Electronic Journal of Qualitative Theory of Differential Equations 2022, No. 35, 1–31; https://doi.org/10.14232/ejqtde.2022.1.35 www.math.u-szeged.hu/ejqtde/ On the cyclicity of Kolmogorov polycycles David Marín1,2 and Jordi Villadelprat B 3 1 Departament de Matemàtiques, Edifici Cc, Universitat Autònoma de Barcelona, 08193 Cerdanyola del Vallès (Barcelona), Spain 2 Centre de Recerca Matemàtica, Edifici Cc, Campus de Bellaterra, 08193 Cerdanyola del Vallès (Barcelona), Spain 3 Departament d’Enginyeria Informàtica i Matemàtiques, ETSE, Universitat Rovira i Virgili, 43007 Tarragona, Spain Received 11 April 2022, appeared 30 July 2022 Communicated by Gabriele Villari Abstract. In this paper we study planar polynomial Kolmogorov’s differential systems ( ẋ = x f ( x, y; µ), Xµ ẏ = yg( x, y; µ), with the parameter µ varying in an open subset Λ ⊂ R N . Compactifying Xµ to the Poincaré disc, the boundary of the first quadrant is an invariant triangle Γ, that we assume to be a hyperbolic polycycle with exactly three saddle points at its vertices for all µ ∈ Λ. We are interested in the cyclicity of Γ inside the family { Xµ }µ∈Λ , i.e., the number of limit cycles that bifurcate from Γ as we perturb µ. In our main result we define three functions that play the same role for the cyclicity of the polycycle as the first three Lyapunov quantities for the cyclicity of a focus. As an application we study two cubic Kolmogorov families, with N = 3 and N = 5, and in both cases we are able to determine the cyclicity of the polycycle for all µ ∈ Λ, including those parameters for which the return map along Γ is the identity. Keywords: limit cycle, polycycle, cyclicity, asymptotic expansion. 2020 Mathematics Subject Classification: 34C07, 34C20, 34C23. 1 Introduction and main results The present paper is motivated by the results obtained by Gasull, Mañosa and Mañosas [8] with regard to the stability of an unbounded polycycle Γ in the Kolmogorov’s polynomial differential systems ( ẋ = x f ( x, y), ẏ = yg( x, y). These systems are widely used in ecology to describe the interaction between two populations, see [18] for instance. That being said, the stability of the polycycle is not the main issue to B Corresponding author. Email: jordi.villadelprat@urv.cat 2 D. Marín and J. Villadelprat which this paper is addressed. Indeed, assuming that the coefficients of the polynomials f and g depend analytically on a parameter µ, we are interested in the cyclicity of the polycycle (see Definition 1.2 below), which roughly speaking is the number of limit cycles that can bifurcate from Γ as we perturb µ. In our main result (Theorem A) we define three functions, d1 (µ), d2 (µ) and d3 (µ), that play the same role for the cyclicity of the polycycle as the first three Lyapunov quantities for the cyclicity of a focus. Recall that the displacement map can be analytically extended to a focus and that the Lyapunov quantities are the coefficients of its Taylor’s series. On the contrary the displacement map has no smooth extension to a polycycle. At best one can hope that it has some asymptotic expansion. This is indeed the case for the polycycle that we study in the present paper and in order to obtain it we strongly rely in our previous results [14–16] about the asymptotic expansion of the Dulac map of an unfolding of hyperbolic saddles. The principal part of the asymptotic expansion of the displacement map is given in a monomial scale containing a deformation of the logarithm, the so-called Ecalle–Roussarie compensator, and the remainder is uniformly flat with respect to the parameters. The functions di (µ) in Theorem A are essentially the coefficients of the first three monomials in the principal part, which explains their relation with the cyclicity. For other results regarding the cyclicity of polycycles and more general limit periodic sets the reader is referred to [5, 6, 11, 22] and references therein. Most of the work on planar polynomial differential systems, including this paper, is related to the questions surrounding Hilbert’s 16th problem (see for instance [10,21,23] and references therein) and its various weakened versions. In this setting it is worth to mention that, using a compactness argument, Roussarie [20] showed that to prove the existential part of Hilbert’s 16th problem in the family Pn of all polynomial vector fields of degree ⩽ n it is sufficient to show that each limit periodic set in Pn has finite cyclicity. Definition 1.1. Let X be a vector field on R2 (or S2 ). A graphic Γ for X is a compact, non- empty invariant subset which is a continuous image of S1 and consists of a finite number of isolated singular points { p1 , . . . , pm , pm+1 = p1 } (not necessarily distinct) and compatibly oriented separatrices {s1 , . . . , sm } connecting them (i.e., such that the α-limit set of s j is p j and the ω-limit set of s j is p j+1 ). A graphic is said to be hyperbolic if all its singular points are hyperbolic saddles. A polycyle is a graphic with a return map defined on one of its sides. The polycycle that we aim to study is unbounded. In order to investigate the behaviour of the trajectories of a polynomial vector field Y near infinity we can consider its Poincaré compactification p(Y ), see [2, §5] for details, which is an analytically equivalent vector field defined on the sphere S2 . The points at infinity of R2 are in bijective correspondence with the points of the equator of S2 , that we denote by ℓ∞ . Furthermore, the trajectories of p(Y ) in S2 are symmetric with respect to the origin and so it suffices to draw its flow in the closed northern hemisphere only, the so called Poincaré disc. Definition 1.2. Let { Xµ }µ∈Λ be a family of vector fields on S2 and suppose that Γ is a polycyle for Xµ0 . We say that Γ has finite cyclicity in the family { Xµ }µ∈Λ if there exist κ ∈ N, ε > 0 and δ > 0 such that any Xµ with ∥µ − µ0 ∥ < δ has at most κ limit cycles γi with dist H (Γ, γi ) < ε. The minimum of such κ when  ε and δ go to zero is called the cyclicity of Γ in { Xµ }µ∈Λ and denoted by Cycl (Γ, Xµ0 ), Xµ . In this paper we consider the family of vector fields { Xµ }µ∈Λ given by Xµ := f ( x, y; µ) x∂ x + g( x, y; µ)y∂y (1.1) On the cyclicity of Kolmogorov polycycles 3 s2 Γ s1 s3 ℓ∞ Figure 1.1: Placement of the hyperbolic saddles and the polycycle Γ in the Poincaré disc. where Λ is an open subset of R N and f and g are polynomials in x and y of degree n ∈ N with the coefficients depending analytically on µ. The standing hypothesis on the family { Xµ }µ∈Λ are the following: H1 f (z, 0; µ) > 0, g(0, z; µ) < 0 and f n − gn (1, z; µ) < 0 for all z > 0 and µ ∈ Λ.  f  f −g  g H2 λ1 (µ) := gn −n f n (1, 0; µ), λ2 (µ) := n gn n (0, 1; µ) and λ3 (µ) := − f (0, 0; µ ), are well defined and strictly positive for all µ ∈ Λ. Here, and in what follows, f n and gn denote, respectively, the homogeneous part of degree n of f and g. Conditions H1 and H2 guarantee that, after compactifying the polynomial vector field Xµ to the Poincaré disc, the boundary of the first quadrant is a polycyle with three hyperbolic saddles, see Figure 1.1, s1 : = { y = 0} ∩ ℓ ∞ , s2 : = { x = 0} ∩ ℓ ∞ and s3 := (0, 0). From now on we shall denote this polycyle by Γ, that we remark is a compact subset of the Poincaré disc. The hyperbolicity ratios of the saddles at its vertices are precisely the ones given in H2. We also define: Z u    Z u    f −g 1 dz fn dz L11 (u) = exp f (1/z, 0) + λ1 z , L12 (u) = exp f n − gn (1, z) + λ1 z , 0 0 Z u    Z u    gn dz g− f L21 (u) = exp gn − f n (z, 1) + 1 λ2 z , L22 (u) = exp (0, 1/z) + λ2 dzz , g 0 0 Z u    Z u    g dz f 1 dz L31 (u) = exp f (z, 0) + λ3 z , L32 (u) = exp g (0, z ) + λ z 0 0 3 together with     L11 (u) g g M1 (u) = − ∂2 (1/u, 0) and M3 (u) = L31 (u) ∂2 (u, 0). (1.2) u f f 4 D. Marín and J. Villadelprat We point out that all these functions depend on the parameter µ. This dependence is omitted for the sake of shortness when there is no risk of confusion. We can now state our main result, which is addressed to the cyclicity of the polycycle Γ inside the polynomial family { Xµ }µ∈Λ . More formally, we should refer to the compactified family { p( Xµ )}µ∈Λ of vector fields on S2 but for the simplicity in the exposition we commit an abuse of language by identifying both families. It is clear that the number of limit cycles of p( Xµ ) and Xµ is the same. In the statement R ( · ; µ) stands for the return map of the vector field Xµ around the polycycle Γ (see Figure 1.1) and we use the notion of functional independence that is given in Definition 2.8. Theorem A. Consider the family of Kolmogorov polynomial vector fields { Xµ }µ∈Λ given in (1.1) and verifying the assumptions H1 and H2. Then, for any µ0 ∈ Λ, the following assertions hold with regard to the cyclicity of the polycycle Γ inside the family: ( a) Cycl (Γ, Xµ0 ), Xµ = 0 if d1 (µ) := 1 − λ1 λ2 λ3 does not vanish at µ0 .  (b) Cycl (Γ, Xµ0 ), Xµ ⩾ 1 if d1 vanishes and is independent at µ0 and R ( · ; µ0 ) ̸≡ Id.       λ2 λ1 λ2 L12 L31 L22 (c) Cycl (Γ, Xµ0 ), Xµ ⩽ 1 if d2 (µ) := log L21  L11 L32 (1) does not vanish at µ0 . (d) Cycl (Γ, Xµ0 ), Xµ ⩾ 2 if d1 and d2 vanish and are independent at µ0 and R ( · ; µ0 ) ̸≡ Id.  In case that λ1 (µ0 ) < 1, λ2 (µ0 ) > 1 and λ3 (µ0 ) > 1 then the following assertions hold as well: (e) Cycl (Γ, Xµ0 ), Xµ ⩽ 2 if d3 (µ) := M̂3 (λ3 , 1) L11 (1) − M̂1 λ11 , 1 L31 (1) does not vanish at µ0 .   ( f ) Cycl (Γ, Xµ0 ), Xµ ⩾ 3 if d1 , d2 and d3 vanish and are independent at µ0 and R ( · ; µ0 ) ̸≡ Id.  