This is a preprint of: “New lower bound for the Hilbert number in piecewise quadratic differential systems”, Leonardo Pereira Costa da Cruz, Douglas D. Novaes, Joan Torregrosa, J. Differential Equations, vol. 266, 4170–4203, 2019. DOI: [10.1016/j.jde.2018.09.032] NEW LOWER BOUND FOR THE HILBERT NUMBER IN PIECEWISE QUADRATIC DIFFERENTIAL SYSTEMS LEONARDO P. C. DA CRUZ, DOUGLAS D. NOVAES, AND JOAN TORREGROSA Abstract. We study the number of limit cycles bifurcating from a piecewise qua- dratic system. All the differential systems considered are piecewise in two zones separated by a straight line. We prove the existence of 16 crossing limit cycles in this class of systems. If we denote by Hp (n) the extension of the Hilbert number to degree n piecewise polynomial differential systems, then Hp (2) ≥ 16. As fas as we are concerned, this is the best lower bound for the quadratic class. Moreover, all the limit cycles appear in one nest bifurcating from the period annulus of some isochronous quadratic centers. 1. Introduction Consider the class of polynomial differential systems of degree n. The maximum number of isolated periodic orbits, the so-called limit cycles, that a polynomial dif- ferential system of degree n can have is called Hilbert number, H(n). It is well known that linear systems have no limit cycles, then H(1) = 0. For n = 2, the problem of estimating H(2) has been studied intensively during the last century. Lower bounds for H(2) can be given by providing concrete examples of polynomial differential sys- tems of degree 2. Up to now, the best result was given by Shi in [31], where he proved the existence of a quadratic system with 4 limit cycles in configuration (3, 1), that is H(2) ≥ 4. We call by M (n) the maximum number of limit cycles bifurcating from a singular point as a degenerate Hopf bifurcation. Clearly, M (n) is a lower bound for H(n). Bautin showed in [2] that M (2) = 3; in [32, 33], Żołądek proved that M (3) ≥ 11; a simpler proof was provided by Christopher in [9]. For n = 3, Li, Liu, and Yang proved in [21] that H(3) ≥ 13. In the last few years there has been an increasing interest in piecewise smooth systems. This interest has been mainly motivated by their wider range of application in modeling real phenomena (see, for instance, [1, 11]). In this paper we shall deal with the following class of piecewise vector fields  + Z (x, y), h(x, y) > 0, Z(x, y) = (1) Z − (x, y), h(x, y) < 0, where Z ± = (X ± , Y ± ) are smooth vector fields and h : R2 → R is a C 1 function for which 0 is a regular value. In the above vector field, the discontinuity curve and the regions where Z ± are denoted by Σ = h−1 (0) and Σ± = {(x, y) ∈ R2 : ±h(x, y) > 0}, respectively. The local trajectories of Z on Σ was stated by Filippov in [12] (see Figure 1). The points on Σ where both vectors fields simultaneously point outward or inward from Σ define the escaping (Σe ) and sliding region (Σs ), respectively. The 2010 Mathematics Subject Classification. Primary: 37G15, 37C27. Secundary: 37G10, 34C07. Key words and phrases. Non-smooth differential system, limit cycles in piecewise quadratic dif- ferential systems, first and second order perturbations of isochronous quadratic systems, Hilbert number for piecewise quadratic differential systems. 1 2 L. P. C. DA CRUZ, D. NOVAES, AND J. TORREGROSA interior of its complement on Σ defines the crossing region (Σc ), and the boundary of these regions is constituted by tangential points of Z ± with Σ. Let Z ± h denote Z + (p) = Z ± (p) Z + (p) Z − (p) Z − (p) Z ± (p) Σ Z ± (p) Σ p Σ p p Z + (p) + + + Σ Σ Σ Z − (p) Σ− Σ− Σ− Figure 1. Definition of the vector field on Σ following Filippov’s con- vention in the sewing, escaping, and sliding regions. the derivative of the function h in the direction of the vector Z ± that is, Z ± h(p) = h∇h(p), Z ± (p)i. Notice that p ∈ Σc provided that Z + h(p) · Z − h(p) > 0, p ∈ Σe ∪ Σs provided that Z + h(p) · Z − h(p) < 0, and p in Σ is a tangential point of Z ± provided that Z + h(p)Z − h(p) = 0. We say that p ∈ Σ is a singularity of Z, if p is either a tangential point or a singularity of Z + or Z − . We call p ∈ Σ an invisible fold of Z + (resp. Z − ) if p is a tangential point of Z + (resp. Z − ) and (Z + )2 h(p) < 0 (resp. (Z − )2 h(p) > 0). Analogously to the smooth case, we denote by Hpc (n) the maximum number of crossing limit cycles that piecewise polynomial differential systems of degree n admit when the curve of discontinuity is a straight line. We also denote by Mpc (n) the maximum number of crossing limit cycles bifurcating from a singular point or sliding set. Up to now, for piecewise linear systems in two zones separated by a straight line, there are no examples with more than 3 limit cycles. An example with 3 limit cycles was firstly detected numerically in [17] by Huan and Yang. Later, it was analytically proved by Llibre and Ponce in [25]. The existence of 3 limit cycles was also obtained from perturbations of a center. For instance, Buzzi et al. in [3] obtained 3 limit cycles after a seventh order piecewise linear perturbation of a linear center, and Llibre et al. in [24] obtained the same result through a first order perturbation of a piecewise linear center. We may also quote Freire et al. [14]. Consequently, for piecewise linear systems in two zones separated by a straight line we have Hpc (1) ≥ 3. The averaging theory of order five for studying piecewise perturbations of the linear center was used by Llibre and Tang in [26] who provided that Hpc (2) ≥ 8 and Hpc (3) ≥ Mpc (3) ≥ 13. These are the best results so far for piecewise quadratic and cubic systems in two zones separated by a straight line. Previously, using the averaging theory of first order for studying piecewise perturbations of some quadratic isochronous systems, Llibre and Mereu in [22] obtained only 5 limit cycles. It is worthwhile to say that for quadratic polynomial systems Chicone and Jacobs in [7] proved that at most 2 limit cycles can bifurcate from any period annulus. In this paper we shall use the averaging theory of first and second order to provide better lower bounds for the maximum number of limit cycles that piecewise quadratic systems can have. More specifically, we shall give examples satisfying Mpc (2) ≥ 16. Consequently, Hpc (3) ≥ Hpc (2) ≥ 16. Table 1 summarizes the results about the Hilbert numbers for lower degree vector fields. Theorem 1.1. There exists a piecewise planar quadratic differential system in two zones separated by a straight line with 16 crossing limit cycles. HILBERT NUMBER IN PIECEWISE QUADRATIC SYSTEMS 3 deg PVF PPVF n = 1 H(1) = 0 Hpc (1) ≥ 3 n = 2 H(2) ≥ 4 Hpc (2) ≥ 16(8) n = 3 H(3) ≥ 13 Hpc (3) ≥ 16(13) Table 1. Summary of Hilbert numbers for polynomial and piecewise polynomial systems of degree n. Listed in parenthesis the known Hilbert numbers so far. In order to prove our main result we shall proceed with a first and second or- der perturbation analysis of quadratic isochronous centers. In [27] the quadratic isochronous centers are classified in four families, namely S1 , S2 , S3 , and S4 . In [28] their isochronicity properties were proved as well as their linearizations. In this paper we consider the first three classes of centers, which are birational equivalent to the linear one. This property does not hold for S4 . It is proved in [5] that any center of families S1 , S2 , and S3 can be transformed, after a birational change of variables, in one of the following centers:  ( (  4 2 ẋ = −y + x − y ,2 2 ẋ = −y + x ,  ẋ = −y − x2 , S1 : S2 : S3 : 3 (2) ẏ = x + 2xy. ẏ = x + xy.   