 ds where h0 := Tb=0 3 h. 0 ∂ϕ On account of this definition and the fact that if ∂ui ≡ 0 then Tumi =0 (ϕψ) = ϕTumi =0 (ψ), we get 1 1 5 g(s; a, b) − g0 (s; a, b) (a, b) − P (a, b) = h(s; a, b) − h0 (s; a, b) s − 2 ds = 5 ds, (1 − s)a s 2 0 0 where a a g(s; a, b) := (1 + bs)3+a − 4s(1 − s) 2 (1 + bs)1+ 2 and g0 := Tb=0 3 g. Therefore the assertion in (i) will follow once we prove that g(x; a, b)  g0 (x; a, b) for all a, b, x ∈ (0, 1). As a first step to this aim let us prove that, setting a (a, y) := (1 + y)3+a − 4(1 + y)1+ 2 and 0 := Ty=0 3 , then g(x; a, b) − g0 (x; a, b)  (a, bx) − 0 (a, bx) for all a, b, x ∈ (0, 1). In order to show this we note that g(x; a, b) − g0 (x; a, b) = (1 + bx)3+a − Tb=0 3 (1 + bx)3+a a  a a  − 4x(1 − x) 2 (1 + bx)1+ 2 − Tb=0 3 (1 + bx)1+ 2  (1 + bx)3+a − Tb=0 3 (1 + bx)3+a  a a  − 4 (1 + bx)1+ 2 − Tb=0 3 (1 + bx)1+ 2 a because x(1 − x) 2  1 and, thanks to the remainder’s formula in Taylor’s Theorem, one can easily verify that (1 + y)η − Ty=0m (1 + y)η  0 for any m odd and η ∈ (1, 2). It is clear then that a sufficient condition for the inequality in (i) to hold is that (a, y) − 0 (a, y)  0 for all 195 D. Marín and J. Villadelprat Journal of Differential Equations 332 (2022) 123–201 a, y ∈ (0, 1). In order to show that this is indeed true we note that, by the remainder’s formula in Taylor’s Theorem again, ∂y4 (a, y0 ) (a, y) − 0 (a, y) = y 4 for some y0 ∈ (0, y), 4! and consequently it suffices to verify that a ∂y4 (a, y) = (a + 3)(a + 2)(a + 1)a(1 + y)a−1 − 14 (a + 2)a(a − 1)(a − 4)(1 + y) 2 −3  0 a 1 (a−2)(a−4) for all a, y ∈ (0, 1). This inequality is equivalent to (1 +y)2+ 2  4 (a+3)(a+1) , which is obviously 2+ a2 true due to (1 + y)  1  14 (a−2)(a−4) (a+3)(a+1) for all a, y > 0. This proves the first assertion of the claim. We now turn to the proof of the inequality in (ii). On account of the definition of the Gauss hypergeometric function, see (47), together with the definition of the function (a, b) given in the statement it easily follows that P (a, b) = Tb=0 3 (a, b) = ρ0 (a) − ρ1 (a)b + ρ2 (a)b2 − ρ3 (a)b3 with     (−3 − a)n (− 32 )n 3 (−1 − a2 )n (− 12 )n a 1 ρn (a) := B 1 − a, − −4 B 1 − ,− . (49) (− 12 − a)n n! 2 ( 12 − a2 )n n! 2 2 Observe that P (a, b) is a polynomial in b for each fixed a. In order to prove the inequality in (ii) we consider P (a, · ) as a polynomial family depending on a parameter a ∈ (0, 1). In doing so it is clear that the following three conditions imply that the number of zeros of b → P (a, b) on the interval (0, 1), counted with multiplicities, is the same for all a ∈ (0, 1): (a) P (a, 0) > 0 for all a ∈ (0, 1), (b) P (a, 1) > 0 for all a ∈ (0, 1) and (c) Discrimb P (a, b) = 0 for all a ∈ (0, 1). Since one can readily show that, for instance, P ( 12 , b) > 0 for all b ∈ (0, 1), it is clear that (ii) will follow once we prove that these three conditions are true. This constitutes our next task. In order to prove the inequality in (a) we first note that, from (46),     3 a 1 P (a, 0) = ρ0 (a) = B 1 − a, − − 4B 1 − , − 2 2 2   √ (1 − a2 ) a + 12 (1 − a) =8 π − , ( 12 − a2 ) 6 ( 12 − a) √ where we use that ( 12 ) = π and (z + 1) = z(z), see [1, §6.1]. Taking this into account, the fact that P (a, 0) > 0 for all a ∈ [ 12 , 1) is clear because (z) is negative for z ∈ (−1, 0) and 196 D. Marín and J. Villadelprat Journal of Differential Equations 332 (2022) 123–201 positive for z > 0 and limz→0 1 (z) = 0, see [1, §6.1] again. To show that this is also true for a ∈ (0, 1 2) we use that then (1 − a2 ) (1 − a) a+ 1 2 (1 − a) > > . ( 12 − a 2) ( 12 − a) 6 ( 12 − a) The second inequality above is obvious whereas the first one follows noting that z → (1−z) ( 12 −z)   is positive and decreasing on (0, 12 ). In its turn this is true due to log (1−z) 1 = ( 12 − z) − ( 2 −z) (1 − z) < 0 for all z ∈ (0, 1 2) since thedigamma function 1  (z) 1 − x z−1 (z) := = −γ + dx (50) (z) 1−x 0 is a well defined monotonous increasing function for z > 0, see [1, §6.3]. Here γ ≈ 0.577 is the Euler-Mascheroni constant. This proves the validity of the inequality in (a). Let us turn next to the proof of the assertion with regard to P (a, 1). To this end, for the sake of convenience, we introduce the function 3 ((a − 1)(a − 3)(a − 5))2 B 1 − a, − 32 2 72 − a2 F (a) := 16 (2a − 3)(4a 2 − 1) B 1 − a2 , − 12 = √  , (51) 2a π 52 − a where the identity follows using the so called duplication formula for , see [1, §6.1.18]. This function will enable us to write P (a, 1) = ρ0 (a) − ρ1 (a) + ρ2 (a) − ρ3 (a) in a more convenient form taking advantage of the fact that each ρn (a) is linear in B 1 − a, − 32 and B 1 − a2 , − 12 , see (49). In doing so, some easy computations using a symbolic manipulator (see [21] for in- stance) show that   40a(a + 2)(3a − 5) a 1   P (a, 1) = B 1 − , − F (a) − g(a) , ((a − 1)(a − 3)(a − 5))2 2 2 where (23a − 94)(a − 1)(a − 3)(a − 4)(a − 5) g(a) := . 160(3a − 5)(a + 2) √ 2 π(1− a2 ) Thus, since B 1 − a2 , − 12 = − is negative for all a ∈ (0, 1), the assertion in (b) will ( 12 − a2 ) follow once we prove that F (a) > g(a) for all a ∈ (0, 1). As an intermediate step to this end we claim that if a ∈ (0, 1) then F (a) > 0, F  (a) < 0 and F  (a) > 0. The first inequality is clear from (51) because (z) > 0 for all z > 0. The second inequality is also easy because some computations show that F  (a) = − h(a) + ln 2 with h(a) := (7/2 − a/2) − (5/2 − a) F (a) 197 D. Marín and J. Villadelprat Journal of Differential Equations 332 (2022) 123–201 Fig. 11. The graphs of the transcendental function F (a) in blue, its tangent lines 0 (a) and 1 (a) at a = 0 and a = 1, respectively, in black and the rational function g(a) in red. and, on the other hand, h(a) > 0 for all a ∈ (0, 1) due to  (z) > 0 for z > 0. Finally, in order to show the third inequality we first note that F  (a) − h (a). 2 = h(a) + log 2 F (a) 3 5 a 5 a Furthermore, due to x 2 −a > x 2 − 2 > 2−n x 2 − 2 for all x ∈ (0, 1) and a > 0, from (50) it turns out that h(n) (a) = (−1/2)n (n) (7/2 − a/2) − (−1)n (n) (5/2 − a) 1   (− log x)n 3 5 a = x 2 −a − 2−n x 2 − 2 dx > 0 for all n ∈ Z≥0 . 1−x 0 Hence h(n) is increasing on (0, +∞) for all n ∈ Z≥0 and, consequently, h(n) (1) > h(n) (a) > h(n) (0) for all a ∈ (0, 1). Thus if a ∈ (0, 1) then  2 F  (a) 2 27 5π 2 − h (1) = 2 > h(0) + log 2 + log 2 + − ≈ 0.46. F (a) 5 8 12 Accordingly, F  (a) > 0 for all a ∈ (0, 1) and this concludes the proof of the claim. We proceed now with the proof of (b), which let us recall that it will follow once we prove that F (a) > g(a) for all a ∈ (0, 1). With this aim in view we take the tangent lines to the graph of F at a = 0 and a = 1, that are given by 0 (a) = 75 16 − ( 15 8 + 75 16 log 2)a and 1 (a) = 4 π + π2 (1 − 6 log 2)(a − 1), 198 D. Marín and J. Villadelprat Journal of Differential Equations 332 (2022) 123–201 respectively, see Fig. 11. Since F is convex, in order to show that F (a) > g(a) for all a ∈ (0, 1), it suffices to verify that max{0 (a), 1 (a)} > g(a) for all