Let us make some remarks with regard to the regularity of the functions d1 , d2 and d3 defined in the statement. On account of the hypothesis H1 and H2 it is evident that d1 is analytic on the whole parameter space Λ. On the other hand, d2 is defined in terms of the functions µ 7→ Lij (1), which in turn are given by some (apparently) improper integrals. By applying the Weierstrass Division Theorem one can easily show that each Lij (1) is an analytic strictly positive function, so that d2 is also analytic on Λ. Finally, d3 is given by means of a sort of incomplete Mellin transform (which is defined in Proposition 2.5) of the functions M1 and M3 in (1.2). One can show that the hypothesis H1 and H2 imply that each Mi (u; µ) is analytic on (−ε, +∞) × Λ for some ε > 0. Taking this into account, by applying (d) in Proposition 2.5 it follows that d3 is a meromorphic function on Λ having poles only at those µ0 such that 1/λ1 (µ0 ) ∈ N or λ3 (µ0 ) ∈ N. Also with regard to the statement of Theorem A, the assertions (e) and ( f ) hold under the assumptions λ1 (µ0 ) < 1, λ2 (µ0 ) > 1 and λ3 (µ0 ) > 1. However, one can always reduce to this case provided that λi (µ0 ) ̸= 1 for i = 1, 2, 3 by means of a rescaling of time and a projective change of coordinates that permute conveniently the three singular points of the polycycle. The paper [8] constitutes an important previous contribution to the study of Kolmogorov polycycles that should be referred. Indeed, following our notations and definitions, the au- thors prove (see [8, Theorem 1]) that if d1 (µ0 ) = 0 then the return map of Xµ0 around the polycycle Γ is of the form R (s; µ0 ) = ∆s + o(s), (1.3) cf. (b) in Theorem 2.6, and they also provide the explicit expression of the coefficient ∆. This coefficient is given as the limit of a sum of three improper integrals, which computed On the cyclicity of Kolmogorov polycycles 5 separately diverge. An easy manipulation of the integrals shows that these divergences cancel each other, yielding to the expression of d2 given in Theorem A. It is important to remark that the expansion in (1.3) can not be used to obtain an upper bound for Cycl (Γ, Xµ0 ), Xµ because the remainder is not uniform with respect to the parameters. It is possible, however, to use it to obtain lower bounds. In this direction the authors prove in [8, Corollary 5] that if d1 vanishes and is independent at µ0 and d2 (µ0 ) ̸= 0 then Cycl (Γ, Xµ0 ), Xµ ⩾ 1. Since  d2 (µ0 ) ̸= 0 implies R ( · ; µ0 ) ̸≡ Id by Theorem 2.6, this lower bound follows by applying (b) in Theorem A. The paper is organised in the following way. Section 2 is entirely devoted to prove Theo- rem A and for that purpose we rely in our previous results about the asymptotic expansion of the Dulac map of a hyperbolic saddle that we obtain in [14–16]. For this reason, before starting the proof of Theorem A we first state these results and introduce the necessary definitions. The asymptotic expansion of the displacement map near the polycycle is given in Theorem 2.6 and constitutes the fundamental tool in order to prove Theorem A. As a by-product of this ex- pansion we obtain a method to study the stability of the polycycle, see Remark 2.7. Section 3 is addressed to the applications. The first one is Theorem 3.1, where we consider a Kol- mogorov’s cubic system depending on three parameters that was previously studied in [8]. The authors in that paper show that there exist parameters for which the cyclicity of the poly- cycle is at least 1. In the present paper we obtain the exact cyclicity of all the parameters in the family (that can be 0 or 1), including the case in which the return map along the polycycle is the identity. We also show that there exists exactly one singularity in the first quadrant, which can be a focus or a center, and we compute its cyclicity. Finally we prove that it is not possible a simultaneous bifurcation of limit cycles from the polycycle and that singularity. We give our second application in Theorem 3.2, where we consider a Kolmogorov’s cubic system depending on five parameters. In this case we also provide the exact cyclicity of Γ for all the parameters in the family, which again can be 0 or 1. 2 Proof of Theorem A In order to tackle the proof of Theorem A we will appeal to some previous results from [14–16] about the asymptotic expansion of the Dulac map. For reader’s convenience we gather these results in Proposition 2.4. To this end it is first necessary to introduce some new notation and definitions. Setting ν̂ := (λ, ν) ∈ Ŵ := (0, +∞) × W with W an open set of R N , we consider the family of vector fields { Xν̂ }ν̂∈Ŵ with Xν̂ ( x1 , x2 ) = x1 P1 ( x1 , x2 ; ν̂)∂ x1 + x2 P2 ( x1 , x2 ; ν̂)∂ x2 (2.1) where • P1 and P2 belong to C ω (U × Ŵ ) for some open set U of R2 containing the origin, • P1 ( x1 , 0; ν̂) > 0 and P2 (0, x2 ; ν̂) < 0 for all ( x1 , 0), (0, x2 ) ∈ U and ν̂ ∈ Ŵ, P (0,0;ν) • λ = − P2 (0,0;ν) . 1 Thus, for all ν̂ ∈ Ŵ, the origin is a hyperbolic saddle of Xν̂ with the separatrices lying in the axis. We point out that here the hyperbolicity ratio of the saddle is an independent parameter, although in the proof of Theorem A we will have λ = λ(ν). The reason for this is that the 6 D. Marín and J. Villadelprat x2 σ1 (s) Σ1 σ1 0 D (s) φ( · , σ1 (s)) σ2 0 s σ2 ( D (s)) x1 Σ2 Figure 2.1: Definition of the Dulac map D ( · ; ν̂), where φ(t, p; ν̂) is the solution of Xν̂ passing through the point p ∈ U at time t = 0. hyperbolicity ratio turns out to be the ruling parameter in our results and, besides, having it uncoupled from the rest of parameters simplifies the notation in the statements. Moreover, for i = 1, 2, we consider a C ω transverse section σi : (−ε, ε) × Ŵ −→ Σi to Xν̂ at xi = 0 defined by  σi (s; ν̂) = σi1 (s; ν̂), σi2 (s; ν̂) such that σ1 (0, ν̂) ∈ {(0, x2 ); x2 > 0} and σ2 (0, ν̂) ∈ {( x1 , 0); x1 > 0} for all ν̂ ∈ Ŵ. We denote the Dulac map of Xν̂ from Σ1 to Σ2 by D ( · ; ν̂), see Figure 2.1. The asymptotic expansion of D (s; ν̂) at s = 0 consists of a remainder and a principal part. The principal part is given in a monomial scale that contains a deformation of the logarithm, the so-called Ecalle-Roussarie compensator, whereas the remainder has good flatness properties with respect to the param- eters. We next give precise definitions of these key notions. Definition 2.1. The function defined for s > 0 and α ∈ R by means of ( −α s −1 α if α ̸= 0, ω (s; α) = − log s if α = 0, is called the Ecalle–Roussarie compensator. Definition 2.2. Consider an open subset U ⊂ Ŵ ⊂ R N +1 . We say that a function ψ(s; ν̂) belongs to the class Cs∞>0 (U ) if there exist an open neighbourhood Ω of {(s, ν̂) ∈ R N +2 ; s = 0, ν̂ ∈ U } = {0} × U in R N +2 such that (s, ν̂) 7→ ψ(s; ν̂) is C ∞ on Ω ∩ (0, +∞) × U .  More formally, the definition of Cs∞>0 (U ) must be thought in terms of germs with respect to relative neighbourhoods of {0} × U in (0, +∞) × U. In doing so Cs∞>0 (U ) becomes a ring. We can now introduce the notion of flatness that we shall use in the sequel. Definition 2.3. Consider an open subset U ⊂ Ŵ ⊂ R N +1 . Given L ∈ R and ν̂0 ∈ U, we say that a function ψ(s; ν̂) ∈ Cs∞>0 (U ) is L-flat with respect to s at ν̂0 , and we write ψ ∈ F L∞ (ν̂0 ), if for each ℓ = (ℓ0 , ℓ1 , . . . , ℓ N +1 ) ∈ Z≥ N +2 0 there exist a neighbourhood V of ν̂0 and C, s0 > 0 such that ∂|ℓ| ψ(s; ν̂) ℓ ℓ ⩽ Cs L−ℓ0 for all s ∈ (0, s0 ) and ν̂ ∈ V. ℓ ∂s 0 ∂ν̂1 · · · ∂ν̂N +1 1 N + 1 If W is a (not necessarily open) subset of U then define F L∞ (W ) := F L∞ (ν̂0 ). T ν̂0 ∈W On the cyclicity of Kolmogorov polycycles 7 Apart from the remainder and the monomial order, the most important ingredient for our purposes is the explicit expression of the coefficients in the asymptotic expansion. In order to give them we introduce next some additional notation, where for the sake of shortness the dependence on ν̂ = (λ, ν) is omitted. We define the functions: Z u  Z u  P1 (0, z) 1 dz P2 (z, 0) dz L1 (u) := exp + L2 (u) := exp +λ 0 P2 (0, z) λ z 0 P1 (z, 0) z     (2.2) P1 P2 M1 (u) := L1 (u)∂1 (0, u) M2 (u) := L2 (u)∂2 (u, 0) P2 P1 On the other hand, for shortness as well, we use the compact notation σijk for the kth derivative at s = 0 of the jth component of σi (s; ν̂), i.e., σijk (ν̂) := ∂ks σij (0; ν̂). Taking this notation into account we also introduce the following real values, where once again we omit the dependence on ν̂:   σ112 σ P1 σ111 S1 : = − 121 (0, σ120 ) − M̂1 (1/λ, σ120 ) 2σ111 σ120 P2 L1 (σ120 )   (2.3) σ222 σ P2 σ221 S2 : = − 211 (σ210 , 0) − M̂2 (λ, σ210 ). 2σ221 σ210 P1 L2 (σ210 ) Here M̂i stands for a sort of incomplete Mellin transform of Mi that will be defined by Propo- sition 2.5 below. We can now state the following result, which gathers Theorem A and Theo- rem 4.1 in [16] and that it will constitute the key tool in order to prove the main result in the present paper. Proposition 2.4. Let D (s; ν̂) be the Dulac map of the hyperbolic saddle (2.1) from Σ1 and Σ2 and define σλ σ120 L2 (σ210 ) ∆0 (ν̂) = λ111 , ∆1 (ν̂) = ∆0 λS1 and ∆2 (ν̂) = −∆20 S2 , L1 (σ120 ) σ221 σ210 λ where ν̂ = (λ, ν) ∈ Ŵ = (0, +∞) × W. Then ∆0 is analytic and strictly positive on Ŵ, ∆1 is meromorphic on Ŵ with poles only at λ ∈ N 1 and ∆2 is meromorphic on Ŵ with poles only at λ ∈ N. Moreover the following assertions hold: (1) If λ0 < 1 then D (s; ν̂) = sλ ∆0 (ν̂) + ∆2 (ν̂)sλ + Fℓ∞ ({λ0 } × W ) for any ℓ ∈ λ0 , min(2λ0 , 1) .    (2) If λ0 = 1 then D (s; ν̂) = sλ ∆0 (ν̂) + ∆λ0 (ω; ν̂)s + Fℓ∞ ({λ0 } × W ) for any ℓ ∈ [1, 2), where  ∆λ0 (ω; ν̂) = ∆1 (ν̂) + ∆2 (ν̂)(1 + αω ), α = 1 − λ and ω = ω (s; α). (3) If λ0 > 1 then D (s; ν̂) = sλ ∆0 (ν̂) + ∆1 (ν̂)s + Fℓ∞ ({λ0 } × W ) for any ℓ ∈ 1, min(λ0 , 2) .    In particular, D (s; ν̂) = sλ ∆0 (ν̂) + Fℓ∞ ({λ0 } × W ) for any ℓ ∈ 0, min(λ0 , 1) .   The flatness ℓ of the remainder can range in a certain interval depending on λ0 . The left endpoint of this interval is only given for completeness to guarantee that all the monomials in the principal part are relevant (i.e., they cannot be included in the remainder). The important 8 D. Marín and J. Villadelprat information about the flatness is given by the right endpoint. A key tool in order to give a closed expression of the coefficients ∆i is the use of a sort of incomplete Mellin transform, which is accurately defined in the next result. For a proof of this result the reader is referred to [16, Appendix B]. Proposition 2.5. Consider an open interval I of R containing x = 0 and an open subset U of R M . ( a) Given f ( x; υ) ∈ C ∞ ( I × U ), there exits a unique fˆ(α, x; υ) ∈ C ∞ ((R \ Z≥0 ) × I × U ) such that x∂ x fˆ(α, x; υ) − α fˆ(α, x; υ) = f ( x; υ). −α (b) If x ∈ I \ {0} then ∂ x ( fˆ(α, x; υ)| x |−α ) = f ( x; υ) |x|x and, taking any k ∈ Z≥0 with k > α, k −1 ∂ix f (0; υ) i Z x  ds fˆ(α, x; υ) = ∑ i!