ẏ = x − 16 xy. 3 Here, the dot denotes the derivative with respect to time. The phase portraits of these systems are depicted in Figure 2. Figure 2. Phase portrait of systems S1 , S2 , and S3 from left to right. The first order averaging method was used in [22] to get 4 and 5 limit cycles by perturbing, respectively, the centers S1 and S2 inside the class of piecewise quadratic systems with two zones separated by the straight line y = 0. Here, due to restric- tions of the employed technique, we take {x = 0} as the curve of discontinuity for the centers S1 and S3 . The first and second order analysis for S1 are performed in Propositions 3.1 and 4.1, where we get 5 and 11 limit cycles, respectively. Analo- gously, for S3 Propositions 3.2 and 4.3 provide 5 and 10 limit cycles, respectively. We shall see that for the center S2 the employed technique works whenever the curve of discontinuity is a straight line passing through the origin. This allows to reach the best result, namely Hpc (2) ≥ 16. In fact, proceeding with a first order analysis Propo- sition 3.3 provides √5, 6, and 8 limit cycles when the curve of discontinuity is {x = 0}, {y = 0}, and {y + 3x = 0}, respectively. Due to the difficulties in the massive com- putations, the second order analysis has been performed only for the case of highest ciclicity at the first order analysis, namely when the curve of discontinuity is given 4 L. P. C. DA CRUZ, D. NOVAES, AND J. TORREGROSA √ by {y + 3x = 0}. In this case, Proposition 4.7 provides 16 limit cycles bifurcating from the origin, that is Mpc (2) ≥ 16. This proves our main result, Theorem 1.1. This work is structured as follows. In Section 2, we present some basic notions and preliminary tools. In Sections 3 and 4, the first and second order analysis are performed, respectively. Finally, last section contains the integrals used in this paper. 2. Preliminaries This section is devoted to present some basic notions and preliminary tools needed to prove our main result. Firstly, we introduce some results on averaging theory of first and second orders. In fact, the limit cycles will appear from the simple zeros of some integrals (see, for instance, [18, 23]). Secondly, we recall the concepts of Extended Complete Chebyshev system (ECT-system) and Chebyshev system with accuracy (see, for instance, [29]). Then, we introduce the concept of pseudo-Hopf bifurcation, which is the birth of a limit cycle when the sliding set changes stability (see, for instance, [12, 13]). Finally, we state the Poincaré–Miranda theory, which is an extension of the intermediate value theorem, see [20]. 2.1. Averaging Theory. Assume that the origin is a center equilibrium point for system (1). Consider the following perturbed piecewise polynomial vector field  + Z (x, y) + ε (P1+ (x, y), Q+ 1 (x, y)) + ε (P2 (x, y), Q2 (x, y)), if h(x, y) > 0, 2 + + ± Zε = Z − (x, y) + ε (P1− (x, y), Q−1 (x, y)) + ε (P2 (x, y), Q2 (x, y)), if h(x, y) < 0, 2 − − (3) where ε is sufficiently small, Pk , Qk are polynomials of degree n in (x, y), for ± ± k = 1, 2, and h(x, y) = y − tan(α)x. After changing to polar coordinates, (x, y) = (r cos θ, r sin θ), system (3) writes  + Zeε (θ, r), if α < θ < α + π, Zeε (θ, r) = (4) Zeε− (θ, r), if α − π < θ < α. Taking θ as the new independent variable, the differential system associated to the vector field (4) becomes the piecewise differential equation dr r0 (θ) = = εF1 (θ, r) + ε2 F2 (θ, r) + O(ε3 ), (5) dθ with  Fi+ (θ, r) if α < θ < α + π, Fi (θ, r) = Fi− (θ, r) if α − π < θ < α, where Fi± : [α − π, α + π] × (0, ρ∗ ) → R are analytical functions 2π–periodic in the variable θ for i = 1, 2. We define F1 , F2 : (0, ρ∗ ) → R as Z α+π  F1 (r) = F1+ (θ, r) + F1− (θ − π, r) dθ, Zαα+π  F2 (r) = F2+ (θ, r) + F2− (θ − π, r) dθ (6) α+π   αZ ∂ + + ∂ − − + F (θ, r)r1 (θ, r) + F1 (θ − π, r)r1 (θ − π, r) dθ. α ∂r 1 ∂r HILBERT NUMBER IN PIECEWISE QUADRATIC SYSTEMS 5 Here, the functions r1± : (−π, π) × R+ → R are defined as Z α+θ ± r1 (θ, r) = F1± (φ, r)dφ. (7) α Theorem 2.1 ([23]). Consider the piecewise differential equation (5). (i) Suppose that for ρ ∈ (0, ρ∗ ) with F1 (ρ) = 0 and F10 (ρ) 6= 0. Then, for |ε| > 0 sufficiently small, there exists a 2π–periodic solution r(θ, ε) of (5) such that r(0, ε) → ρ when ε → 0. (ii) Assume that F1 = 0. Suppose that for ρ ∈ (0, ρ∗ ) with F2 (ρ) = 0 and F20 (ρ) 6= 0. Then, for |ε| > 0 sufficiently small, there exists a 2π–periodic solution r(θ, ε) of (5) such that r(0, ε) → ρ when ε → 0. 2.2. ECT-Systems. Let F = [u0 , . . . , un ] be an ordered set of functions of class C ∞ on the closed interval [a, b]. We denote by Z(F) the maximum number of zeros counting multiplicity that any nontrivial function v ∈ Span(F) can have. Here, Span(F) is the set of functions generated by linear combinations of elements of F, that is v(s) = a0 u0 (s) + a1 u1 (s) + · · · + an un (s) where ai , for i = 0, 1, . . . , n, are real numbers. The theory of Chebyshev systems is a classical tool to study the quantity Z(F). In fact, when Z(F) ≤ n, F is called an Extended Chebyshev system or ET-system on [a, b], see [19]. We say that F is an Extended Complete Chebyshev system or an ECT- system on a closed interval [a, b] if and only if for any k, 0 ≤ k ≤ n, [u0 , u1 , . . . , uk ] is an ET-system. In order to prove that F is a ECT-system on [a, b] it is sufficient and necessary to show that W (u0 , u1 , . . . , uk )(t) 6= 0 on [a, b] for 0 ≤ k ≤ n, see also [19]. Here, W (u0 , u1 , . . . , un )(t) denotes the Wronskian of F with respect to t. That is,   u0 (t) · · · un (t)  u00 (t) · · · u0n (t)    Wn (t) = W (u0 , . . . , un )(t) = det  .. .. .. .  . . .  (n) (n) u0 (t) · · · un (t) Furthermore, the sufficient condition to be an ECT-system also provides that each configuration of m ≤ n zeros, taking into account their multiplicity, is realizable. The next theorem, proved in [29], extends the results for ECT-systems when some of the Wronskian vanish. Theorem 2.2 ([29]). Let F = [u0 , u1 , . . . , un ] be an ordered set of analytic functions on [a, b]. Assume that all the νi zeros of the Wronskian Wi are simple for i = 0, . . . , n. Then, the number of isolated zeros for every element of Span(F) does not exceed n + νn + νn−1 + 2(νn−2 + · · · + ν0 ) + νn−1 + · · · + ν3 where νi = min(2νi , νi−3 + · · · + ν0 ), for i = 3, . . . , n − 1. 2.3. Pseudo-Hopf Bifurcation. In the well-known Hopf bifurcation (see, for in- stance, [15]) a limit cycle arises from an equilibrium point when it changes its sta- bility. In piecewise differential systems, the pseudo-Hopf bifurcation describes the same phenomenon but when the sliding segment changes its stability. Analogously to the classical Hopf bifurcation, the proof is a direct consequence of the generalized Poincaré–Bendixson Theorem for piecewise differential systems (see, for instance, [4]). Proposition 2.3. Let Z ± = (X ± (x, y), Y ± (x, y)) be a C 1 piecewise differential system in two zones separated by the straight line y = 0. Additionally, the origin is a stable 6 L. P. C. DA CRUZ, D. NOVAES, AND J. TORREGROSA monodromic equilibrium point and a = (∂Y + /∂x)|(0,0) > 0. Given a real number b, we consider the perturbed system Zb± = (Xb± (x, y), Yb± (x, y)) defined by Xb± (x, y) = X ± (x, y), and Yb− (x, y) = Y − (x, y) and Yb+ = Y + + b. Then, for b small enough, the system Zb± exhibits a pseudo-Hopf bifurcation at b = 0 when ab > 0. See Figure 3. b<0 b=0 b>0 Figure 3. Pseudo-Hopf bifurcation. 2.4. Poincaré–Miranda Theorem. The next result is a generalization of the in- termediate value theorem. It was conjectured by Poincaré in 1883 and proved by Miranda in 1940 (see, for instance, [20] and the references therein). Theorem 2.4 ([20]). Let a be a positive real number and B = [−a, a]n the n- dimensional cube. Let f = (f1 , . . . , fn ) : B → Rn be a continuous function such that fi (Bi− ) < 0 and fi (Bi+ ) > 0, for each i ≤ n, where Bi± = {(x1 , . . . , xn ) ∈ B : xi = ±a}. Then, there exists a point c ∈ B such that f (c) = 0. 3. First order perturbation In this section the first order averaging method is used to study the limit cycles of the perturbed piecewise vector field (3) when the unperturbed vector field Z0 is a quadratic isochronous center in one of the families S1 , S2 , or S3 . Regarding (3) we shall denote Zε = Zi,ε and Zε± = Zi,ε± in order to indicate that Z0 ∈ Si , for i = 1, 2, 3. Here, it is only considered quadratic polynomial perturbations, that is j 2 X j 2 X X X Pk± (x, y) = p± i j−i k,i,j−i x y and Q± k (x, y) = ± qk,i,j−i xi y j−i . j=0 i=0 j=0 i=0 The first order analyses for families S1 , S3 , and S2 are performed in Propositions 3.1, 3.2, and 3.3, respectively. For the families S1 and S2 , we shall also use the ECT- system properties to study the bifurcation of limit cycles in the global interval of definition. Accordingly, Propositions 3.1 and 3.2 are concerned about upper bounds (up to a first order analysis) for the maximum number of limit cycles bifurcating from the period annulus (the so-called medium amplitude limit cycles). In the third result, Proposition 3.3, a local analysis is performed around the center point. In this case, we also see how the number of limit cycles changes when we consider different lines of discontinuity. Before stating the main results of this section we briefly discuss the choosing of the lines of discontinuity. The birational linearizations of families S1 and S3 (see, for instance, [5]) transform the straight line {x = 0} into another straight line passing through the origin. Moreover, {x = 0} is the unique straight line for which this hap- pens. This is the main reason for choosing Σ = {x = 0} as the curve of discontinuity. The birational linearization of the family S2 transforms straight lines passing through the origin into straight lines passing through the origin, so that we are allowed to choose any straight line passing through the origin as the curve of discontinuity. Nev- ertheless, in this last case, since the computations are more intricate we only study HILBERT NUMBER IN PIECEWISE QUADRATIC SYSTEMS 7 the limit cycles bifurcating from the origin. We anticipate that all the conclusions of this section will be improved by results of the next section. Proposition 3.1. For |ε| > 0 sufficiently small the averaging method of first order predicts at most 5 crossing limit cycles for the piecewise quadratic vector field Z1,ε when the curve of discontinuity is the straight line {x = 0}. Moreover, this number is reached. Proof. In order to apply Theorem 2.1, we have to write the vector field (4) as a differential equation (5). So, we first proceed with the change of variables (see, for instance, [5]) v u2 + v 2 − u x=− 2 and y = − , (8) v + (u − 1)2 v 2 + (u − 1)2 which has the following rational inverse x2 + y 2 + y x u= and v = − . (9) x2 + y 2 + 2y + 1 x2 + y2 + 2y + 1 With this change of variables the differential equation S1 becomes the linear center (u0 , v 0 ) = (−v, u) and the line of discontinuity becomes v = 0. Then, we change to polar coordinates u = r cos θ and v = r sin θ. Taking θ as the new independent variable, (4) becomes ṙ A(r cos θ, r sin θ) r0 (θ) = =ε + O(ε2 ), (10) θ̇ C(θ, r) where C(θ, r) = (2r cos θ − r2 − 1)2 and A is the piecewise function  + A (r cos θ, r sin θ) if 0 < θ ≤ π, A(r cos θ, r sin θ) = (11) A− (r cos θ, r sin θ) if π < θ ≤ 2π, being A± polynomials of degree 3. From here we want to use the integral formulas of Section 5 to compute the averaged function F1 , as stated in (6), for α = 0. The denominators of F1+ (θ, r) and F1− (θ−π, r) write (2r cos θ − r2 − 1)2 and (2r cos θ + r2 + 1)2 , respectively. In order to use the integrals we must apply a transformation on r in order to get the denominators written in a standard form.√ Firstly, take r = (−1 + 1√− R2 )/R. The denominator of F1+ (θ, r) is transformed into 2R2 (R cos θ + 1)2 (R2 + 2 1 − R2 − 2). Hence, the first part of the first averaged function Z π √ F1+ (θ, (−1 + 1 − R2 )/R)dθ 0 can be computed using the integrals (25) for α = 0, ` = 2, and k ∈ {0, 1, 2, 3}. We shall suppress it here. The original √ variable r is recovered by taking R = −2r/(r2 +1). Secondly, take r = (1 − 1 − R2 )/R. √ The denominator of F1 (θ − π, r) is trans- − formed into 2R (R cos θ + 1) (R + 2 1 − R2 − 2). Hence, the second part of the first 2 2 2 averaging function Z π √ F1− (θ − π, (1 − 1 − R2 )/R)dθ 0 can be computed also using the integrals (25) for α = 0, ` = 2, and k ∈ {0, 1, 2, 3}. The original variable r is recovered by taking R = 2r/(r2 + 1). 