a ∈ (0, 1). To see this we consider the unique solution of 0 (a) = 1 (a), which one can check that it is given by 75π − 32 − 192 log 2 a = â := ≈ 0.45. (75π − 192) log 2 + 30π + 32 pi (a) One can also verify that, for i = 0, 1, i (a) − g(a) = 160π i (2+a)(5−3a) with p0 (a) = 23a 5 − 393a 4 + (3479 + 2250 log 2) a 3 + (−9957 + 750 log 2) a 2 + (7688 − 7500 log 2) a + 1860 and p1 (a) = 23πa 5 − 393πa 4 + (−960 + 2579π + 5760 log 2) a 3 − (8007π + 1280 + 3840 log 2) a 2 + (11438π + 2880 − 21120 log 2) a + 3200 − 5640π + 19200 log 2. By applying Sturm’s Theorem we can assert that p0 is positive on (0, 0.46) and that p1 is positive on (0.44, 1), which imply that max{0 (a), 1 (a)} > g(a) for all a ∈ (0, 1) as desired. This proves (b). Our last task is to prove the assertion in (c). To this end we use a symbolic manipulator in order to show that −2(a + 2)B(1 − a2 , − 12 )4 Discrimb P (a, b) = R a, F (a) , 3((a − 1)(a − 3)(a − 5))8 where R(a, t) = − 16384(2a − 1)(8a 6 + 36a 5 − 126a 4 − 413a 3 + 429a 2 + 576a − 512)(a + 3)2 (2a − 3)2 t 4 + 3072(a − 1)(a − 3)(a − 5)(2a − 3)(a + 3) 48a 8 + 252a 7 − 1904a 6 − 2305a 5 + 11568a 4 − 2566a 3 − 14160a 2 − 2784a + 11520 t 3 − 24a(a + 2) 768a 9 − 7808a 8 + 3616a 7 + 135520a 6 − 221032a 5 − 557976a 4 + 823685a 3 + 1082256a 2 − 894960a − 915840)(a − 5)2 (a − 1)2 (a − 3)2 t 2 − 4(a − 4)(a + 2)(320a 8 − 1400a 7 + 1830a 6 − 5491a 5 + 4678a 4 + 32889a 3 − 4482a 2 − 47520a − 64800 (a − 1)3 (a − 3)3 (a − 5)4 t + 15a(a − 2)(a − 4)(a + 2)(5a 4 − 15a 3 − 5a 2 + 27a + 36)(a − 1)4 (a − 3)5 (a − 5)6 . Let us mention that in order to ease this computation and introduce F we use that the coefficients of P (a, b) = ρ0 (a) − ρ1 (a)b + ρ2 (a)b2 − ρ3 (a)b3 , see (49), are linear in B 1 − a, − 32 and B 1 − a2 , − 12 and that, on the other hand, the discriminant of a third degree polynomial is a 199 D. Marín and J. Villadelprat Journal of Differential Equations 332 (2022) 123–201 homogeneous polynomial of degree 4 in its coefficients. On account of the above expression it is clear that (c) will follow once we prove that R(a, F (a)) = 0 for all a ∈ (0, 1). To this end we note that F (0) = 75 16 and F (1) = π . Therefore, see Fig. 11, the graph t = F (a) for 4 a ∈ (0, 1) verifies max{0 (a), π4 } < F (a) < 75  16 because we previously proved that F < 0 and  F > 0 on the interval (0, 1). Accordingly it suffices to show that R(a, t) = 0 for all (a, t) inside the trapezium given by max{0 (a), π4 } < t < 75 16 and a ∈ (0, 1). We will prove this taking t ∈ ( π4 , 75 ) as a fixed parameter and showing that the polynomial a → R(a, t) has not any root 16   −1 −1 on (0 (t), 1), where 0 (t) = 30−75 log 2 . To this effect we show first that R 30−75 75−16t 75−16t log 2 , t , R(1, t) and Discrima R(a, t) do not vanish for all t ∈ ( π4 , 75 16 ). This implies that the number of −1 roots of R(a, t) = 0 on (0 (t), 1) does not change for t ∈ ( π4 , 75 16 ). Taking this into account the desired result follows by checking, for instance, that this number is zero for t = 2 ∈ ( π4 , 75 16 ). All these assertions can be checked systematically by applying Sturm’s Theorem because only one variable polynomials are involved. This shows the validity of (c) and so the inequality in (ii) is true. This concludes the proof of the result.  References [1] M. Abramowitz, I.A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, Dover, New York, 1992, reprint of the 1972 edition. [2] G.E. Andrews, R. 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