(i − α) x + | x |α 0 f (s; υ) − T0k−1 f (s; υ) |s|−α , s i =0 where T0k f ( x; υ) = ∑ik=0 i!1 ∂ix f (0; υ) xi is the k-th degree Taylor polynomial of f ( x; υ) at x = 0. (c) For each (i0 , x0 , υ0 ) ∈ Z≥0 × I × W the function (α, x, υ) 7→ (i0 − α) fˆ(α, x; υ) extends C ∞ at (i0 , x0 , υ0 ) and, moreover, it tends to i10 ! ∂ix0 f (0; υ0 ) x0i0 as (α, x, υ) → (i0 , x0 , υ0 ). (d) If f ( x; υ) is analytic on I × U then fˆ(α, x; υ) is analytic on (R \ Z≥0 ) × I × U. Finally, for each (α0 , x0 , υ0 ) ∈ Z≥0 × I × U the function (α, x, υ) 7→ (α0 − α) fˆ(α, x; υ) extends analytically to ( α 0 , x 0 , υ0 ) . On account of this result for each Mi (u; ν̂) in (2.2) we have that (α, u; ν̂) 7→ M̂i (α, u; ν̂) is a well defined meromorphic function with poles only at α ∈ Z≥0 . Accordingly, see (2.3), M̂1 (1/λ, σ120 ) and M̂2 (λ, σ210 ) are the values (depending on ν̂) that we obtain by tak- ing M̂1 (α, u; ν̂) with α = 1/λ and u = σ120 (ν̂) and by taking M̂2 (α, u; ν̂) with α = λ and u = σ210 (ν̂), respectively. At this point we get back to the setting treated in the present paper and from now on we recover the original notation for the parameters in the family under consideration, see (1.1). In order to study the Dulac maps of the hyperbolic saddles at the vertices of the polycycle Γ we take three local transverse sections Σ1 , Σ2 , and Σ3 parametrised, respectively, by s 7→ (1, s), s 7→ (1/s, 1/s) and s 7→ (s, 1) with s > 0. We define D1 (s; µ) to be the Dulac map of Xµ from Σ1 to Σ2 , D2 (s; µ) to be the Dulac map of Xµ from Σ2 to Σ3 and, finally, D3 (s; µ) to be the Dulac map of − Xµ from Σ1 to Σ3 , see Figure 2.2 x It is then clear that the limit cycles of Xµ near Γ are in one to one correspondence with the isolated positive zeroes of D (s; µ) := D2 ◦ D1 − D3 (s; µ)  near s = 0. The proof of Theorem A strongly relies in our next result, where we get the asymptotic expansion of D (s; µ) at s = 0 and we compute its coefficients. In its statement di (µ), for i = 1, 2, 3, are the functions defined in Theorem A and R{µ}µ0 stands for the local ring of convergent power series at µ0 . Theorem 2.6. Let us fix any µ0 ∈ Λ and set λ0i := λi (µ0 ) for i = 1, 2, 3. ( a) If λ01 λ02 λ03 ̸= 1 then, for any ℓ1 ∈ min(λ01 λ02 , 1/λ03 ), min(λ01 + λ01 λ02 , 1 + λ01 λ02 , 2λ01 λ02 , 1 + 1/λ03 , 2/λ03 ) , D (s; µ) = a1 (µ)sλ1 λ2 − a2 (µ)s1/λ3 + Fℓ∞1 (µ0 ), where a1 and a2 are analytic and strictly positive functions on Λ. On the cyclicity of Kolmogorov polycycles 9 s2 ℓ∞ D2 Σ2 Σ3 D1 D3 Σ1 s3 s1 Figure 2.2: Auxiliary Dulac maps for the definition of D = D2 ◦ D1 − D3 in Theorem 2.6. The return map in Theorem A, with respect to the transverse section Σ1 , would be R = D3−1 ◦ D2 ◦ D1 = D3−1 ◦ D + Id. (b) If λ01 λ02 λ03 = 1 then, for any ℓ2 ∈ (0, min(1, λ01 , λ01 λ02 )), D (s; µ) = b1 (µ) ω s; α(µ) + b2 (µ) + Fℓ∞2 (µ0 ) s1/λ3 ,   where α = 1/λ3 − λ1 λ2 , b1 = αa1 and b2 = a1 − a2 . Moreover the equalities (b1 ) = (d1 ) and (b1 , b2 ) = (d1 , d2 ) between ideals over the ring R{µ}µ0 are verified. Assuming λ01 > 1, λ02 > 1 and λ03 < 1 additionally then, for any ℓ3 ∈ 1, min(2, λ01 , 1/λ03 ) , D (s; µ) = b1 (µ) ω s; α(µ) + b2 (µ) + b3 (µ)s + Fℓ∞3 (µ0 ) s1/λ3 U (s; µ),   where b3 is an analytic function at µ0 verifying that (b1 , b2 , b3 ) = (d1 , d2 , d3 ) over the ring R{µ}µ0 and U is an analytic function such that U (0; µ0 ) = 1. Proof. In order to study the Dulac map D1 from Σ1 to Σ2 we compactify Xµ by means of y the coordinate change { x1 = x , x2 = 1x }. One can easily verify that the new vector field is orbitally conjugated to (2.1) particularised with P1 ( x1 , x2 ) = x2n f − g)( x12 , xx21 ) and P2 ( x1 , x2 ) = x2n f ( x12 , xx12 ), whereas in these coordinates the transverse sections Σ1 and Σ2 are parametrised by σ1 (s) = (s, 1) and σ2 (s) = (1, s), respectively. The hyperbolicity ratio of the saddle at f the origin is λ1 = − PP12 (0, 0) = gn −n f n (1, 0). Therefore, by applying Proposition 2.4 we can   assert that D1 (s) = ∆10 sλ1 1 + Fℓ∞1 (µ0 ) , with ∆10 := L12 L11 − λ1   (1) (2.4) and any 0 < ℓ1 < min(1, λ01 ). Recall here that λ0i = λi (µ0 ) for i = 1, 2, 3 by definition. 10 D. Marín and J. Villadelprat Next, to analyse the Dulac map D2 from Σ2 to Σ3 we compactify Xµ performing the change of coordinates given by { x1 = y1 , x2 = yx }. One can check that the new vector field is orbitally conjugated to (2.1) with P1 ( x1 , x2 ) = x1n g( xx21 , x11 ) and P2 ( x1 , x2 ) = x1n g − f )( xx21 , x11 ) and that in these coordinates the transverse sections Σ2 and Σ3 are parametrised by σ1 (s) = (s, 1) and σ2 (s) = (1, s), respectively. The hyperbolicity ratio of the saddle at the origin is λ2 = − PP12 (0, 0) = f n g−ngn (0, 1). Thus, by Proposition 2.4 again,   D2 (s) = ∆20 sλ2 1 + Fℓ∞2 (µ0 ) , − λ2  with ∆20 := L22 L21  (1) (2.5) and any 0 < ℓ2 < min(1, λ02 ). Finally, to study the Dulac map D3 of from Σ1 to Σ3 we make the reflection { x1 = y, x2 = x }, which brings − Xµ to (2.1) with P1 ( x1 , x2 ) = − g( x2 , x1 ) and P2 ( x1 , x2 ) = − f ( x2 , x1 ). In these coordinates the transverse sections Σ1 and Σ3 are parametrised by σ1 (s) = (s, 1) and σ2 (s) = (1, s), respectively, and the hyperbolicity ratio of the saddle is λ13 = − PP21 (0, 0) = − gf (0, 0). Hence by Proposition 2.4 once again,  D3 (s) = ∆30 s1/λ3 1 + Fℓ∞3 (µ0 ) , −1/λ3  with ∆30 := L32 L31  (1) (2.6) and any 0 < ℓ3 < min(1, 1/λ03 ). Consequently, from (2.4), (2.5) and (2.6) we get that D (s) = D2 ◦ D1 − D3 (s) = ∆20 ∆10 1 + Fℓ∞1 (µ0 ) 2 1 + Fℓ∞4 (µ0 ) − ∆30 s1/λ3 1 + Fℓ∞3 (µ0 )  λ2 λ1 λ2 λ   s = ∆20 ∆10 1 + Fℓ∞1 (µ0 ) 1 + Fℓ∞4 (µ0 ) − ∆30 s1/λ3 1 + Fℓ∞3 (µ0 ) λ2 λ1 λ2    s = ∆20 ∆10 λ2 λ1 λ2 s − ∆30 s1/λ3 + Fℓ∞5 (µ0 ), where, by [15, Lemma A.2], we can take ℓ4 and ℓ5 to be any numbers such that 0 < ℓ4 < min(λ01 , λ01 λ02 ) and min(λ01 λ02 , 1/λ03 )) < ℓ5 < min(λ01 + λ01 λ02 , 1 + λ01 λ02 , 2λ01 λ02 , 1 + 1/λ03 , 2/λ03 ), respectively. Thus, setting a1 := ∆20 ∆10 λ2 and a2 := ∆30 , we obtain D (s) = a1 sλ1 λ2 − a2 s1/λ3 + Fℓ∞5 (µ0 ) (2.7) and this proves ( a) because each ∆i0 is a strictly positive analytic function by Proposition 2.4. Let us proceed next with the proof of the assertions in (b), thus from now on we assume that λ01 λ02 λ03 = 1. Then, setting α = 1/λ3 − λ1 λ2 and taking any ℓ6 ∈ (0, min(1, λ01 , λ01 λ02 )), from the above expression we get D (s) = s1/λ3 a1 sλ1 λ2 −1/λ3 − a2 + Fℓ∞6 (µ0 )  = s1/λ3 a1 (1 + αω (s; α)) − a2 + Fℓ∞6 (µ0 )  = s1/λ3 b1 ω (s; α) + b2 + Fℓ∞6 (µ0 ) ,  where b1 := αa1 and b2 := a1 − a2 . Here we also take Definition 2.1 into account. Since a1 is an analytic non-vanishing function at µ0 and d1 = λ3 α, we obtain the equality (b1 ) = (d1 ) between ideals over the local ring R{µ}µ0 . In order to show that (b1 , b2 ) = (d1 , d2 ) holds as On the cyclicity of Kolmogorov polycycles 11 well we note that, from (2.4), (2.5) and (2.6) again, b2 (µ) = a1 − a2 = ∆20 ∆10 λ2 − ∆30 − λ2 λ2 − λ1 λ2 −1/λ3 = ( L22 L21 L12 L11 − L32 L31 ) u =1 ! 1/λ λ −1/λ3 L22 L31 3 L122 = L32 L31 −1 L32 Lλ1 λ2 Lλ2 21 11 u =1 ! ! λ λ λ −1/λ3 L22 L311 2 L122 = L32 L31 α L31 −1 α + ( L31 − 1) L32 Lλ1 λ2 Lλ2 11 21 u =1 = κ 1 ( µ ) d2 ( µ ) + α ( µ )κ 2 ( µ ), L32 L31 −1 α L32 e d2 −1 where κ1 = 1/λ u =1 d2 and κ2 = 1/λ3 α u =1 are analytic functions at µ0 because so it L31 3 L31 is each Lij by [16, Lemma 2.3]. Thus, on account of d1 = λ3 α, we get that b2 = κ1 d2 + κ3 d1 . Hence (b1 , b2 ) = (d1 , d2 ) over the ring R{µ}µ0 since κ1 (µ0 ) > 0 and we have already shown that (b1 ) = (d1 ). So far we have proved the first assertion in (b). To show the second one, besides λ01 λ02 λ03 = 1, we assume λ01 > 1, λ02 > 1 and λ03 < 1. On account of this we can apply point (3) in Proposition 2.4 to conclude that D1 (s) = sλ1 ∆10 + ∆11 s + Fℓ∞7 (µ0 ) for any ℓ7 ∈ 1, min(λ01 , 2) ,    (2.8) D2 (s) = sλ2 ∆20 + ∆21 s + Fℓ∞8 (µ0 ) for any ℓ8 ∈ 1, min(λ02 , 2) ,    (2.9) and D3 (s) = s1/λ3 ∆30 + ∆31 s + Fℓ∞9 (µ0 ) for any ℓ9 ∈ 1, min(1/λ03 , 2) .    (2.10) Here the first order coefficients ∆10 , ∆20 and ∆30 are the ones already defined in (2.4), (2.5) and (2.6), respectively. With regard to the second order coefficients, only the ones of D1 and D3 are relevant for our purposes, which are given by λ1 ∆10 M̂1 (1/λ1 , 1) ∆30 M̂3 (λ3 , 1) ∆11 := − and ∆31 := − , (2.11) L11 (1) λ3 L31 (1) respectively. In each case, on account of (1.2), this follows easily from the formula ∆1 = λ∆0 S1 given in Proposition 2.4 and taking S1 in (2.3) particularised to σ1 (s) = (s, 1). From (2.8) and (2.9), by applying [15, Lemma A.2] we can assert that D2 ◦ D1 (s) = sλ1 λ2 ∆10 + ∆11 s + Fℓ∞7 2 ∆20 + ∆21 sλ1 ∆10 + ∆11 s + Fℓ∞7 + Fℓ∞10  λ   = sλ1 λ2 ∆10 λ2 −1 + λ2 ∆10 ∆11 s + Fℓ∞7 ∆20 + ∆21 sλ1 ∆10 + ∆11 s + Fℓ∞11 + Fℓ∞10 λ2    = sλ1 λ2 ∆10 λ2 −1 + λ2 ∆10 ∆11 s + Fℓ∞7 ∆20 + ∆10 ∆21 sλ1 + ∆11 ∆21 sλ1 +1 + Fℓ∞12 λ2   for any ℓ10 ∈ [λ01 , λ01 min(λ02, 2) in the first equality, any ℓ11 ∈ λ01 + 1, λ01 + min(λ01 , 2) in the    second one and any ℓ12 ∈ λ01 + 1, min(2λ01 , λ01 + 2, λ01 λ02 ) in the third one. Furthermore, in the second equality we use that, for any η = η (µ), (1 + as + Fℓ )η = (1 + as)η + Fℓ = 1 + aηs + F2−ε + Fℓ , (2.12) 12 D. Marín and J. Villadelprat for any ε > 0, which in turn follows noting that Fℓ η  (1 + as + Fℓ )η − (1 + as)η = (1 + as)η (1 + 1+ as ) − 1  = (1 + as)η (1 + Fℓ )η − 1 = (1 + as)η Fℓ = Fℓ . Consequently D2 ◦ D1 (s) = sλ1 λ2 ∆20 ∆10 λ2 −1 + λ2 ∆11 ∆20 ∆10 s + Fℓ∞7 ,  λ2  where we use again [15, Lemma A.2] taking λ01 > 1 and λ02 > 1 into account. Hence, from (2.10) and plug in s−α = 1 + αω (s; α) as before, we get D (s) = D2 ◦ D1 − D3 (s)    = s1/λ3 s−α ∆20 ∆10 λ2 −1 + λ2 ∆11 ∆20 ∆10 s + Fℓ∞7 − ∆30 − ∆31 s − Fℓ∞9 λ2   = s1/λ3 (1 + αω (s; α))∆20 ∆10 λ2 U (s) − ∆30 − ∆31 s − Fℓ∞9 , −1 where we define U (s) = 1 + λ2 ∆11 ∆10 s + Fℓ∞7 . The application of the formula given in (2.12) −1 with η = −1 shows that U (s)−1 = 1 − λ2 ∆11 ∆10 s + Fℓ7 . Thus one can easily verify that the above expression yields to D (s) = s1/λ3 U (s) b1 ω (s; α) + b2 + b3 s + Fℓ∞13  −1 with ℓ13 ∈ [1, min(λ01 , 1/λ03 , 2)) and b3 := λ2 ∆11 ∆30 ∆10 − ∆31 . Let us recall here that b1 = αa1 = α∆20 ∆10 and b2 = a1 − a2 = ∆20 ∆10 − ∆30 , where a1 and a2 are the analytic and strictly positive λ2 λ2 1 functions in (2.7). On account of the assumptions λ01 > 1 and λ03 < 1 we have λ1 (µ) ∈ / N and λ3 (µ) ∈/ N for µ ≈ µ0 , which imply respectively that ∆11 and ∆31 are analytic at µ0 by Proposition 2.4. Consequently b3 is an analytic function at µ0 . That being said we claim that the equality (b1 , b2 , b3 ) = (d1 , d2 , d3 ) between ideals over the local ring R{µ}µ0 is true. In order to prove this, for the sake of shortness in the next computation we follow the convention that κ stands for an analytic function at µ0 and κ̂ stands for an analytic strictly positive function at µ0 . Some easy computations following this convention yield −1 −1 b3 = ∆10 (λ2 ∆11 ∆30 − ∆31 ∆10 ) = ∆10 (λ2 ∆11 ∆30 − ∆31 ∆10 )     = −∆30 λ2 λ1 M̂1L(111/λ1 ,1) 1 M̂3 (λ3 ,1) 1  (1) − λ3 L31 (1) = − κ̂ λ1 λ2 λ3 M̂1 λ1 , 1 L31 (1) − M̂3 (λ3 , 1) L11 (1)   = −κ̂ (1 − d1 ) M̂1 λ11 , 1 L31 (1) − M̂3 (λ3 , 1) L11 (1)    = κ̂ M̂3 (λ3 , 1) L11 (1) − M̂1 λ11 , 1 L31 (1) + κd1 .  