8 L. P. C. DA CRUZ, D. NOVAES, AND J. TORREGROSA Adding up the above integrals, we get the averaged function F1 (r). Proceeding with the change of parameters k0 p− + 1,0,0 = p1,0,0 − 2k4 + , 2 k2 + k0 p− − + + 1,0,1 = p1,0,2 + p1,0,1 − p1,0,2 − 2k4 + 2k5 + , 2 − − + + q1,0,2 + q1,2,0 + q1,0,2 + q1,2,0 k1 − k3 p− 1,1,0 + − + = −p1,1,0 + 2(q1,0,0 + q1,0,0 ) − + , 2 π − + (12) q1,1,1 − q1,1,1 p− 1,2,0 + = p1,2,0 + − 2k4 − 2k5 , 2 − − + + − − + + q1,0,2 + q1,2,0 + q1,0,2 + q1,2,0 k1 + 2k3 q1,0,1 = 2q1,0,0 + 2q1,0,0 − q1,0,1 + + , 2 π + − + − q1,1,1 − q1,1,1 + k2 + k0 q1,1,0 = q1,1,0 + , 2 we get 5 X F1 (r) = kn fn (r), (13) n=0 where f0 (r) = 1, f1 (r) = r, f2 (r) = r2 , f3 (r) = r3 , 1 − r2 f4 (r) = L(r), f5 (r) = r(1 − r2 ) L(r), r and  1−r L(r) = log . (14) 1+r Clearly, from (12), the parameters kn can be chosen arbitrarily. The maximum number of simple zeros that (13) can have follows by studying the Wronskians of the ordered set [f0 , f1 , . . . , f5 ]. Straightforward computations show that W0 (r) = 1, W1 (r) = 1, W2 (r) = 2, W3 (r) = 12, 288 9216(r2 + 5) W4 (r) = W 4 (r), W5 (r) = W 5 (r), r5 (1 − r2 )4 r6 where 2 r (7r2 − 8r2 + 3) 2 r (3r2 − 22r2 + 15) W 4 = L(r) − , W 5 = L(r) − . 3 (r2 − 1)3 3 (r2 − 1)2 (r2 + 5) Clearly, W0 , W1 , W2 , and W3 do not vanish in (0, 1). Now, computing the derivative 0 4r4 (5r2 + 1) W 4 (r) = > 0, (r2 − 1)4 as W 4 (0) = 0, also W4 (r) is does not vanish in (0, 1). The same argument applies for W5 (r), but using 0 64r6 W 5 (r) = 2 . (r − 1)3 (r2 + 5)2 So, the proof follows by noticing that the ordered set of functions [f0 , . . . , f5 ] is an ECT-system.  HILBERT NUMBER IN PIECEWISE QUADRATIC SYSTEMS 9 Proposition 3.2. For |ε| > 0 sufficiently small the averaging method of first order predicts at most 5 crossing limit cycles for the piecewise quadratic vector field Z3,ε when the curve of discontinuity is the straight line {x = 0}. Moreover, this number is reached . Proof. We shall follow the same procedure of the proof of Proposition 3.1. The linearization stated in [5] is given by 3u 3(4u2 + 8v 2 + v) x= y= , 8v + 1 (8v + 1)2 which has the following rational inverse 3x −4x2 + 3y u= 2 and v = . 32x − 24y + 9 32x2 − 24y + 9 Then, applying the change of variables u = r sin θ and v = −r cos θ, and taking θ as the new independent variable, equation (5) becomes ṙ A(θ, r) r0 (θ) = =ε + O(ε2 ), θ̇ C(θ, r) where C(θ, r) = 8r cos θ − 1 and A is the piecewise function  + A (r sin θ, −r cos θ) if 0 < θ ≤ π, A(θ, r) = A− (r sin θ, −r cos θ) if π < θ ≤ 2π, being A± polynomials of degree 6. Now we compute the averaged function (6) for α = 0. As in the proof of Proposi- tion 3.1, the denominators of F1+ (θ, r) and F1− (θ − π, r) are not written in a standard form in order to use directly the integrals of Section 5. Firstly, take r = −R/8. The denominator of F1+ (θ, r) in (6) becomes (R cos θ + 1)4 . Hence, the integral Z π F1+ (θ, −R/8)dθ 0 can be computed using (25) for α = 0, ` = 4, and k ∈ {0, 1, . . . , 6}. The original variable r is recovered taking R = −8r. Secondly, take r = R/8. The denominator of F1− (θ − π, r) in (6) also becomes (R cos θ + 1)4 . Hence, the integral Z π F1− (θ − π, R/8)dθ, 0 can be also computed using (25) for α = 0, ` = 4, and k ∈ {0, 1, . . . , 6}. The original variable r is recovered taking R = 8r. Adding up the √ above integrals we obtain the first averaged function F1 (r), which depends on r, 1 − r2 , and L(r) defined in (14). Proceeding with the change r = 2ρ/(1 + ρ2 ), (15) the averaged function writes 5 X Fe1 (ρ) = kn fn (ρ), (16) n=0 10 L. P. C. DA CRUZ, D. NOVAES, AND J. TORREGROSA where f0 (ρ) = ρ, f1 (ρ) = ρ2 , f2 (ρ) = ρ3 , f3 (ρ) = ρ4 + 1, f4 (ρ) = ρ5 , f5 (ρ) = L(ρ)/ρ. We remark that L(r) = 2L(ρ). The maximum number of simple zeros that (16) can have follows by studying the Wronskians of the ordered set [f0 , f1 , . . . , f5 ]. Straightforward computations show that. W0 (ρ) = ρ, W1 (ρ) = ρ2 , W2 (ρ) = 2ρ3 , W3 (ρ) = 12(ρ4 − 1), 207360(1 − ρ4 ) W4 (ρ) = 288ρ(ρ4 − 5), W5 (ρ) = W 5 (ρ), ρ5 where ρ (75 ρ12 − 175 ρ10 + 61 ρ8 + 95 ρ6 − 230 ρ4 + 140 ρ2 − 30) W 5 (ρ) = L(ρ) − . 15 (ρ2 − 1)6 (ρ2 + 1) Clearly W0 , W1 , W2 , W3 , and W4 do not vanish in (0, 1). The last Wronskian does not vanish either because W 5 (0) = 0 and the derivative 0 ρ4 (ρ4 − 5)(105ρ8 + 105ρ6 + 175ρ4 − 5ρ2 + 4) W 5 (ρ) = 15 (ρ2 −1)7 (ρ2 +1)2 is positive for every ρ ∈ (0, 1). So, the proof follows by noticing that the ordered set of functions [f0 , . . . , f5 ] is an ECT-system.  The global analysis performed in the previous results cannot be performed in a straightforward way for the family S2 . Hence, for this family we provide only a local analysis around the origin. Proposition 3.3. For |ε| > 0 sufficiently small and under the condition P ± (0, 0) = Q± (0, 0) = 0, the averaging method of first order predicts at most 4, 5, or 7 limit cycles bifurcating from the origin for the quadratic vector field√Z2,ε when the curve of discontinuity is the straight line {x = 0}, {y = 0}, or {y + 3x = 0}, respectively. Moreover, these numbers are reached. Proof. The linearization stated in [5] for family S2 is given by u v x=− and y = − , v−1 v−1 which has the following rational inverse x y u= and v = . y+1 y+1 As we have commented before, straight lines passing through the origin are trans- formed into straight lines passing through the origin. Firstly, assume that Σ = {x = 0}. Applying the change of variable (u, v) = (r sin θ, −r cos θ) and taking θ as the new independent variable we obtain the equiv- alent functions (10) and (11). Here, A± are cubic polynomials and the denominator becomes C(θ, r) = 1 + r cos θ. For this case, the first averaged function F1 is given by (6) for α = 0. Since C(θ, r) = 1 + r cos θ is the denominator of F1+ (θ, r) in (6), the integrals (25) can be used directly. Nevertheless, the denominator of F1− (θ − π, r) in (6) is given by C(θ − π, r) = 1 − r cos θ, so it is necessary to proceed with the change r = −R in order to use the integrals (25). Applying the integrals (25) for ` = 1 HILBERT NUMBER IN PIECEWISE QUADRATIC SYSTEMS 11 and k ∈ {0, 1, 2, 3}, and going back to the original variable r we have computed the first averaged function F1 (r). Finally, with the change (15) and after some algebraic manipulations, we get 4 X Fe1 (ρ) = kn fn (ρ), n=0 with ρ ρ2 f0 (ρ) = , f 1 (ρ) = , (ρ2 + 1)2 (ρ2 + 1)2 3 ρ4 + 3 ρ3 + ρ2 + 3 ρ4 − ρ2 + 1 f2 (ρ) = + L(ρ), 3 (ρ2 + 1)2 2 (ρ2 + 1) ρ 3ρ2 3ρ ρ5 f3 (ρ) = − − L(ρ), f4 (ρ) = , 4 (ρ2 + 1)2 8 (ρ2 + 1) (ρ2 + 1)2 and L is defined in (14). Moreover, the parameters kn can be chosen arbitrarily. The first part of the statement follows because, in a neighborhood of the origin, fi (ρ) = ρi+1 + O(ρi+2 ). Now, assume that Σ = {y = 0}. The procedure for this case is similar to the previous case. We only detail the differences. The functions F1 and Fe1 are obtained from (6) and (25), but now for α = −π/2. Thus, after some algebraic manipulations we get X5 e F1 (ρ) = kn fn (ρ), n=0 with ρ ρ2 ρ3 f0 (ρ) = , f1 (ρ) = , f 2 (ρ) = , (ρ2 + 1)2 (ρ2 + 1)2 (ρ2 + 1)2 3ρ2 3ρ(ρ2 − 1)2  π  ρ5 f3 (ρ) = 2 − φ ρ, , f 4 (ρ) = , (17) 4(ρ + 1)2 8(ρ2 + 1)3 2 (ρ2 + 1)2 525 ρ4 − 490 ρ2 + 525 (175 ρ4 + 70 ρ2 + 175) (ρ2 − 1)  π  2 f5 (ρ) = − φ ρ, . 768 (ρ2 + 1)2 512 ρ (ρ2 + 1)3 2 Here, the function φ is defined as r  !! 1 1−r θ φ(r, θ) = √ θ − 2 arctan tan , 1 − r2 1+r 2 and the parameters kn , n = 0, 1, . . . , 5, are arbitrary real numbers. The functions (17) also write fi (ρ) = ρi+1 + O(ρi+2 ). Consequently, the second part of the proof follows. √ Finally, assume that Σ = {y + 3x = 0}. Again, the procedure for this case is similar to the previous cases and we shall only detail the differences. The functions F1 and Fe1 are obtained from (6) and (25), but now for α = −π/3. Thus, after some algebraic manipulations we get 7 X Fe1 (ρ) = kn fn , n=0 12 L. P. C. DA CRUZ, D. NOVAES, AND J. TORREGROSA with ρ ρ2 ρ3 ρ5 f0 (ρ) = , f1 (ρ) = , f2 (ρ) = , f 4 (ρ) = , ρ2 + 1 (ρ2 + 1)2 (ρ2 + 1)2 (ρ2 + 1)2 5(54733ρ4 + 94452ρ2 + 54733) 15(1366ρ4 + 1847ρ2 + 1366) e f3 (ρ) = + L(ρ) 6912(ρ2 + 1)2 1024(ρ2 + 1)ρ √ 25 3(236ρ4 − 247ρ2 + 236)(ρ2 − 1)2 e + φ(ρ), 82944ρ(ρ2 + 1)3 35(21835ρ4 + 40596ρ2 + 21835) 105(550ρ4 + 797ρ2 + 550) e f5 (ρ) = − − L(ρ) 6912(ρ2 + 1)2 1024(ρ2 + 1)ρ √ 175 3(176ρ4 − 181ρ2 + 176)(ρ2 − 1)2 e − φ(ρ), (18) 82944ρ(ρ2 + 1)3 245(227ρ4 + 444ρ2 + 227) 315(122ρ4 + 181ρ2 + 122) e f6 (ρ) = + L(ρ) 768(ρ2 + 1)2 1024(ρ2 + 1)ρ √ 35 3(116ρ4 − 115ρ2 + 116)(ρ2 − 1)2 e + φ(ρ), 9216ρ(ρ2 + 1)3 385(77ρ4 + 156ρ2 + 77) 3465(2ρ4 + 3ρ2 + 2) e f7 (ρ) = − − L(ρ) 2304(ρ2 + 1)2 1024(ρ2 + 1)ρ √ 385 3(8ρ4 − 7ρ2 + 8)(ρ2 − 1)2 e − φ(ρ). 27648ρ(ρ2 + 1)3 Here,  2      e ρ −ρ+1 e 2ρ 2π 2ρ 2π L(ρ) = log , φ(ρ) = φ , −φ − 2 , . ρ2 + ρ + 1 ρ2 + 1 3 ρ +1 3 Analogously to the previous cases, kn , n = 0, 1, . . . , 7, are arbitrary real numbers. Here, fi (ρ) = ρi+1 + O(ρi+2 ) for i = 0, 1, . . . , 5, f6 (ρ) = ρ8 + O(ρ9 ), and f7 (ρ) = ρ10 + O(ρ11 ). Therefore, the ordered set of functions [f0 , f1 , . . . , f7 ] is an ECT-system in a neighborhood of the origin. This completes the proof of the last case.  Following the ideas of [10], the previous local result can be numerically improved to a global one. Taking linear combinations of the functions (18) one may try to get an ordered set of 8 functions which is an ECT-system with accuracy (see, for instance, [29]). For instance, it can be checked numerically that the ordered set [f0 , f1 , f2 , f3 , f4 , f6 + f7 , f7 , f5 ] has all Wronskians non-vanishing except W5 , W6 which vanish once. From Theorem 2.2, we conclude that F has at most 9 simple zeros. We shall see that a second order analysis allow us to overcome this number of limit cycles. The next result is a technical lemma describing the existence of a pseudo-Hopf bifurcation for Z2,ε . Lemma 3.4. Consider the piecewise vector fields Zi,ε , i = 1, 2, 3, under the assump- tion P ± (0, 0) = Q± (0, 0) = 0. For all curve of discontinuity given by {h(x, y) = Ax + By = 0}, there exists a constant perturbation such that a small limit cycle bifurcates from the origin in a pseudo-Hopf bifurcation. Proof. The unperturbed vector fields have a monodromic equilibrium point. This property remains under the assumption P ± (0, 0) = Q± (0, 0) = 0. Then, the proof follows directly from Proposition 2.3.  HILBERT NUMBER IN PIECEWISE QUADRATIC SYSTEMS 13 The conclusions on hyperbolic limit cycles of Z2,ε provided by Proposition 3.3 have assumed that P ± (0, 0) = Q± (0, 0) = 0. So, from Lemma 3.4, the parameters P ± (0, 0) and Q± (0, 0) can be used to get a pseudo-Hopf bifurcation for Z2,ε ± , which adds an extra limit cycle to each case of Proposition 3.3. This is the content of the next result. It is worthwhile to say that this is the best result so far obtained after a first order analysis for piecewise quadratic system in two zones separated by a straight line. Corollary 3.5. For |ε| > 0 sufficiently small, the maximum number of limit cycles that the system Z2,ε can have in any neighborhood of origin √is at least 5, 6, and 8 when the curve of discontinuity is {x = 0}, {y = 0}, and {y + 3x = 0}, respectively. 4. Second order perturbation In this section, in order to extend the previous results, we perform a second order analysis on piecewise quadratic perturbations of quadratic isochronous centers from the families S1 , S2 , and S3 (see (2)). More specifically, we shall apply the averaging method of second order to study the limit cycles of Zi,ε , i = 1, 2, 3. Due to the difficulties in the massive second order computations, we only perform a local study. Despite this, we shall get the best lower bounds so far for the maximum number of limit cycles of Zi,ε , i = 1, 2, 3, which are 11, 16, and 10, respectively. This proves our main result, Theorem 1.1. In Propositions 4.1 and 4.3, we provide conditions such that the second averaged functions associated to Z1,ε and Z3,ε are linear with respect to the parameters and have the highest possible rank. Under these conditions the origin is a zero of maximal finite multiplicity for F2 . Moreover, we shall see that F2 satisfies the versal unfolding property at the origin guaranteeing then the existence of the highest possible number of simple zeros near the origin and, consequently, limit cycles for Z1,ε and Z3,ε . The second order analysis for centers of the family S2 is much more difficult and the procedure used for the families S1 and S3 cannot be followed straightly for S2 . In this case, some computer assisted analyses will be needed in order to use the Poincaré– Miranda theorem, that is Theorem 2.4, to obtain analytically the existence of 16 limit cycles of Z2,ε bifurcating from the origin. This is the content of Proposition 4.7. Proposition 4.1. For |ε| > 0 sufficiently small, the maximum number of limit cycles that Z1,ε can have in any neighborhood of the origin is at least 11 when the curve of discontinuity is {x = 0}. Proof. Assume that P ± (0, 0) = Q± (0, 0) = 0. Under such condition, as in Proposi- tion 2.3 or Lemma 3.4, an extra limit cycle can always be obtained from a pseudo-Hopf bifurcation. So, the rest of the proof consists in applying the second order averaging method, Theorem 2.1(ii), to get at least 10 limit cycles bifurcating from the origin. We notice that such special condition on the perturbation will guarantee that the averaged functions F1 and F2 are well defined at the origin. The proof is structured in two parts. Firstly, we provide the expression of the function F2 (r). Secondly, we study the Taylor series of F2 around r = 0 in order to obtain the highest number of independent monomials. The first part will follow the same steps as in the proof of Proposition 3.1. In fact, the function F1 is given by (13). Then, imposing conditions such that F1 ≡ 0, that is k0 = · · · = k5 = 0 in (12), we compute the second averaged function F2 from (6) for α = 0. Proceeding with the changes of variables (8) and (9) the denominators of 14 L. P. C. DA CRUZ, D. NOVAES, AND J. TORREGROSA the functions F1± and F2± write (1 + R cos ψ)2 . Hence, the integrals Z θ √ + r1 (θ, R) = F1+ (θ, (−1 + 1 − R2 )/R)dθ, 0 Z θ √ r1− (θ − π, R) = F1− (θ − π, (1 − 1 − R2 )/R)dθ, 0 can be computed using the expressions {S, C}α=0 k,` (see (25)) for ` = 2, and k ∈ (1) (2) {0, 1, 2, 3}. We notice that F2 = F2 + F2 in (6) has two summands. The first one, which depends linearly on the parameters of second order terms (p± 2,i,j , q2,i,j in ± F2± ), has the same form as F1 . Indeed, changing the first index 1 to 2 of all the (1) (1) parameters p± 1,i,j , q1,i,j in (12) we see that F2 becomes F1 . Consequently, F2 writes ± as (13) for some new parameters k0 , k1 , . . . , k5 . The second summand, which depends quadratically on the remaining parameters of first order terms (p± 1,i,j , q1,i,j in F1 ), can ± (2) also be obtained using the integrals from Section 5. Indeed, in order to get F2 the integrals Z π √  + ∂ + 2 + G (R) = F (θ, (−1 + 1 − R )/R)r1 (θ, R) dθ, 0 ∂R 1 Z π √  − ∂ − − G (R) = F (θ − π, (1 − 1 − R2 )/R)r1 (θ − π, R) dθ, 0 ∂R 1 can be computed using the expressions {s, c}λk,` , {s, c}φk,` , {s, c}θk,` , {s, c}α=0 k,` , for ` = 3 α=0 with k ∈ {0, 1, 2, 3, 4}, and {s, c}k,` for ` = 4 with k ∈ {0, 1, 2, 3, 4, 5, 6}. Finally, taking R = −2r/(1 + r2 ) and R = 2r/(1 + r2 ) in G + (R) and G − (R), respectively, we get back the original variable r. Hence, the second averaged function writes (1) (2) (1) F2 (r) = F2 (r) + F2 (r) = F2 (r) + G + (−2r/(1 + r2 )) + G − (2r/(1 + r2 )). (19) Now, from Lemma 5.1, expression (13), and applying the change of parameters c0 =p− + − + 1,1,1 + p1,1,1 + 2 q1,2,0 + 2 q1,2,0 , − − c1 =2 p+ + + + 1,0,1 − 2 p1,0,2 − 2 p1,2,0 + 2 q1,1,0 − q1,1,1 + q1,1,1 , c2 =p+ + 1,1,0 + q1,0,1 , c3 = − 2 p + + + + 1,1,0 + 2 q1,0,1 − 2 q1,0,2 − 2 q1,2,0 , the second averaged function (19) writes   H0 (r) (1 − r2 )H1 (r) 1−r F2 (r) = 2 + log r r3 1+r     (20) (1 − r ) (1 + r2 ) 2 2 0 2r + H2 (r) log + H3 (r)Φ0 , (1 + 6r2 + r4 )3 r2 + 1 where Φ00 is defined in (27), H0 (r) and H1 (r) are polynomials of degrees 7 and 6, respectively, satisfying 8 H0 (0) = 2H1 (0), H00 (0) = 2H10 (0), H000 (0) = − H1 (0) + 2H100 (0). (21) 3 HILBERT NUMBER IN PIECEWISE QUADRATIC SYSTEMS 15 Moreover, H0 and H1 depend quadratically on the parameters ci , p± 1,i,j , q1,i,j and lin- ± early on the new parameters ki . The other two functions write 2 (r2 − 1) π H2 (r) = − c0 c1 , 16r r2 − 1 H3 (r) = c0 (c3 r4 − 2 c3 r2 − 2 c2 ). 8r (r2 + 1) The above conditions imply that F2 (r) = O(r). This concludes the first part of this proof. Now, we compute the Taylor series of F2 given in (20) around r = 0. So, n X F2 (r) = fi ri+1 + O(rn+2 ). i=0 Here, the coefficients fi are quadratic functions in the variables ci , p± 1,i,j , q1,i,j and linear ± in the variables ki , p+ 1,0,1 , q1,0,2 , q1,2,0 }. Assuming that c0 = 1, 2c2 +c3 −2p1,1,1 −4q 1,0,2 − + − + − + − 4q1,0,2 = 1, 116c2 +58c3 −116p+ + + 1,1,1 −230q1,2,0 −230q1,2,0 −59 = 1, −2c2 −c3 +2p1,1,1 +5 = 1, the system of equations {f0 = c4 , f1 = c5 , f2 = c6 , f3 = c7 , f4 = c9 , f5 = c8 , f7 = c10 , f9 = c11 } has a unique solution. Accordingly, all the perturbation parameters depend only on {c1 , . . . , c11 }. So, the second order averaged function writes 13 X F2 (r) = gi ri+1 + O(r15 ), i=0 with g0 = c4 , g1 = c5 , g2 = c6 , g3 = c7 , g4 = c9 , g5 = c8 , g7 = c10 , g9 = c11 g6 = 3517699860675πc1 + c4 − c6 + c9 , g8 = −12593243758095πc1 , g10 = 61075412843445πc1 , 63045632 3632 3632 1211 109 7 g11 = −786432c2 + c3 + c5 − c7 + c8 − c10 + c11 , 3 1287 1287 429 39 3 g12 = −304692133550805πc1 , 7088319 7088319 g13 = −2484794504 c2 + 16745167364 c3 + c5 − c7 36608 36608 56833457 1230915 209569 + c8 − c10 + c11 . 292864 6656 2048 Notice that (g0 , g1 , . . . , g13 ) is a linear function on the variable (c1 , . . . , c11 ). Since its rank with respect to (c1 , . . . , c11 ) is 11, there exists a change of variables (c1 , . . . , c11 ) 7→ (d1 , . . . , d11 ) such that X11 F2 (r) = r( di Uai (r) + O(r14 )), i=1 where Uai (r) = rai + O(r14 ), ai = i − 1, i = 1, 2, . . . , 8, a9 = 9, a10 = 11, and a11 = 13. Since F2 is analytic at r = 0, the Weierstrass Preparation Theorem (see, for instance, 16 L. P. C. DA CRUZ, D. NOVAES, AND J. TORREGROSA [8]) implies that there exists an analytic function F such that F (0) 6= 0 and 11 X F2 (r) = rF (r) di rai , i=1 This proof follows by noticing that the parameters di , i = 1, 2, . . . , 11, can be chosen (small) in order that the function F2 (r) has 10 simple zeros near the origin.  Remark 4.2. There are two main difficulties in studying the maximum number of simple zeros of (20). First, Φ00 is an integral function that cannot be expressed with simple functions. Second, the parameter coefficients of the polynomials H0 and H1 have a quadratic dependence on the parameters of Z1,ε , consequently the ECT-systems theory cannot be directly applied. Similar difficulties as pointed out by Remark 4.2 will also appear in the next two propositions. Proposition 4.3. For |ε| > 0 sufficiently small, the maximum number of limit cycles that Z3,ε can have in any neighborhood of the origin is at least 10 when the curve of discontinuity is {x = 0}. Proof. The proof follows the same steps as the proof of Proposition 4.1. We recall that the first order analysis has been performed in Proposition 3.2. Again, an extra limit cycle can be obtained from a pseudo-Hopf bifurcation, so we may assume that P ± (0, 0) = Q± (0, 0) = 0. Then, the proof will consists in applying the second order averaging method to get at least 9 limit cycles bifurcating from the origin. Firstly, using (7) for α = 0, the functions r1+ (θ, R) and r1− (θ − π, R) write Z θ + r1 (θ, R) = F1+ (θ, −R/8)dθ, 0 Z θ r1− (θ − π, R) = F1− (θ − π, R/8)dθ. 0 The above integrals can be computed using the expressions from Section 5, {S, C}α=0 k,` for ` = 4, and k = 0, 1, . . . , 6. Then, the second summand of the second averaged (2) function, F2 , follows from the integrals Z π  + ∂ + + G (R) = F (θ, −R/8)r1 (θ, R) dθ, 0 ∂R 1 Z π  − ∂ − − G (R) = F (θ − π, R/8)r1 (θ − π, R) dθ, 0 ∂R 1 which can be computed using the expressions from Section 5, {s, c}λk,` , {s, c}φk,` , {s, c}θk,` , {s, c}α=0 k,` for ` = 5 with