where in the third and fifth equalities we use (2.11) and d1 := 1 − λ1 λ2 λ3 , respectively. Hence b3 = κ̂d3 + κd1 since d3 := M̂3 (λ3 , 1) L11 (1) − M̂1 ( λ11 , 1) L31 (1). On account of (b1 ) = (d1 ) and (b1 , b2 ) = (d1 , d2 ), this shows the validity of the claim and completes the proof of the result. Remark 2.7. There are two important observations to be made about Theorem 2.6: ( a) The statement claims that the equalities (b1 ) = (d1 ), (b1 , b2 ) = (d1 , d2 ) and (b1 , b2 , b3 ) = (d1 , d2 , d3 ) between ideals over the local ring R{µ}µ⋆ are satisfied. As a matter of fact, in the proof we show a stronger property, namely that the following holds:      d1 κ1 0 0 b1  d2  =  ∗ κ2 0   b2  , d3 ∗ ∗ κ3 b3 On the cyclicity of Kolmogorov polycycles 13 where all the entries in the matrix are analytic functions on Λ and each κi is strictly positive. (b) From the dynamical point of view it is interesting to point out that the asymptotic expan- sion of the displacement map D (s; µ0 ) given in Theorem 2.6 provides a method to study the stability of the polycycle Γ. Indeed, taking also the previous observation into account, it shows that 1. if d1 (µ0 ) < 0 (respectively, > 0) then Γ is asymptotically stable (respectively, unstable), 2. if d1 (µ0 ) = 0 and d2 (µ0 ) < 0 (respectively, > 0) then Γ is asymptotically stable (respec- tively, unstable), and 3. if d1 (µ0 ) = d2 (µ0 ) = 0 and d3 (µ0 ) < 0 (respectively, > 0) then Γ is asymptotically stable (respectively, unstable). Of course this is relevant because we have an explicit expression of these functions by Theorem A. In this regard let us note that the first assertion is well known since d1 (µ0 ) < 0 is equivalent to require that λ01 λ02 λ03 > 1, while the second assertion was already proved by Gasull et al. in [8], see Theorem 1. On the contrary the third assertion constitutes a new result to the best of our knowledge. □ We give at this point the precise definition of independence of functions that we use in the present paper. Definition 2.8. Let us consider the functions gi : Λ −→ R for i = 1, 2, . . . , k. The real variety V ( g1 , g2 , . . . , gk ) is defined to be the set of µ ∈ Λ such that gi (µ) = 0 for i = 1, 2, . . . , k. We say that g1 , g2 , . . . , gk are independent at µ⋆ ∈ V ( g1 , g2 , . . . , gk ) if the following conditions are fulfilled: (1) Every neighbourhood of µ⋆ contains two points µ1 , µ2 ∈ V ( g1 , . . . , gk−1 ) verifying that gk (µ1 ) gk (µ2 ) < 0 (if k = 1 then we set V ( g1 , . . . , gk−1 ) = V (0) = Λ for this to hold). (2) The varieties V ( g1 , . . . , gi ), 2 ⩽ i ⩽ k − 1, are such that if µ0 ∈ V ( g1 , . . . , gi ) then every neighbourhood of µ0 contains two points µ1 , µ2 ∈ V ( g1 , . . . , gi−1 ) so that gi (µ1 ) gi (µ2 ) < 0. (3) If µ0 ∈ V ( g1 ) then every neighbourhood of µ0 contains two points µ1 , µ2 such that g1 (µ1 ) g1 (µ2 ) < 0. It is clear that if gi ∈ C 1 (Λ) for i = 1, 2, . . . , k and the gradients ∇ g1 (µ⋆ ), ∇ g2 (µ⋆ ) . . . , ∇ gk (µ⋆ ) are linearly independent vectors of R N +1 then there exists a neighbourhood U⋆ of µ⋆ such that the restrictions of g1 , g2 , . . . , gk to U⋆ are independent at µ⋆ . Lemma 2.9. Suppose that the equalities (c1 , . . . , ck ) = (d1 , . . . , dk ) between ideals over the local ring R{µ}µ⋆ hold for k = 1, 2, . . . , n, where µ⋆ ∈ V (c1 , . . . , cn ) = V (d1 , . . . , dn ). Then c1 , . . . , cn are independent at µ⋆ if, and only if, d1 , . . . , dn are independent at µ⋆ . Proof. Let us assume for instance that c1 , . . . , cn are independent at µ⋆ and prove that then d1 , . . . , dn are also independent. To this aim we note that the equalities (c1 , . . . , ck ) = (d1 , . . . , dk ) for k = 1, 2, . . . , n imply the existence of two triangular matrices A = ( aij ) and B = (bij ) with 14 D. Marín and J. Villadelprat coefficients in R{µ}µ⋆ such that ···     a11 0 0   d1 c1  ..   ∗ a22 ··· 0  ..   . =    .. .. .. ..  .   . . . .  dn cn ∗ ··· ∗ ann and ···     b11 0 0   c1 d1  ..   ∗ b22 ··· 0  ..  .  . =    .. .. .. ..  .   . . . .  cn dn ∗ ··· ∗ bnn Clearly R = (rij ) := BA is also a triangular matrix with coefficients in the local ring R{µ}µ⋆ and r11 0 · · · 0       c1  ∗ r22 · · · 0  c1  ..    .   .  =  .. .. .. ..   ..  .  . . . .  cn cn ∗ · · · ∗ rnn We claim that, since c1 , . . . , cn are independent at µ⋆ , then rkk (µ⋆ ) = 1 for all k = 1, 2, . . . , n. The fact that this is true for k = 1 follows easily by continuity. Let us prove by contradiction that this is also true for k ⩾ 2. So assume that rkk (µ⋆ ) ̸= 1 for some k ∈ {2, . . . , n}. Then the equality ck = rk1 c1 + . . . + rkk ck implies that ck = α1 c1 + . . . + αk−1 ck−1 where each αi := rki 1−rkk is an analytic function at µ⋆ . This clearly contradicts the assumption that c1 , . . . , cn are independent at µ⋆ (see Definition 2.8). Hence the claim is true and, consequently, det( R) = det( A)det( B) = 1 at µ = µ⋆ . This shows, in particular, that A is an invertible matrix in the local ring R{µ}µ⋆ and so there exists a neighbourhood U of µ⋆ such that akk (µ) ̸= 0 for all µ ∈ U and k = 1, 2, . . . , n. On account of this, the fact that d1 , . . . , dn are independent at µ⋆ follows easily noting that if we take any two points µ1 , µ2 ∈ U ∩ V (c1 , . . . , ci−1 ) = U ∩ V (d1 , . . . , di−1 ) verifying ci (µ1 )ci (µ2 ) < 0 then we have that di (µ1 )di (µ2 ) = aii (µ1 ) aii (µ2 )ci (µ1 )ci (µ2 ) < 0. This completes the proof of the result. Proof of Theorem A. Let us fix any µ0 ∈ Λ and set λ0i := λi (µ0 ) for i = 1, 2, 3. Recall that the limit cycles of Xµ near Γ are in one to one correspondence with the isolated positive zeros of D (s; µ) = D2 ◦ D1 − D3 (s; µ)  near s = 0. If d1 (µ0 ) = 1 − λ01 λ02 λ03 is not zero then by applying ( a) in Theorem 2.6 we have that, for any ℓ1 ∈ min(λ01 λ02 , 1/λ03 ), min(λ01 + λ01 λ02 , 1 + λ01 λ02 , 2λ01 λ02 , 1 + 1/λ03 , 2/λ03 ) , D (s; µ) = a1 (µ)sλ1 λ2 + a2 (µ)s1/λ3 + Fℓ∞1 (µ0 ), where a1 and a2 are analytic and strictly positive functions on Λ. Thus lim s−λ1 λ2 D (s; µ) = a1 (µ0 ) in case that λ01 λ02 < λ03 (s,µ)→(0,µ0 ) and lim s−1/λ3 D (s; µ) = a2 (µ0 ) in case that λ01 λ02 > λ03 . (s,µ)→(0,µ0 ) On the cyclicity of Kolmogorov polycycles 15 Since ai (µ0 ) ̸= 0 for i = 1, 2, this implies the existence of an open neighbourhood U of µ0 and  such that D (s; µ) ̸= 0 for all µ ∈ U and s ∈ (0, ε) when λ1 λ2 λ3 ̸= 1. Hence ε > 0 small enough 0 0 0 Cycl (Γ, Xµ0 ), Xµ = 0 and the assertion in ( a) is true. In order to prove (b) we note that if d1 (µ0 ) = 1 − λ01 λ02 λ03 is equal to zero then, by (b) in Theorem 2.6, we can assert that, for any ℓ2 ∈ (0, min(1, λ01 , λ01 λ02 )), s−1/λ3 D (s; µ) = b1 (µ) ω s; α(µ) + b2 (µ) + Fℓ∞2 (µ0 ),  (2.13) where α = 1/λ3 − λ1 λ2 and b1 and b2 are analytic functions at µ0 such that (b1 ) = (d1 ) and (b1 , b2 ) = (d1 , d2 ) over the ring R{µ}µ0 . The assumptions in this case imply that D (s; µ0 ) ̸≡ 0, b1 (µ0 ) = 0 and, thanks to Lemma 2.9, that b1 is independent at µ0 . Thus, given any ε > 0, there exists s1 ∈ (0, ε) such that D (s1 ; µ0 ) ̸= 0. Let us assume for instance that D (s1 ; µ0 ) > 0, the other case follows verbatim. Then, thanks to (1) in Definition 2.8, there exists µ1 ≈ µ0 with b1 (µ1 ) < 0 and, by continuity, such that D (s1 ; µ1 ) > 0. Moreover by applying Lemmas A.3 and A.4 in [14], s−1/λ3 D (s; µ) b2 (µ) Z1 (s; µ) :=  = b1 (µ) +  + Fℓ∞2 −δ (µ0 ) → κ (µ) as s → 0, (2.14) ω s; α(µ) ω s; α(µ) where κ (µ) := b1 (µ) − b2 (µ) min(α(µ), 0)). Here we use that 1/ω (s; α(µ)) ∈ F−∞δ (µ0 ) for any δ > 0 and that lims→0 1/ω (s; α) = max(−α, 0) by assertions ( a) and (b) in [14, Lemma A.4], respectively. Note on the other hand that, by (b) in Theorem 2.6, b1 = αa1 and b2 = a1 − a2 , where each ai is an analytic strictly positive function. Thus κ (µ) = α(µ) a2 (µ) if α(µ) < 0 and κ (µ) = α(µ) a1 (µ) if α(µ) ⩾ 0. Therefore, since b1 = αa1 , we can write κ = b1 η with η > 0. (2.15) Hence, on account of b1 (µ1 ) < 0, we can assert that κ (µ1 ) < 0, which in turn, from (2.14), guar- antees the existence of some s2 ∈ (0, s1 ) such that Z1 (s2 ; µ1 ) < 0. Thus D (s1 ; µ1 )D (s2 ; µ1 ) <  0 and, by continuity, D (ŝ; µ1 ) = 0 for some ŝ ∈ (s2 , s1 ) ⊂ (0, ε). This implies Cycl (Γ, Xµ0 ), Xµ ⩾ 1 as desired. Let us prove (c) next, i.e., that if d2 (µ0 ) ̸= 0 then Cycl (Γ, Xµ0 ), Xµ ⩽ 1. Note first that to  this end we can also assume that d1 (µ0 ) = 0, otherwise Cycl (Γ, Xµ0 ), Xµ = 0. Consequently the expression in (2.13) is valid. That being said, the proof will follow by applying the well- known derivation-division algorithm. In doing so, from (2.14) and the fact that ∂s ω (s; α) = − s − α −1 , b2 (µ) ∂s Z1 (s; µ) = α(µ)+1 2  + Fℓ∞2 −1−δ (µ0 ), s ω s; α(µ) where the flatness of the remainder follows from ( f ) in Lemma A.3 of [14]. Therefore sα(µ)+1 ω 2 s; α(µ) ∂s Z1 (s; µ) = b2 (µ) + Fℓ∞2 −4δ (µ0 ) → b2 (µ0 )  as (s, µ) → (0, µ0 ). Recall at this point that (b1 ) = (d1 ) and (b1 , b2 ) = (d1 , d2 ) over the local ring R{µ}µ0 by Theorem 2.6. Thus, the assumptions d1 (µ0 ) = 0 and d2 (µ0 ) ̸= 0 imply that b2 (µ0 ) ̸= 0. Ac- cordingly, on account of the above limit and by Bolzano’s Theorem, we obtain ε > 0 such that if ∥µ − µ0 ∥ < ε then Z1 ( · ; µ), and so D ( · ; µ), has at most one zero for s ∈ (0, ε), multiplicities taking into account. Hence Cycl (Γ, Xµ0 ), Xµ ⩽ 1 and (c) follows. 16 D. Marín and J. Villadelprat Let us turn next to the proof of (d), in which the assumptions are R ( · ; µ0 ) ̸≡ Id and that d1 and d2 vanish and are independent at µ0 . Then D ( · ; µ0 ) ̸≡ 0 and, due to (b1 ) = (d1 ) and (b1 , b2 ) = (d1 , d2 ) once again, b1 and b2 vanish and are independent at µ0 by Lemma 2.9. Thus, given any ε > 0, there exists s1 ∈ (0, ε) such that, for instance, D (s1 ; µ0 ) < 0. Then, by continuity and condition (1) in Definition 2.8, there exists µ1 ≈ µ0 such that b2 (µ1 ) > 0, b1 (µ1 ) = 0 and D (s1 ; µ1 ) < 0. Hence, from (2.13), s−1/λ3 D (s; µ1 ) = b2 (µ1 ) + Fℓ∞2 (µ0 ) → b2 (µ1 ) as s → 0, which shows the existence of s2 ∈ (0, s1 ) such that D (s2 ; µ1 ) > 0. For the same reasons we can choose µ2 ≈ µ1 satisfying D (s1 ; µ2 ) < 0 and D (s2 ; µ2 ) > 0 together with b1 (µ2 ) < 0. Then, from (2.14) and (2.15), lims→0 Z1 (s; µ2 ) = b1 (µ2 )η (µ2 ) < 0 and so there exists s3 ∈ (0, s2 ) verifying that D (s3 ; µ2 ) < 0. By continuity there exist ŝ1 , ŝ2 ∈ (0, ε) with D (ŝ1 ; µ2 ) = D (ŝ2 ; µ2 ) = 0. Accordingly Cycl (Γ, Xµ0 ), Xµ ⩾ 2.  