k = 0, 1, . . . , 7, and {s, c}α=0 k,` with ` = 8 and k = 0, 1, . . . , 12. So, going back to the original variable r we get (1) (2) (1) F2 (r) = F2 (r) + F2 (r) = F2 (r) + G + (−8r) + G − (8r). Again, from Lemma 5.1 and expression (16) we get   H0 (r) H1 (r) 1−r F2 (r) = 2 + log . r (1 − r4 )2 r3 (1 + r2 ) 1+r Here, H0 and H1 are polynomials of degree 13 and 8, respectively, and satisfy the relations (21). As previously, with these conditions, we have F2 (r) = O(r). HILBERT NUMBER IN PIECEWISE QUADRATIC SYSTEMS 17 Now, computing the Taylor series of F2 around r = 0 we get n X F2 (r) = fi ri + O(rn+1 ). (22) i=1 In order to simplify the expression of F2 (r), we introduce the new parameters, − − + + c1 = 16 q1,1,0 + 3 q1,1,1 − 16 q1,1,0 − 3 q1,1,1 , − − c2 = −16 q1,1,0 + 4 p+ + + 1,0,2 − 3 q1,1,1 + 16 q1,1,0 + 3 q1,1,1 , − − c3 = −48 q1,1,0 + 12 p+ + + + 1,0,2 + 8 p1,2,0 − 9 q1,1,1 + 48 q1,1,0 + 7 q1,1,1 , − c4 = 184 p+ + 1,1,1 + 21 q1,0,2 + 71 q1,0,2 , − + c5 = q1,0,2 + q1,0,2 , − + c6 = q1,2,0 + q1,2,0 , − c7 = 200 p+ + + 1,1,0 + 579 p1,1,1 − 825 q1,2,0 − 1300 q1,0,1 , c8 = 16 p− + − + 1,0,1 − 16 p1,0,1 − 3 q1,1,1 + 3 q1,1,1 , c9 = 16 p+ + 1,0,1 − 3 q1,1,1 , + c10 = q1,1,0 + p+ 1,0,1 , c11 = 800 p+ + + 1,1,0 − 1413 p1,1,1 − 800 q1,0,1 , 789 + 1 71 184 4 2 c12 = − p1,1,1 − c4 + c5 + c6 + c7 − c11 . 6125 24500 24500 18375 67375 28875 We see that the coefficients fi in (22) depend linearly on the second order parameters, 2,i,j and q2,i,j , and quadratically on the new parameters ci . Under the assumption p± ± P ± (0, 0) = Q± (0, 0) = 0, the first averaged function, studied in Proposition 3.2, provides only the first 5 linearly independent coefficients. Thus, there exists a trans- formation on the parameters space such that fi = di for i = 1, . . . , 5, and, under the condition c1 = 1, c5 = c6 = c8 = c10 = 0, the system {f6 = d6 , f7 = d7 , f8 = d8 , f9 = d9 , f10 = d10 } has a unique solution with respect to {c2 , c3 , c4 , c7 , c9 }. Therefore, we get 10 X F2 (r) = di ri + O(r11 ). i=1 In this case, the multiplicity of the origin cannot be increased because the coefficients d11 and d12 depend linearly on the parameters {d1 , . . . , d10 }. In fact, d11 = d5 + d7 − d9 76 37 502 116 2 d12 = − d2 − d4 + d6 + d8 − d10 . 715 65 715 65 5 Finally, this proof follows by noticing that the parameters di , i = 1, 2, . . . , 9, can be chosen (small) in order that the function F2 (r) has 9 simple zeros near the origin.  Next technical results, whose proofs are straightforward, provide lower and upper bounds for the values that a polynomial of n variables take in a n dimensional poly- hedron. Moreover, they will be useful for proving the last proposition of this section concerning the second order analysis of the system Z2,ε . 18 L. P. C. DA CRUZ, D. NOVAES, AND J. TORREGROSA Lemma 4.4. Consider h > 0, p > 0, q real numbers such that p ∈ [p, p], with pp > 0, and q ∈ [q, q], with qq > 0. (i) Then, σ ` (q, p) ≤ qp ≤ σ r (q, p), ( ( q p, if q > 0, q p, if q > 0, where σ ` (q, p) = and σ r (q, p) = q p, if q < 0, q p, if q < 0. (ii) If uj ∈ [−h, h], for j = 1, . . . , n, and denoting ui = ui11 ui22 · · · uinn for i = (i1 , . . . , in ) 6= 0, we have χ` (q, ui ) ≤ qui ≤ χr (q, ui ), where   0, if q > 0 and ik even for all k = 1, . . . , n, ` i χ (q, u ) = −q h 1 i +···+i n , if q > 0 and ik odd for some k = 1, . . . , n,  q hi1 +···+in , if q < 0, and   q h 1 i +···+in , if q > 0, r i χ (q, u ) = 0, if q < 0 and ik even for all k = 1, . . . , n,  −q hi1 +···+in , if q < 0 and i odd for some k = 1, . . . , n. k Furthermore, χ (q, 1) = q and χ (q, 1) = q. ` r Lemma 4.5. Let h > 0 and pj be a positive non rational numbers such that pj ∈ [pj , pj ] with pj , pj rational numbers satisfying pj pj > 0, for j = 1, . . . , m. Consider the polynomial M X X m  U(u1 , . . . , un ) = Uj,i pj ui , (23) i1 +···+in =0 j=1 with ui = ui11 · · · uinn , i = (i1 , . . . , in ), and Uj,i rational numbers. Then, m X Ui` ≤ Uj,i pj ≤ Uir j=1 Pm P with Ui` = j=1 Uj,i · σ ` (Uj,i , pj ) and Uir = m j=1 Uj,i · σ (Uj,i , pj ). Moreover, if uj ∈ r [−h, h], for j = 1, . . . , n, and Ui` · Uir > 0 then M X M X ` U= χ (Ui` , ui ) ≤ U(u1 , . . . , un ) ≤ χr (Uir , ui ) = U. i1 +···+in =0 i1 +···+in =0 The next example shows how the above two technical lemmas can be used to get rational lower and upper bounds for the values that a function takes in a given 3D- √ polyhedron. In this example the lower and upper bounds for the values of π and 3 are chosen from their continued fraction. Example 4.6. Consider the polynomial P (u, v, w) = P0 + P1 u + P2 v 2 w2 + P3 uv 2 w4 , (24) √ √ √ √ with P0 = π−5 3+4, P1 = −π 2 +3 3−3, P2 = −2π √ − 3+70, and P3 = 4π+ 3+7. 3 Consider the following intervals containing π and 3,       333 355 √ 5 7 π ∈ p, p = , and 3 ∈ [s, s] = , . 106 113 3 4 HILBERT NUMBER IN PIECEWISE QUADRATIC SYSTEMS 19 Then, for u, v, w ∈ [−1/9, 1/9], we have   8036904331130 5753192708807927 P (u, v, w) ∈ − ,− . 3236907751533 18184947748112394 Proof.√Following the notation of Lemma 4.5, we take p1 = π, p2 = π 2 , p3 = π 3 , and p4 = 3. Then, the coefficients of (24) write as P0 = p1 − 5p4 + 4, P1 = −p2 + 3p4 − 3, P2 = −2p3 − p4 + 70, and P3 = 4p1 + p4 + 7. The intervals given in the statement provide that {p1 , p4 , p2 } ⊂ [p2 , p2 ] and p3 ∈ [p3 , p3 ]. So, these new variables allow us to define the coefficients Uj,i in (23). From Lemma 4.4(i) we have 341 ` r 404 − = p − 5s + 4 = U000 ≤P0 ≤ U000 = p − 5s + 4 = − , 212 339 100487 21402 − = −p2 + 3s − 3 = U100 ` ≤P1 ≤ U100r = −p2 + 3s − 3 = − , 12769 2809 35999881 11301029 = −2p3 − s + 70 = U022 ` ≤P2 ≤ U022r = −2p3 − s + 70 = , 5771588 1786524 3376 ` r 9635 = 4p + s + 7 = U124 ≤P3 ≤ U124 = 4p + s + 7 = . 159 452 Finally, applying Lemma 4.4(ii) we get the following lower and upper bounds for P, ` ` r U000 + U100 h − U124 h7 = U ≤ P (u, v, w) ≤ U = U000 r ` − U100 r h + U022 h4 + U124 r h7 . The proof follows by substituting h = 1/2 in the above expression.  