From now on, in order to prove (e) and ( f ), we assume 0 0 0 0 0  λ1 < 1, λ2 > 1 and λ3 > 1. Then by applying Theorem 2.6, for any ℓ3 ∈ 1, min(2, λ1 , 1/λ3 ) , D (s; µ) = b1 (µ) ω s; α(µ) + b2 (µ) + b3 (µ)s + Fℓ∞3 (µ0 ) s1/λ3 U (s; µ),   (2.16) where b3 is an analytic function at µ0 verifying that (b1 , b2 , b3 ) = (d1 , d2 , d3 ) over the ring R{µ}µ0 and U is an analytic function such that U (0; µ0 ) = 1. Hence D (s; µ) Z2 (s; µ) := s1/λ3 ω (s; α(µ))U (s) (2.17) 1 s = b1 (µ) + b2 (µ) + b3 (µ) + Fℓ∞3 −δ (µ0 ), ω (s; α(µ)) ω (s; α(µ)) where we use once again that 1/ω (s; α(µ)) ∈ F−∞δ (µ0 ) for any δ > 0. Note furthermore that, for µ ≈ µ0 , the positive zeros of D ( · ; µ) and Z2 ( · ; µ) near s = 0 are in one to one correspondence because s1/λ3 ω 1(s;α(µ)) tends to +∞ as (s, µ) → (0, µ0 ). That being stablished we begin first with the proof of assertion (e) and to this aim, besides d3 (µ0 ) ̸= 0, we can also suppose d1 (µ0 ) = d2 (µ0 ) = 0, otherwise Cycl (Γ, Xµ0 ), Xµ ⩽ 1 by ( a) or (b), which  already have been proved. In this case, since (b1 ) = (d1 ), (b1 , b2 ) = (d1 , d2 ) and (b1 , b2 , b3 ) = (d1 , d2 , d3 ), it turns out that b3 (µ0 ) ̸= 0. As in the proof of (c), we will apply to steps of the derivation-division algorithm in (2.17). In doing so we obtain that Z3 (s; µ) := sα(µ)+1 ω 2 (s; α(µ))∂s Z2 (s; µ) = b2 (µ) + b3 (µ) s + sα(µ)+1 ω (s; α(µ)) + Fℓ∞3 −4δ (µ0 ),  where the flatness of the remainder follows by applying Lemmas A.3 and A.4 in [14] as before and we use that ∂s ω (s; α) = −s−α−1 . Note also that the positive zeros of Z3 ( · ; µ) and ∂s Z2 ( · ; µ) near s = 0 are in one to one correspondence for µ ≈ µ0 because ω (s; α(µ)) tends to +∞ as (s, µ) → (0, µ0 ). Finally ∂s Z3 (s; µ) = (α(µ) + 1)b3 (µ) + Fℓ∞3 −1−6δ (µ0 ) → b3 (µ0 ) ̸= 0 as (s, µ) → (0, µ0 ). s µ) ω (s; α(µ)) α ( By applying twice Bolzano’s Theorem, we can assert the existence of some ε > 0 such that if ∥µ − µ0 ∥ < ε then Z2 ( · ; µ), and so D ( · ; µ), has  at most two zeros for s ∈ (0, ε), multiplicities taking into account. Hence Cycl (Γ, Xµ0 ), Xµ ⩽ 2, which proves (e). Finally, in order to prove ( f ) we suppose that R ( · ; µ0 ) ̸≡ Id and that d1 , d2 and d3 vanish and are independent at µ0 . Consequently D ( · ; µ0 ) ̸≡ 0 and, due to the equality between the On the cyclicity of Kolmogorov polycycles 17 corresponding ideals over the local ring, b1 , b2 and b3 vanish and are independent at µ0 by Lemma 2.9. Thus, given any ε > 0, there exists s1 ∈ (0, ε) such that, for instance, D (s1 ; µ0 ) < 0. Then, by continuity and condition (1) in Definition 2.8, there exists µ1 ≈ µ0 such that b3 (µ1 ) > 0, b1 (µ1 ) = b2 (µ1 ) = 0 and D (s1 ; µ1 ) < 0. Hence, from (2.16), D (s; µ1 ) + = b3 (µ1 ) + Fℓ∞3 −1 (µ0 ) → b3 (µ1 ) > 0 as s → 0, s 1/λ3 U (s; µ) 1 which shows the existence of s2 ∈ (0, s1 ) such that D (s2 ; µ1 ) > 0. For the same reasons we can choose µ2 ≈ µ1 satisfying D (s1 ; µ2 ) < 0 and D (s2 ; µ2 ) > 0 together with b1 (µ2 ) = 0 and b2 (µ2 ) < 0. Accordingly, from (2.16) again, D (s; µ2 ) = b2 (µ2 ) + b3 (µ2 )s + Fℓ∞3 −1 (µ0 ) → b2 (µ2 ) < 0 as s → 0, s1/λ3 U (s; µ) which shows the existence of s3 ∈ (0, s2 ) such that D (s3 ; µ2 ) < 0. In the final step we take µ3 ≈ µ2 satisfying D (s1 ; µ3 ) < 0 and D (s2 ; µ3 ) > 0 and D (s3 ; µ3 ) < 0 together with b1 (µ3 ) > 0. Then, from (2.14) and (2.15), lims→0 s−1/λ3 Z1 (s; µ3 ) = b1 (µ3 )η (µ3 ) > 0 and so there exists s4 ∈ (0, s3 ) such that D (s3 ; µ3 ) > 0. Bycontinuity there exist ŝ1 , ŝ2 , ŝ3 ∈ (0, ε) with D (ŝi ; µ3 ) = 0 for i = 1, 2, 3. Hence Cycl (Γ, Xµ0 ), Xµ ⩾ 3 and this completes the proof of the result. □ 3 Applications We begin this section by revisiting in Theorem 3.1 a family of Kolmogorov differential systems that was first studied in [8], where the authors (following the notation in our statement) prove that if µ0 = ( a0 , p0 , q0 ) verifies p0 + q0 = 0 and a0 ̸= 0 then Cycl (Γ, Xµ0 ), Xµ ⩾ 1, cf.  assertion (b). Theorem 3.1. Consider the family of Kolmogorov differential systems ( ẋ = x (1 + x + x2 + axy + py2 ), Xµ ẏ = y(−1 − y + qx2 + axy − y2 ), where µ = ( a, p, q) ∈ R3 with p < −1 and q > 1 and let us fix any µ0 = ( a0 , p0 , q0 ). Then, compactifying Xµ to the Poincaré disc, the boundary of the first quadrant is a polycycle Γ such that: ( a) Cycl (Γ, Xµ0 ), Xµ = 0 if p0 + q0 ̸= 0.  (b) Cycl (Γ, Xµ0 ), Xµ = 1 if p0 + q0 = 0 and a0 ̸= 0.  (c) The return map of Xµ0 along Γ is the identity if, and only if, a0 = p0 + q0 = 0. In this case Γ √ 1+ −3−4p0 is the outer boundary of the period annulus of a center at ( x0 , y0 ) with x0 = y0 = − 2(1+ p ) 0 that foliates the first quadrant and, moreover, Cycl (Γ, Xµ0 ), Xµ = 1.  On the other hand the vector field Xµ has a unique singularity Qµ = (υ1 , υ2 ) in the first quadrant, which is either a focus or a center, and has trace equal to τ (µ) = υ1 + 2υ12 + 2aυ1 υ2 − υ2 − 2υ22 . Furthermore, the following holds:  (d) If τ (µ0 ) ̸= 0 then Cycl ( Qµ0 , Xµ0 ), Xµ = 0 and a sufficient condition for τ (µ0 ) ̸= 0 to hold is that p0 + q0 = 0 and a0 ̸= 0. 18 D. Marín and J. Villadelprat  (e) If τ (µ0 ) = 0 and p0 + q0 ̸= 0 then Qµ0 is a weak focus of order 1 and Cycl ( Qµ0 , Xµ0 ), Xµ = 1. ( f ) If τ (µ0 ) = 0 and p0 + q0 = 0 then a0 = 0. In addition Qµ0 is a center if, and only if, p0 + q0 = a0 = 0 and, in this case, Cycl ( Qµ0 , Xµ0 ), Xµ = 1. Finally it is not possible a simultaneous bifurcation of limit cycles from Γ and Qµ . Proof. The assertions in ( a) and (b) follow directly by applying Theorem A. Indeed, in this case, following the notation in (1.1), f ( x, y) = 1 + x + x2 + axy + py2 and g( x, y) = −1 − y + qx2 + axy − y2 , so that f 2 ( x, y) = x2 + axy + py2 and g2 ( x, y) = qx2 + axy − y2 . Taking this into account, together with p < −1 and q > 1, one can easily check that the assumptions H1 and H2 are verified. As a matter of fact the first assumption holds not only for z > 0 but for all z ∈ R, and this implies that the boundary of each quadrant is a monodromic polycycle for the compactified vector field. Hence, by the Poincaré–Bendixson theorem (see [21] for instance), there exists at least one singularity of Xµ inside each one of the four quadrants. Due to deg( f ) = deg( g) = 2, by Bézout’s theorem there exists exactly one in each quadrant. From now on we denote the singularity of Xµ in the first quadrant by Qµ . That being said, the hyperbolicity ratios of the saddles at Γ are λ1 = q−1 1 , λ2 = −( p + 1) and λ3 = 1. Consequently the first assertion follows from ( a) in Theorem A because p+q d1 ( µ ) = 1 − λ1 λ2 λ3 = . (3.1) q−1 The second assertion will follow by applying (b) and (c) in Theorem A. To show this we first recall that       L12 L22 L31 d2 (µ) = λ2 log (1) + log (1) + λ1 λ2 log (1) L21 L32 L11 and this leads us to the computation of the following improper integrals:   Z 1      L12 f2 g2 1 dz ∆1 (µ) := log (1) = (1, z) + λ1 − (z, 1) − L21 0 f 2 − g2 g2 − f 2 λ2 z Z 1 g− f        L22 f 1 dz ∆2 (µ) := log (1) = (0, 1/z) + λ2 − (0, z) − L32 0 g g λ3 z Z 1   f −g      L31 g 1 dz ∆3 (µ) := log (1) = (z, 0) + λ3 − (1/z, 0) − L11 0 f f λ1 z These expressions have to be computed assuming that p + q = 0, i.e., λ1 λ2 = 1. In doing so we obtain that Z 1 2a zdz aπ ∆1 ( a, p, − p) = = 1−q 0 z2 + 1 2( p + 1) and Z 1 zdz ( p + 1) π ∆2 ( a, p, − p) = −∆3 ( a, p, − p) = −( p + 1) =− √ . 0 z2 +z+1 3 3 Therefore d2 ( a, p, − p) = − aπ 2 is zero if, and only if, a = 0. Taking this  into account, the combination of (b) and (c) in Theorem A shows that Cycl (Γ, Xµ0 ), Xµ = 1 for any µ0 = On the cyclicity of Kolmogorov polycycles 19 ( a0 , p0 , − p0 ) with a0 ̸= 0, as desired. It is important to remark for the forthcoming analysis that by applying the Weierstrass Division Theorem (see for instance [9, 12]) we can assert that aπ d2 ( µ ) = − + ( p + q)h(µ) (3.2) 2 for some analytic function h. Next we proceed with the proof of (c).To this aim we fix any µ0 = ( a0 , p0 , q0 ) and apply Theorem 2.6, which gives the asymptotic expansion of D (s; µ) at s = 0 for µ ≈ µ0 . This result, taking (3.1) and (3.2) into account, shows that if D (s; µ0 ) ≡ 0 then a0 = p0 + q0 = 0. In order to prove the converse observe that if µ0 = (0, p0 , − p0 ) then the vector field Xµ0 writes as ( ẋ = x (1 + x + x2 + p0 y2 ), ẏ = −y(1 + y + p0 x2 + y2 ). One can easily check that Qµ0 , the only singularity of Xµ0 in the first quadrant, is a weak focus √ 1+ −3−4p0 at the point ( x0 , y0 ) with x0 = y0 = − 2(1+ p ) . Furthermore, setting σ( x, y) = (y, x ), it 0 turns out that σ⋆ Xµ0 = − Xµ0 and so the vector field is reversible with respect to the straight line y = x. Hence Qµ0 is a center and a straightforward application of the Poincaré–Bendixson theorem shows that its period annulus fills the first quadrant, which in particular implies that D (s; µ0 ) ≡ 0. So far we have proved that the return map of Xµ0 along Γ is the identity if, and only if, µ0 = (0, p0 , − p0 ). Our next task is to show that, in this case, Cycl (Γ, Xµ0 ), Xµ = 1. With this aim in view we apply (b) in Theorem 2.6, which shows that if µ ≈ µ0 then D (s; µ) = b1 (µ) ω s; α(µ) + b2 (µ) + r (s; µ) s1/λ3 ,   (3.3) where α = 1/λ3 − λ1 λ2 , r ∈ Fℓ∞ (µ0 ) with ℓ ∈ 0, min(1, p−+11 ) and, in addition,  (b1 ) = (d1 ) and (b1 , b2 ) = (d1 , d2 ) over the local ring R{µ}µ0 . Consequently if µ = ( a, p, q) satisfies a = p + q = 0 then b1 (µ) = b2 (µ) = 0 and r (s; µ) ≡ 0. Furthermore, since the vectors ∇d1 (µ0 ) and ∇d2 (µ0 ) are linearly independent, see (3.1) and (3.2), the above equalities between ideals show that this is also the case of ∇b1 (µ0 ) and ∇b2 (µ0 ). We can thus take (η1 , η2 , η3 ) ∈ / ⟨∇b1 (µ0 ), ∇b2 (µ0 )⟩ and define b3 (µ) := η1 a + η2 ( p − p0 ) + η3 (q + p0 ) so that ν = Ψ(µ) := b1 (µ), b2 (µ), b3 (µ) is a local analytical change of coordinates in a neighbourhood of µ = µ0 . Note that Ψ maps µ0 to 03 := (0, 0, 0) and { a = p + q = 0} to {ν1 = ν2 = 0} and in addition R1 (s; ν) := s−1/λ3 D (s; µ) = ν1 ω (s; α̂) + ν2 + r̂ (s; ν), µ = Ψ −1 ( ν ) where α̂ = α̂(ν) := α Ψ−1 (ν) and r̂ (s; ν) := r s; Ψ−1 (ν) ∈ Fℓ∞ (03 ). The key point is that   r̂ (s; 0, 0, ν3 ) ≡ 0 implies, thanks to [17, Lemma 4.1], that r̂ (s; ν) = ν1 h1 (s; ν) + ν2 h2 (s; ν) with hi ∈ Fℓ∞ (03 ). Accordingly R1 (s; ν) = ν1 ω (s; α̂) + h1 (s; ν) + ν2 1 + h2 (s; ν) .   (3.4) Observe that if s → 0+ and ν → (0, 0, 0) then the factor multiplying ν1 tends to +∞, whereas the factor multiplying ν2 tends to 1. Here we use Definition 2.3 and that lim(s,α)→(0,0) ω (s; α) = +∞. We claim that there exists s0 > 0 and an open neighbourhood U of ν = (0, 0, 0) such that R1 (s; ν) 1 + h2 (s; ν) R2 (s; ν) := = ν1 + ν2 ω (s; α̂) + h1 (s; ν) ω (s; α̂) + h1 (s; ν) 20 D. Marín and J. Villadelprat has at most one zero on (0, s0 ), counted with multiplicities, for all ν = (ν1 , ν2 , ν3 ) ∈ U with ν1 + ν2 ̸= 0. This will imply that Cycl (Γ, Xµ0 ), Xµ ⩽ 1 because R2 (s; 0, 0, ν3 ) ≡ 0, so that it 2 2  has not any isolated zero. The claim is clear in case that ν2 = 0. To tackle the case ν2 ̸= 0 we compute the derivative with respect to s to obtain that 1 + Fℓ∞ 1 + Fℓ∞   ! ′ R2 (s; ν) = ν2 ∂s = ν2 ∂s ω (s; α̂) + Fℓ∞ ∞ ) ω (s; α̂)(1 + Fℓ− ε  1 + Fℓ− ∞  ν2 ∞ ν2 = ν2 ∂s ε = (1 + Fℓ− ε) + F∞ ω (s; α̂) sα̂+1 ω 2 (s; α̂) ω (s; α̂) ℓ−ε−1 ν2 ∞ α̂+1 ∞  ν2 ∞  = + 1 + Fℓ− ε+s ω (s; α̂)Fℓ− ε −1 = + 1 + Fℓ− 3ε . s ω 2 (s; α̂) α̂ 1 s ω 2 (s; α̂) α̂ 1 Here, in the second equality we apply first assertion (c) of Lemma A.4 in [14] to get that 1/ω (s; α̂) ∈ F−∞ε (03 ) for all ε > 0 small enough, due to α̂(03 ) = 0, and use next that F−∞ε Fℓ∞ ⊂ ∞ from ( g ) of Lemma A.3 in [14] . In the third equality, on account of 1 − 1 ∈ F ∞ and by Fℓ− ε 1+ s 1 (h) of Lemma A.3 in [14] , we use first the inclusion 1+F1 ∞ ⊂ 1 + Fℓ− ∞ . Then, by using ( d ) and ε ℓ−ε ( g) of Lemma A.3 in [14] , we expand the numerator to get that (1 + Fℓ∞ )(1 + Fℓ− ∞ ) ⊂ 1+ F∞ . ε ℓ−ε Next, in the fourth equality we use that ∂s ω (s; α) = s − α − 1 and assertion ( f ) of Lemma A.3 in ∞ ⊂ F∞ [14] to deduce that ∂s Fℓ− ε ℓ−ε−1 . Finally in the last equality we apply ( c ) of Lemma A.4 in [14] to get that s ω (s; α̂) ∈ F1∞−2ε and we use again that F1∞−2ε Fℓ− α̂ + 1 ∞ ∞ ε−1 ⊂ Fℓ−3ε . On account of Definition 2.3 we can assert the existence of some s0 ∈ (0, 1) and a neighbourhood U of ν = (0, 0, 0) such that R2′ (s; ν) ̸= 0 for all s ∈ (0, s0 ) and ν ∈ U with ν2 ̸= 0. Hence the application of Rolle’s theorem shows that the claim is true for ν2 ̸= 0 as well. So far we have proved that Cycl (Γ, Xµ0 ), Xµ ⩽ 1. The fact that this upper bound is attained follows by applying the assertion in (b) taking µ0 = ( a0 , p0 , − p0 ) with a0 ≈ 0 but different from zero. This completes the proof of (c). Let us turn now to the proof of the assertions regarding the singularity of Xµ at Qµ . The approach here is rather standard and the technical difficulty is that we do not dispose of a feasible expression of the coordinates of Qµ . To overcome this problem we shall parametrise the family of vector fields more conveniently. For reader’s convenience we summarise the chain of reparametrisations that we shall perform: µ = ( a, p, q) → ( a, p, ε) → (υ1 , υ2 , ε) → (υ1 , υ2 , τ ) → µ̂ = (ρ, σ, τ ). For the first one we simply introduce ε = p + q. In the second one we take the coordinates of the singular point Qµ = (υ1 , υ2 ) as new parameters, i.e., we isolate a and p from ( 1 + υ1 + υ12 + aυ1 υ2 + pυ22 = 0, (3.5) 1 + υ2 + ( p − ε)υ12 − aυ1 υ2 + υ22 = 0, to obtain υ12 υ22 ε + υ14 − υ24 + υ13 − υ23 + υ12 − υ22 υ12 ε − υ12 − υ22 − υ1 − υ2 − 2 a=− and p= . υ1 υ2 υ12 + υ22 υ12 + υ22  In this respect we point out that υ1 and υ2 are strictly positive because Qµ is inside the first quadrant for all admissible µ. More important, the map φ : ( a, p, ε) 7→ (υ1 , υ2 , ε) is smooth and, taking (3.5) into account, injective. The smoothness follows by the Inverse Function On the cyclicity of Kolmogorov polycycles 21 Theorem since one can check that the determinant of the Jacobian of (υ1 , υ2 , ε) 7→ ( a, p, ε) is non-zero at the image by φ of any admissible parameter. Then one can check that the trace of the Jacobian of the vector field at ( x, y) = (υ1 , υ2 ) is x f x ( x, y) + ygy ( x, y) ( x,y)=(υ1 ,υ2 ) = υ1 + 2υ12 + 2aυ1 υ2 − υ2 − 2υ22 a= a(υ1 ,υ2 ,ε) 2ευ12 υ22 + (υ1 − υ2 )(υ1 + 2 + υ2 )(υ1 + υ2 ) =− , υ22 + υ12 and we introduce τ = τ (υ1 , υ2 , ε) isolating ε from 2ευ12 υ22 + (υ1 − υ2 )(υ1 + 2 + υ2 )(υ1 + υ2 ) = τ. (3.6) In other words, τ is (up to a non-vanishing factor) the trace of the vector field. Finally, for convenience, we define ρ = υ1 −2 υ2 and σ = υ1 +2 υ2 . Observe then that { p + q = a = 0} becomes {ρ = τ = 0}. In what follows, setting µ̂ = (ρ, σ, τ ) for shortness, we denote the vector field by Xµ̂ . Let us also remark that the map µ 7→ µ̂ is smooth and injective as a consequence of the previous discussion. At this point we claim that Qµ is either a focus or a center. To show this we will check that the discriminant Dµ of the characteristic polynomial of the Jacobian matrix of Xµ at Qµ is strictly negative for all admissible parameter. Indeed, one can verify that Dµ expressed in terms of (υ1 , υ2 , ε) can be written as 4 ( υ1 υ2 ) 4 ε 2 + A 1 ( υ1 , υ2 ) ε + A 0 ( υ1 , υ2 ) Dµ = , (υ12 + υ22 )2 where Ai are polynomials of degree 7. Thus Dµ = 0 gives two roots ε = ε̂ i (υ1 , υ2 ), for i = 1, 2, that one can check to be well-defined continuous functions on V := {(υ1 , υ2 ) ∈ R2 : υ1 > 0, υ2 > 0}. To see the claim we first prove that, for i = 1, 2, ( p + 1)(q − 1) ε=ε̂ i > 0 for all (υ1 , υ2 ) ∈ V. (3.7) This implies that Dµ can not vanish at an admissible parameter due to the assumptions p < −1 and q > 1. In this regard we note that the product ( p + 1)(q − 1) expressed in terms of (υ1 , υ2 , ε) is given by (υ1 υ2 )2 ε2 + B1 (υ1 , υ2 )ε + B0 (υ1 , υ2 ) ( p + 1)(q − 1) = , (υ12 + υ22 )2 where deg( B1 ) = 3 and deg( B0 ) = 2. A computation shows that the resultant (see [4, Chapter 3] for instance) between the numerators of Dµ and ( p + 1)(q − 1) with respect to ε is a polyno- mial in υ1 and υ2 with all the coefficients being natural numbers. Consequently the resultant does not vanish on V and, accordingly, ( p + 1)(q − 1)|ε=ε̂ i ̸= 0 for all (υ1 , υ2 ) ∈ V. Thus, since V is arc-connected and the function (υ1 , υ2 ) 7→ ( p + 1)(q − 1)|ε=ε̂ i is continuous on V, it suf- fices to verify (3.7) at some particular choice of parameter. For instance, taking υ1 = υ2 = 1 we obtain that ( p + 1)(q − 1)|ε=ε̂ i = 92 for i = 1, 2. Therefore Dµ ̸= 0 at any admissible parameter. Thus, exactly as before, since µ 7→ Dµ is continuous and the set of admissible parameters is arc-connected, the claim will follow once we verify its validity√ at some particular parameter. 67+25 5 For instance the choice µ = (0, −2, 2) yields Dµ = − 2 < 0. 22 D. Marín and J. Villadelprat  We proceed now with the study of the cyclicity of Qµ . The fact that Cycl ( Qµ0 , Xµ0 ), Xµ = 0 when τ (µ0 ) ̸= 0 is well-known. On the other hand, if p + q = 0 then ε = 0 and τ (υ1 , υ2 , 0) = (υ1 − υ2 )(υ1 + 2 + υ2 )(υ1 + υ2 ), which vanishes at an admissible parameter if, and only if, υ1 = υ2 . For this to happen, see (3.5) with ε = 0, it is necessary that a = 0. This proves the validity of (d). We shall next solve the center-focus problem in the family. With this aim in view, taking a local transversal section at Qµ we consider the displacement map D (s; µ̂), which extends analytically to s = 0, so that we can compute its Taylor’s expansion D (s; µ̂) = η1 (µ̂)s + η2 (µ̂)s2 + η3 (µ̂)s3 + sR(s; µ̂), where the remainder R is o(s2 ). Recall that the trace of Xµ̂ at Qµ̂ is equal to τu1 (µ̂), where u1 is a unity. The coefficients ηi are called the Lyapunov quantities of the focus. We have in particular (see for instance [19, p. 94]) that η1 (µ̂) = eτu1 (µ̂) − 1 = τu2 (µ̂), where u2 is again a unity. Since the first nonzero coefficient of the expansion is the coefficient of an odd power of s, see [19, p. 94] again, we get that η2 (µ̂) = τ ℓ1 (µ̂) for some analytic function ℓ1 . In order to obtain η3 we shall appeal to the well-known relation between the Lyapunov and focus quantities which, following the notation in [19, Theorem 6.2.3], we denote by gii . The first ones are the coefficients in the Taylor’s expansion of the displacement map that we already introduced, while the second ones are the obstructions for the existence of a first integral. It occurs that η2i+1 − πgii ∈ ( g11 , . . . , gi−1,i−1 ) and, more important for our purposes, that η3 = πg11 . On account of this we can compute g11 instead of η3 , which is easier to obtain, and in doing so (see [3, p. 29]) we get that 2ρ (ρ − σ) (σ + 1) 4 + 34σ + 29σ2 + 8σ3 − 3ρ2  η3 (µ̂) τ =0 = π 2 . 3 (ρ + σ)3 (ρ + 2σ + 2σ2 + 2ρ2 + 2) In this respect we claim that η3 (µ̂) τ =0 = ρh(ρ, σ) with h(ρ, σ) ̸= 0 in case that |ρ| < σ, which corresponds to the admissible values υ1 , υ2 > 0 due to ρ = υ1 −2 υ2 and σ = υ1 +2 υ2 . Indeed, it is clear that the factor (ρ − σ)(σ + 1) does not vanish inside the admissible set, while the other one does not vanish neither because 4 + 34σ + 29σ2 + 8σ3 − 3ρ2 > 4 + 34σ + 26σ2 + 8σ3 > 0, where the first inequality follows using that |ρ| < σ and the second one the fact that σ > 0. Hence the claim is true. Therefore, if Qµ is a center then τ = ρ = 0, and the assertion in (c) shows that these two conditions are also sufficient because { p + q = a = 0} = {τ = ρ = 0}. Observe moreover that we can write η3 (µ̂) = τ ℓ2 (µ̂) + ρh(ρ, σ) for some analytic function ℓ2 . On the other hand, due to R(s; µ̂)|ρ=τ =0 ≡ 0, we can also write R = τR1 + ρR2 with Ri ∈ o(s2 ) and, accordingly, D (s; µ) = τs u2 + ℓ1 s + ℓ2 s2 + R1 ) + ρs hs2 + R2 .  (3.8) Note that if τ (µ) = 0 and ε = p + q ̸= 0 then ρ = υ1 −2 υ2 must be different from zero because otherwise, from (3.6), we would get that ε = 0. Consequently, due  to h(ρ, σ) ̸= 0 for all ad- missible ρ and σ, the equality in (3.8) implies Cycl ( Qµ0 , Xµ0 ), Xµ ⩽ 1 in case that τ (µ0 ) = 0 and p0 + q0 ̸= 0. The fact that this upper bound is attained follows by means of an easy per- turbative argument using that ∂ε τ (µ) = −2υ12 υ22 ̸= 0. This proves the validity of the assertion in (e). On the cyclicity of Kolmogorov polycycles 23 In order to prove ( f ) note that if τ (µ) = 0 and ε = p + q = 0 then, from (3.6), ρ = υ1 −2 υ2 = 0. Hence, from (3.5), 2aυ1 υ2 = 0, which implies a = 0 and shows the first assertion. That being stablished, we have already proved that Qµ0 is a center if, and only if p0 + q0 = a0 = 0. We show next that, in this case, Cycl ( Qµ0 , Xµ0 ), Xµ ⩽ 1. Indeed, since u2 is a unity we can consider D (s; µ̂) hs2 + R2 D1 (s; µ̂) := = τ + ρ . s ( u2 + ℓ1 s + ℓ2 s2 + R1 ) u2 + ℓ1 s + ℓ2 s2 + R1 The upper bound for the cyclicity of Qµ0 in the center case will follow once we prove that there exist s0 > 0 and an open neighbourhood U of (ρ, σ, τ ) = (0, σ̂, 0) such that D1 (s; µ̂) has at most one zero on (0, s0 ), counted with multiplicities, for all µ̂ ∈ U with ρ2 + 2 √τ ̸= 0. Recall 1+ −3−4p0 in this regard that D1 (s; µ̂)|ρ=τ =0 ≡ 0 and that, on account of (c), σ̂ = − 2(1+ p ) > 0 is 0 the first component of Qµ0 . The idea to show this is exactly the same as in the proof of (c) but with less technicalities because the involved functions are analytic at s = 0. The desired property is evident when ρ = 0. In case that ρ ̸= 0 we compute the derivative of D1 with respect to s to obtain ∂s D1 (s; µ̂) = ρs 2h/u2 + o(1) .  Since h(ρ, σ) ̸= 0 in case that |ρ| < σ and u2 = u2 (µ̂) is a unity, the existence of the desired s0 > 0 and the open neighbourhood  U follows by Rolle’s theorem. So far we have proved that Cycl ( Qµ0 , Xµ0 ), Xµ ⩽ 1 if p0 + q0 = a0 = 0. The fact that this upper bound is attained follows noting that we can take µ = ( a, p, q) with τ (µ) = 0 and p + q ̸= 0 arbitrarily close to µ0 = (0, p0 , − p0 ) and apply then the assertion in (e). This proves ( f ). Let us turn now to the proof of the last assertion in the statement. Observe in this respect that the combination of ( a) and (b) together with (d) and (e) shows that a simultaneous bifurcation of limit cycles from Γ and Qµ can only occur if we perturb some µ⋆ = ( a⋆ , p⋆ , q⋆ ) with a⋆ = p⋆ + q⋆ = 0. We shall prove by contradiction that this is neither possible. So assume that for each n ∈ N there exist µn = ( an , pn , qn ) and two limit cycles γn and γn′ of the vector field Xµn in the first quadrant such that the Hausdorff distances d H (γn , Γ) and d H (γn′ , Qµn ) tend to zero and µn tends to µ⋆ as n → +∞. Let us consider the asymptotic expansion of the displacement map of Xµ at the polycycle Γ that we compute in (3.3) and denote it by D p (s; µ). We also consider its Taylor’s expansion near the focus Qµ given in (3.8) and denote it by Dc (s′ ; µ). Then the assumption implies the existence of two sequences sn → 0+ and s′n → 0+ such that D p (sn ; µn ) = 0 and Dc (s′n ; µn ) = 0 for all n ∈ N. We claim that the first equality implies that pn + qn lim = 0. (3.9) n→+∞ an Indeed, from (3.4) we have that R1 (sn ; ν) ν=Ψ(µ ) = b1 (µn ) ω (sn ; α(µn )) + h1 (sn ; ν) + b2 (µn ) 1 + h2 (s; ν) ν=Ψ(µ ) = 0   n n for all n ∈ N. Thus, due to limn→+∞ ω (sn ; α(µn )) = +∞ and hi ∈ Fℓ∞ (03 ), we obtain that b (µ ) = −∞. Moreover, since (b1 ) = (d1 ) and (b1 , b2 ) = (d1 , d2 ) with ∇d1 and ∇d2 limn→+∞ b2 (µnn ) 1 independent at µ = µ⋆ , we can write b2 κ d + κ̂2 d2 = 1 1 b1 κ̂3 d1 d (µ ) with κ̂i (µ⋆ ) ̸= 0 and, consequently, limn→+∞ d2 (µnn ) = ∞. This, on account of (3.1) and (3.2), 1 gives the limit in (3.9) and so the claim is true. Recall on the other hand that in order to study 24 D. Marín and J. Villadelprat the displacement map near the focus Qµ we use a more convenient parametrisation given by µ̂ := (ρ, σ, τ ) = ϕ(µ). That being said, setting (ρn , σn , τn ) := ϕ( an , pn , qn ), similarly as we argue before, the fact that Dc (s′n ; µn ) = 0 for all n ∈ N implies from (3.8) that τn lim = 0. (3.10) n→+∞ ρn Let us remark that here we also take into account that u2 is a unity. We next arrive to contra- √ (3.9) and (3.10) cannot hold simultaneously. Indeed, one can verify that, diction showing that 1+ −3−4p⋆ setting σ⋆ = − 2(1+ p ⋆ ) , pn + qn ρ2n + σn2 τn + ρn σn (σn + 2) =4 an µn =ϕ−1 (µ̂n ) ρn − σn 2τn − ρn (2σn + 1)(ρ2n + σn2 ) 2 2 ρ2n + σn2 τn /ρn + σn (σn + 2) 4(σ⋆ + 2) =4 2 2 2 2 → ̸= 0 as n → +∞. ρn − σn 2τn /ρn − (2σn + 1)(ρn + σn ) σ⋆ (2σ⋆ + 1) Here, in addition to (3.10), we use that if p + q and a tend to zero then ρ → 0 and σ → σ⋆ , where σ⋆ is precisely the first component of the center at Qµ⋆ (which is in the diagonal of the first quadrant). This shows that (3.9) and (3.10) cannot occur simultaneously, which yields to the desired contradiction and finishes the proof of the result. The following is our second example of application of Theorem A. In this case the family of Kolmogorov’s systems is five-parametric and to the best of our knowledge it has not been studied previously. Theorem 3.2. Consider the family of Kolmogorov differential systems ( ẋ = x (c + x2 + axy − ( p + 1)y2 ), Xµ ẏ = y(−1 + (q + 1) x2 + ( a − b) xy − y2 ), √ where µ = ( a, b, c, p, q) ∈ R5 with c > 0, p > 0, q > 0 and b < 2 pq and let us fix any µ0 = ( a0 , b0 , c0 , p0 , q0 ). Then there exists a unique singular point Qµ in the first quadrant, which is either a center, a focus or a node. Moreover, compactifying Xµ to the Poincaré disc, the boundary of the first quadrant is a polycycle Γ such that: ( a) Cycl (Γ, Xµ0 ), Xµ = 0 if p0 − c0 q0 ̸= 0.  (b) Cycl (Γ, Xµ0 ), Xµ = 1 if p0 − c0 q0 = 0 and 2c0 q0 a0 − (c0 q0 − c0 + 1)b0 ̸= 0.  (c) The return map of Xµ0 along Γ is the identity if, and only if, p0 − c0 q0 = 2c0 q0 a0 − (c0 q0 − c0 + 1)b0 = 0. In this case Qµ0 is a center with first integral q0 ( x2 + c0 (y2 + 1)) − b0 xy H ( x, y) = 2 , ( xyc0 ) c0 q0 +c0 +1 which foliates the first quadrant. Moreover Γ is the outer boundary of its period annulus and, in addition, Cycl (Γ, Xµ0 ), Xµ = 1.  On the cyclicity of Kolmogorov polycycles 25 Remark 3.3. In contrast to the family of Kolmogorov’s cubic systems studied in Theorem 3.1, for the family in Theorem 3.2 there exist parameters µ0 with d1 (µ0 ) = 0 and d2 (µ0 ) ̸= 0, so that Cycl (Γ, Xµ0 ), Xµ = 1, and satisfying additionally that the unique singular point Qµ0 in  the first quadrant is a non-degenerate node. Hence, for appropriate µ ≈ µ0 we will have a limit cycle γµ with a non-monodromic singular point Qµ as unique singularity in its interior. For instance, the choice µ0 = (−800.01, −900.99999, 1000, 1, 0.001) leads to this phenomenon with Qµ0 = (0.1, 10). A similar occurrence is observed in [2, p. 203] to take place in the family of cubic Liénard systems studied in [7]. □ Proof of Theorem 3.2. In this case, following the notation in (1.1), we have that f ( x, y; µ) := c + x2 + axy − ( p + 1)y2 and g( x, y; µ) := −1 + (q + 1) x2 + ( a − b) xy − y2 . Since f (z, 0; µ) = c + z2 , g(0, z; µ) = −1 − z2 and ( f 2 − g2 )(1, z; µ) = −q + bz − pz2 , one can check that the hypothesis H1 and H2 are satisfied for the admissible parameters, i.e., c > 0, √ p > 0, q > 0 and b < 2 pq. Moreover the hyperbolicity ratios are λ1 = 1/q, λ2 = p and λ3 = 1/c. (3.11) Then Γ is a polycycle and by applying the Poincaré–Bendixson theorem we deduce the ex- istence of at least one singular point of Xµ in the first quadrant. We claim that there exists exactly one. In order to show this we suppose that (υ1 , υ2 ) is a singular point of Xµ in the first quadrant and solve f (υ1 , υ2 ; µ) = 0 and g(υ1 , υ2 ; µ) = 0 for a and b as a function of c, p, q, υ1 and υ2 . In doing so we obtain that pυ22 − υ12 + υ22 − c pυ22 + qυ12 − c − 1 a= and b= . υ1 υ2 υ1 υ2 The substitution of these values in f + cg, which is homogeneous of degree 2 in x and y, yields x2 (υ2 − rυ1 ) rυ2 (c + p + 1) + υ1 (cq + c + 1)   f + cg ( x, y; µ) y=rx = . υ1 υ2 It is clear then that the vanishing of the above numerator provides the possible values of r such that Xµ has a singular point at the straight-line y = rx, namely, υ2 υ1 (cq + c + 1) r1 = >0 and r2 = − < 0. υ1 υ2 ( c + p + 1 ) Since x 7→ f ( x, r1 x ) = c + x2 (1 + ar1 − ( p + 1)r12 ) vanishes at x = υ1 > 0, it must have another real zero, which has to be negative. Therefore Xµ has exactly one singular point in the first quadrant and exactly one singular point in the third quadrant, showing in particular the validity of the claim. An easy computation shows that the determinant of the Jacobian of Xµ at Qµ = (υ1 , υ2 ) is equal to 2υ12 (cq + c + 1) + 2υ22 ( p + c + 1) > 0, so that it can be a center, a focus or a node. So far we have proved that the first assertion in the statement is true. Let us turn to the proof of the assertions in ( a), (b) and (c). The first one follows from ( a) in Theorem A because cq − p d1 ( µ ) = 1 − λ1 λ2 λ3 = . (3.12) cq 26 D. Marín and J. Villadelprat The second assertion will follow by applying (b) and (c) in Theorem A. In this regard let us recall that       L12 L22 L31 d2 (µ) = λ2 log (1) + log (1) + λ1 λ2 log (1). (3.13) L21 L32 L11 On account of the definition of each Lij , see (1.2), we easily obtain that 1 + c ( q + 1) log L11 (1) = log(c + 1), 2c 1 + c ( q + 1) log L31 (1) = log (1/c + 1) , (3.14) 2c log 2 log L22 (1) = log L32 (1) = ( p + c + 1). 2 Moreover mz + n′ Z 1 Z 1 1 mz + n 1 log L12 (1) = dz and log L21 (1) = dz, q 0 − pz2 + bz − q p 0 −qz2 + bz − p where m := −( pq + p + q), n := qa + b and n′ := p(b − a) + b. (3.15) The explicit integration of these functions leads to several cases depending on the parameters. To avoid this we note that mz + n m −2pz + b 1 mb + 2np 2 =− + , − pz + bz − q 2p − pz + bz − q 2p − pz2 + bz − q 2 so that Z 1 p+q−b   m mb + 2np dz log L12 (1) = − log + . 2pq q 2pq 0 − pz2 + bz − q It is clear that the same formula holds for log L21 (1) replacing p, q and n by q, p and n′ , respectively. On account of this and the fact that, from (3.15), mb + 2n′ q = −mb − 2np, we get     L12 m p log (1) = log L12 (1) − log L21 (1) = − log + (mb + 2np)Φ(µ), (3.16) L21 2pq q where Z 1  1 1 1 Φ(µ) := + dz. 2pq 0 − pz2 + bz − q −qz2 + bz − p Notice, and this is the key point in the forthcoming arguments, that Φ is a non-vanishing function because, thanks to property H1, Φ(µ) < 0 at any admissible parameter µ. On the other hand, from (3.14),     L31 (1 + c + cq) L22 log (1) = − log c and log (1) = 0. (3.17) L11 2c L32 Accordingly the substitution of (3.16) and (3.17) in (3.13) yields   m p p(1 + c + cq) d2 (µ) = − log + p(mb + 2np)Φ(µ) − log c 2q q 2qc   pq + p + q p p(1 + c + cq) + p 2pqa − ( pq − p + q)b Φ(µ) −  = log log c, 2q q 2qc On the cyclicity of Kolmogorov polycycles 27 where in the first equality we set the values of the hyperbolicity ratios given in (3.11) and in the second one the expressions of m and n defined in (3.15). Observe at this point, see (3.12), that d1 (µ) = 0 if, and only if, p = cq. Moreover the two logarithmic summands in the above expression of d2 (µ) cancel each other after the substitution p = cq, so that d2 (µ) p=cq = cq2 2cqa − (cq − c + 1)b Φ( a, b, c, cq, q).  Thus, by the Weierstrass Division Theorem (see [9, 12]), there exists an analytic function κ1 such that  d2 (µ) = d1 (µ)κ1 (µ) + 2cqa − (cq − c + 1)b κ2 (µ), (3.18) where κ2 (µ) := cq2 Φ(µ) is a unity in the admissible set. This expression shows that if we take an admissible parameter µ0 = ( a0 , b0 , c0 , p0 , q0 ) such that p0 − c0 q0 = 0 and 2c0 q0 a0 − (c0 q0 − c0 + 1)b0 ̸= 0 then d2 (µ0 ) ̸= 0, which by (c) in Theorem A implies that Cycl (Γ, Xµ0 ), Xµ ⩽ 1. The fact that Cycl (Γ, Xµ0 ), Xµ = 1 follows by applying (b) in Theorem A because ∇d1 (µ0 )  is not the zero vector, see (3.12). This proves the validity of the assertion in (b). To show (c) we take any µ0 = ( a0 , b0 , c0 , p0 , q0 ) satisfying p0 − c0 q0 = 2c0 q0 a0 − (c0 q0 − c0 + 1)b0 = 0. Then one can verify that the function q0 ( x2 + c0 (y2 + 1)) − b0 xy H ( x, y) = 2 ( xyc0 ) c0 q0 +c0 +1 is a first integral of Xµ0 , which is clearly analytic on the whole first quadrant. (For reader’s convenience let us mention that we found this first integral looking for an integrating factor of the form xr ys with r, s ∈ R.) Thus, since the determinant of the Jacobian of Xµ0 at Qµ0 is strictly positive, we can assert that it is a center. A straightforward application of the Poincaré-Bendixson theorem shows that Γ is the outer boundary of its period annulus, which fills the first quadrant. This proves that if p0 − c0 q0 = 2c0 q0 a0 − (c0 q0 − c0 + 1)b0 = 0 then the displacement map D ( · ; µ0 ) of Xµ0 along Γ is identically zero. The converse follows by Theorem 2.6 noting that d1 (µ) = d2 (µ) = 0 if, and only if, p − cq = 2cqa − (cq − c + 1)b = 0. It only remains to be proved that Cycl (Γ, Xµ0 ), Xµ = 1. This follows verbatim the proof of the same fact in assertion (c) of Theorem 3.1 and so we shall omit the details for the sake of shortness. Indeed, by (b) in Theorem 2.6 we have that if µ ≈ µ0 then D (s; µ) = (b1 (µ)ω (s; α(µ)) + b2 (µ) + r (s; µ))sc , where α = c − p/q and r ∈ Fℓ∞ (µ0 ) with ℓ ∈ (0, min(1, p)). Here we use that the three hyperbolicity ratios are λ1 = 1/q, λ2 = p and λ3 = 1/c. We also have that (b1 ) = (d1 ) and (b1 , b2 ) = (d1 , d2 ) over the local ring R{µ}µ0 . Therefore, if µ = ( a, b, c, p, q) verifies p − cq = 2cqa − (cq − c + 1)b = 0 then b1 (µ) = b2 (µ) = 0 and r (s; µ) ≡ 0. Moreover, since ∇d1 (µ0 ) and ∇d2 (µ0 ) are linearly independent, see (3.12) and (3.18), this is also the case of ∇b1 (µ0 ) and ∇b2 (µ0 ). We can thus take three linear functions, say b3 (µ), b4 (µ) and b5 (µ), such that ν = Ψ(µ) := b1 (µ), b2 (µ), b3 (µ), b4 (µ), b5 (µ) is a local analytic change of  coordinates in a neighbourhood of µ = µ0 with Ψ(µ0 ) = 05 := (0, 0, 0, 0, 0). Notice then that Ψ maps { p − cq = 2cqa − (cq − c + 1)b = 0} to {ν1 = ν2 = 0} and, moreover, R1 (s; ν) := s−c D (s; µ) = ν1 ω (s; α̂) + ν2 + r̂ (s; ν), µ = Ψ −1 ( ν ) where α̂ = α̂(ν) := α(Ψ−1 (ν)) and r̂ (s; ν) := r (s; Ψ̂−1 (ν)) ∈ Fℓ∞ (05 ). Due to r̂ (s; 0, 0, ν3 , ν4 , ν5 ) ≡ 0, by applying [17, Lemma 4.1] we can write the remainder as r̂ (s; ν) = ν1 h1 (s; ν) + ν2 h2 (s; ν) 28 D. Marín and J. Villadelprat with hi ∈ Fℓ∞ (05 ) and, consequently, R1 (s; ν) = ν1 ω (s; α̂) + h1 (s; ν) + ν2 1 + h2 (s; ν) .   From this expression we conclude that there exists s0 > 0 and an open neighbourhood U of ν = 05 such that R1 (s; ν) that has at most one zero on (0, s0 ), counted with multiplicities, for all ν = (ν1 , ν2 , ν3 , ν4 , ν5 ) ∈ U with ν1 + ν2 ̸= 0, which implies that Cycl (Γ, Xµ0 ), Xµ ⩽ 1. The 2 2  proof of this follows exactly as we argue to show the same factin Theorem 3.1, cf. (3.4), and it is omitted for brevity. Finally the fact that Cycl (Γ, Xµ0 ), Xµ ⩾ 1 follows taking µ1 ≈ µ0 with p1 − c1 q1 = 0 and 2c1 q1 a1 − (c1 q1 − c1 + 1)b1 ̸= 0, and applying the assertion in (b). This completes the proof of the result. □ Remark 3.4. In order to prove Theorem 3.2 it is only necessary to compute the functions d1 and d2 in Theorem A, which give the conditions for cyclicity 0 and 1, respectively. Let us explain that, as a matter of fact, we computed the function d3 as well, realizing that it vanishes when d1 = d2 = 0. It was this fact that lead us to investigate if the return map along the polycycle is the identity in that case. For completeness let us explain succinctly the computations that involve the obtention of d3 for the Kolmogorov’s family considered in Theorem 3.2. Recall, see (e) in Theorem A, that  d3 (µ) := M̂3 (λ3 , 1) L11 (1) − M̂1 1/λ1 , 1 L31 (1). In this case, cf. (3.14), we have that 1+(q+1)c  1+(2cq+1)c L11 (u) = (1 + cu2 ) 2c and L31 (u) = 1 + u2 /c and then, from the definition in (1.2), 1+(q−3)c M1 (u) = (1 + cu2 ) aq + b + (cb − (c + 1) a)u2  2c and  1+(2cq−3)c M3 (u) = −u 1 + u2 /c ( aq + b)u2 + cb − (c + 1) a /c2 .  In order to proceed with the computation of M̂1 λ11 , 1 and M̂3 λ3 , 1 we note that if   J ( x; η, r ) := (1 + ηx2 )r with η > 0 and r ∈ R then Z 1 1 Ĵ (α, 1; η, r ) = (1 + ηx2 )r x −α−1 dx = − 2 F1 (−r, − α/2; 1 − α/2; − η ) for all α < 0, 0 α where in the first equality we apply (b) in Proposition 2.5 with k = 0 and in the second one we use the equality in [1, 15.3.1] to express the definite integral as a hypergeometric function. In principle the above equality is only true provided that α < 0. However, its validity can be extended to any α ∈ / N thanks to the meromorphic properties of the functions 2 F1 and Ĵ established, respectively, by [17, Lemma B.2] and (d) in Proposition 2.5. Consequently, thanks to this observation and applying twice the above formula, we get aq + b cb − (c + 1) a M̂1 (1/λ1 , 1) = − φ1 (c, q) + φ2 (c, q), q 2−q where  φ1 (c, q) := 2 F1 (3 − q)/2 − 1/(2c), −q/2; 1 − q/2; −c ,  φ2 (c, q) := 2 F1 (3 − q)/2 − 1/(2c), 1 − q/2; 2 − q/2; −c . On the cyclicity of Kolmogorov polycycles 29 Here we also use that if h = f + g then ĥα = fˆα + ĝα and that if f ( x ) = x n g( x ) then fˆα ( x ) = x n ĝα−n ( x ), see [16, Corollary B3]. Similarly aq + b cb − (c + 1) a M̂3 (λ3 , 1) = φ3 (c, q) + φ4 (c, q), c(1 − 3c) c (1 − c ) where  φ3 (c, q) := 2 F1 (3 − q)/2 − 1/(2c), 3/2 − 1/(2c); 5/2 − 1/(2c); −1/c ,  φ4 (c, q) := 2 F1 (3 − q)/2 − 1/(2c), 1/2 − 1/(2c); 3/2 − 1/(2c); −1/c . In the proof of Theorem 3.2 we show that d1 (µ) = d2 (µ) = 0 if, and only if, µ = ( a, b, c, p, q) b(1−c+cq) verifies p = cq and a = 2cq . Long but easy computations show that, under these two conditions, d3 (µ) = 0 if, and only if, c−1   q − 1+2c c+cq φ3 (c, q) − φ4 (c, q) + c φ1 (c, q) + φ2 (c, q) = 0. 1 − 3c q−2 This is an equation for b, c and q that involves four hypergeometric functions. Surprisingly enough it turns out, by applying the formula in [1, 15.3.7], that the function on the left hand side of the above equation is identically zero. In other words, d1 (µ0 ) = d2 (µ0 ) = 0 implies d3 (µ0 ) = 0. □ Acknowledgements The authors want to thank Joan Torregrosa, who kindly lent us his Maple [13] procedure to compute the focus quantities that we used to prove (e) in Theorem 3.1. References [1] M. Abramowitz, I. A. Stegun, Handbook of mathematical functions with formulas, graphs, and mathematical tables, Dover, NewYork, 1992, reprint of the 1972 edition. MR0757537 [2] J. C. Artés, F. Dumortier, J. Llibre, Qualitative theory of planar differential sys- tems, Universitext, Springer-Verlag, Berlin, 2006. https://doi.org/10.1007/978-3-540- 32902-2; MR2256001 [3] C. Christopher, Estimating limit cycle bifurcations from centers, in: Differential equations with symbolic computation, Trends Math., Birkhäuser, Basel, 2005, pp. 23–35. https://doi. org/10.1007/3-7643-7429-2_2; MR2187371 [4] D. Cox, J. Little, D. O’Shea, Ideals, varieties, and algorithms. An introduction to computa- tional algebraic geometry and commutative algebra, Undergrad. Texts Math., Springer, New York, 2007. https://doi.org/10.1007/978-3-319-16721-3; MR3330490 [5] P. De Maesschalck, F. Dumortier, R. Roussarie, Cyclicity of common slow-fast cycles, Indag. Math. (N.S.) 22(2011), 165–206. https://doi.org/10.1016/j.indag.2011.09.008; MR2853605 30 D. Marín and J. Villadelprat [6] F. Dumortier, R. Roussarie, C. Rousseau, Elementary graphics of cyclicity 1 and 2, Nonlinearity 7(1994), 1001–1043. https://doi.org/10.1088/0951-7715/7/3/013 [7] F. Dumortier, C. Li, Quadratic Liénard equations with quadratic damping, J. Differential Equations 139(1997), 41–59. https://doi.org/10.1006/jdeq.1997.3291; MR1467352 [8] A. Gasull, V. Mañosa, F. Mañosas, Stability of certain planar unbounded polycycles, J. Math. Anal. Appl. 269(2002), 332–351. https://doi.org/10.1016/S0022-247X(02)00027- 6; MR1907889 [9] G.-M. Greuel, C. Lossen, E. Shustin, Introduction to singularities and deformations, Springer Monographs in Mathematics, Springer, Berlin, 2007. https://doi.org/10. 1007/3-540-28419-2; MR2290112 [10] Y. Ilyashenko, Centennial history of Hilbert’s 16th problem, Bull. Amer. Math. Soc. (N.S.) 39(2002), 301–354. https://doi.org/10.1090/S0273-0979-02-00946-1 ; MR1898209 [11] R. Huzak, Cyclicity of canard cycles with hyperbolic saddles located away from the critical curve, J. Differential Equations 320(2022), 479–509. https://doi.org/10.1016/j. jde.2022.02.050; MR4393885 [12] S. G. Krantz, H. R. Parks, A primer of real analytic functions, Birkhäuser Advanced Texts, Birkhäuser Basel, 2002. https://doi.org/10.1007/978-0-8176-8134-0; MR1916029 [13] Maple 2021. Maplesoft, a division of Waterloo Maple Inc., Waterloo, Ontario. [14] D. Marín, J. Villadelprat, Asymptotic expansion of the Dulac map and time for un- foldings of hyperbolic saddles: local setting, J. Differential Equations 269(2020), 8425–8467. https://doi.org/10.1016/j.jde.2020.06.024; MR4113207 [15] D. Marín, J. Villadelprat, Asymptotic expansion of the Dulac map and time for un- foldings of hyperbolic saddles: general setting, J. Differential Equations 275(2021), 684–732. https://doi.org/10.1016/j.jde.2020.11.020; MR4191338 [16] D. Marín, J. Villadelprat, Asymptotic expansion of the Dulac map and time for un- foldings of hyperbolic saddles: coefficient properties, preprint. [17] D. Marín, J. Villadelprat, The criticality of reversible quadratic centers at the outer boundary of its period annulus, J. Differential Equations, 332(2022), 123–201. https:// doi.org/10.1016/j.jde.2022.05.026; MR4437712 [18] J. D. Murray, Mathematical biology, Biomathematics, Vol. 19, Springer-Verlag, Berlin, 1989. https://doi.org/10.1007/978-3-662-08539-4; MR1239892 [19] V. Romanovski, D. Shafer, The center and cyclicity problems: a computational algebra ap- proach, Birkhäuser Boston, Ltd., Boston, MA, 2009. https://doi.org/10.1007/978-0- 8176-4727-8; MR2500203 [20] R. Roussarie, A note on finite cyclicity and Hilbert’s 16th problem, in: Dynamical systems, Valparaiso 1986, Lecture Notes in Mathematics, Vol. 1331, Springer, Berlin, 1988, pp. 161– 168. https://doi.org/10.1007/BFb0083072; MR961099 On the cyclicity of Kolmogorov polycycles 31 [21] R. Roussarie, Bifurcations of planar vector fields and Hilbert’s sixteenth problem, reprint of the 1998 edition, Modern Birkhäuser Classics, Birkhäuser/Springer, Basel, 2013. https: //doi.org/10.1007/978-3-0348-8798-4 [22] C. Rousseau, G. Świrszcz, H. Żoładek, ˛ Cyclicity of graphics with semi-hyperbolic points inside quadratic systems, J. Dynam. Control Systems 4(1988), 149–189. https: //doi.org/10.1023/A:1022887001627; MR1626537 [23] Y. Q. Ye, S. L. Cai, L. S. Chen, K. C. Huang, D. J. Luo, Z. E. Ma, E. N. Wang, M. S. Wang, X. A. Yang, Theory of limit cycles, Translations of Mathematical Monographs, Vol. 66, American Mathematical Society, Providence, RI, 1984. MR0854278