The last proposition deals with second order perturbation of family S2 , which exhibits the highest number of limit cycles found in this paper. Theorem 1.1 is a direct consequence of it. In Proposition 3.3 we have studied the zeros of the first averaged function for three different √ straight lines of discontinuity. The best result was obtained when Σ = {y + 3x = 0}. So, we shall perform the second order analysis only in this case. Proposition 4.7. For |ε| > 0 sufficiently small, the maximum number of limit cycles that Z2,ε can have in √ any neighborhood of the origin is at least 16 when the curve of discontinuity is {y + 3x = 0}. Proof. The proof follows basically the same steps as the proofs of Propositions 4.1 and 4.3. Nevertheless, in this case, some of the integrals of F2 cannot be explicitly obtained. Then, since the functions are analytic near the origin, we compute the Taylor series of the integrand before integrating. We recall that the first order analysis has been performed in Proposition 3.3. Again, an extra limit cycle can be obtained from a pseudo-Hopf bifurcation, so we may assume that P ± (0, 0) = Q± (0, 0) = 0. Then, the proof will consists in applying the second order averaging method to get at least 15 limit cycles bifurcating from the origin. Using (7) for α = −π/3, the functions r1+ (θ, R) and r1− (θ − π, R) write Z −π/3+θ + r1 (θ, R) = F1+ (θ, R)dθ, −π/3 and Z −π/3+θ r1− (θ − π, R) = F1− (θ − π, −R)dθ. −π/3 20 L. P. C. DA CRUZ, D. NOVAES, AND J. TORREGROSA α=−π/3 The above integrals can be computed using the expressions for {S, C}k,` , for ` = 1 and k = 0, 1, 2, 3, from Section 5. The second summand of the second averaged (2) function, F2 , follows from the integrals Z 2π/3   + ∂ + + G (R) = F (θ, R)r1 (θ, R) dθ, −π/3 ∂R 1 Z 2π/3   − ∂ − − G (R) = F (θ − π, −R)r1 (θ − π, −R) dθ. −π/3 ∂R 1 We point out that the integrands of the above integrals are rational functions with denominators (1 + R cos θ)2 and numerators depending on {R, θ, cos θ, sin θ, λ(R, −π/3 + θ), λ(R, −4π/3 + θ), φ(R, −π/3 + θ), φ(R, −4π/3 + θ)}. Computing the Taylor series around R = 0, integrating on the interval [−π/3, 2π/3], and going back to the original variable r, we get the Taylor series around r = 0 of (2) (1) F2 (r). The Taylor series of F2 (r) is obtained analogously to the series of F1 (r) in Proposition 3.3. Accordingly, the second averaged function writes Xn F2 (r) = fi ri + O(rn+1 ). i=1 The coefficients fi ’s depend linearly on {p± 2,i,j , q2,i,j } and quadratically on {p1,i,j , q1,i,j }. ± ± ± The rest of the proof is devoted to show that there exists a transformation on the parameters such that the above function becomes 16 X F2 (r) = di ri + O(rn+1 ), i=1 where d1 , . . . , d16 are independent parameters. In fact, we shall prove the existence of a transversal curve of weak foci of order 16. The transversality also guarantees the unfolding of 15 simple zeros near the origin because our function, also the per- turbed one, vanishes at zero. The existence of such curve is obtained in two steps. Firstly, we analyse the maximal rank (f1 , . . . , fn ) with respect to the linear parameters {p±2,i,j , q2,i,j }. Secondly, proceeding with a change of parameters, which eliminates the ± linear terms, we study the quadratic terms regarding {p± 1,i,j , q1,i,j } from (f1 , . . . , fn ). ± We shall see that these quadratic terms are homogeneous and we show the existence of a transversal straight line such that these terms vanish on it. The described pro- cedure is detailed in [9, Theorems. 2.1 and 3.1]. These ideas have been originally introduced in [6, 7] for quadratic vector fields and have also been employed in [16] for Liénard families. Firstly, we see that the system of equations {f1 = d1 , . . . , f6 = d6 , f8 = d8 , f10 = d10 } has a unique solution with respect to the variables p+ + + + + + 2,1,0 , p2,0,2 , p2,1,1 , p2,2,0 , q2,2,0 , q2,0,1 , 2,1,1 , and q2,2,0 . Hence, after this change, the rest of coefficients remains quadratic. p− − Secondly, assuming f16 6= 0 we shall obtain a transversal solution of the quadratic system S : {f7 = f9 = f11 = f12 = f13 = f14 = f15 = 0}. Since there are more parameters than necessary, we impose that {p− + 1,2,0 = p1,1,1 = − − − − − + + + + + q1,0,1 = q1,0,2 = q1,1,0 = q1,1,1 = q1,2,0 = q1,0,1 = q1,0,2 = q1,1,0 = q1,1,1 = q1,2,0 = HILBERT NUMBER IN PIECEWISE QUADRATIC SYSTEMS 21 0, p+1,2,0 = 1}. Furthermore, it is not restrictive to assume that the first parameters d1 , d2 , . . . , d6 , d8 , and d10 vanish. For the sake of simplicity, we change the names of the remaining parameters [p− 1,0,1 , p1,1,0 , p1,0,1 , p1,1,0 , p1,0,2 , p1,1,1 , p1,0,2 ] to [z1 , . . . , z7 ]. In − + + − − + order to solve the quadratic system S we consider two quadratic subsystems, namely S1 = {f7 = f9 = f11 = f13 = f15 = 0} and S2 = {f12 = f14 = f15 = 0}. Then, we study the intersection between their solutions. Using the condition f15 = 0, the subsystem S1 can be rewritten in order that all the equations depend linearly on z1 , z2 , z3 , and z4 . So, solving S1 in these parameters we get ζi (z5 , z6 , z7 ) zi = , for i = 1, 2, 3, 4, η(z5 , z6 , z7 ) where ζi are polynomials of degree 5 and η is a polynomial of degree 4. Later on, we shall see that η does not vanish at the intersection point. Accordingly, the parameters f12 , f14 , and f15 write fe12 (z5 , z6 , z7 ) fe14 (z5 , z6 , z7 ) fe15 (z5 , z6 , z7 ) f12 = , f 14 = , f 15 = . (η(z5 , z6 , z7 ))2 (η(z5 , z6 , z7 ))2 (η(z5 , z6 , z7 ))2 where fe12 , fe14 , and fe15 are polynomials of degree 10. So, on the variety provided by S1 , the subsystem S2 is equivalent to Se2 : {fe12 (z5 , z6 , z7 ) = 0, fe13 (z5 , z6 , z7 ) = 0, fe14 (z5 , z6 , z7 ) = 0}, provided that η(z5 , z6 , z7 ) 6= 0. Consequently, the system S is reduced to Se2 whenever η(z5 , z6 , z7 ) 6= 0. Although Se2 has only 3 equations and 3 unknowns, the high degree of these equations is a barrier for solving the system. Furthermore, the algebraic varieties provided by each equation of Se2 are numerically close to each other (see Figure 4), which adds an extra numerical difficult. z5 z6 z7 Figure 4. Plot of the varieties fe12 = 0, fe14 = 0, and fe15 = 0 in a cube centered at (z5∗ , z6∗ , z7∗ ) with edges of length 10−2 . They are depicted in red, blue, and green, respectively. In order to overcome these difficulties, we shall first work with numerical approx- imations of the solutions. Then, using Lemmas 4.4 and 4.5, and Theorem 2.4 we prove analytically the numerical results. Working with enough precision we get the following numerical solution of the system Se2 , (z5∗ , z6∗ , z7∗ ) ≈ (−0.260976000571, 0.111582119099, −0.84487667629841). 22 L. P. C. DA CRUZ, D. NOVAES, AND J. TORREGROSA Moreover, we see that η(z5∗ , z6∗ , z7∗ ) ≈ 1.85317498452382 · 10−11 and fe16 (z5∗ , z6∗ , z7∗ ) = 8.8767062451915 · 10−26 , so f16 = fe16 (z5 , z6 , z7 )/(η(z5 , z6 , z7 ))2 6= 0. Additionally, the intersection is transversal because the determinant of the Jacobian matrix of f = (fe12 , fe14 , fe15 ) with respect to (z5 , z6 , z7 ) evaluated at the solution (z5∗ , z6∗ , z7∗ ) does not vanish. In fact, Jf (z5∗ , z6∗ , z7∗ ) ≈ −5.379835263496 · 10−67 . It is worthwhile to mention that although the values for η, fe16 , and Jf are very small at (z5∗ , z6∗ , z7∗ ), we were able to observe that they remain fixed when we increase the precision of the computations, while the values for fe12 , fe14 , and fe15 decrease to zero. Finally, we shall prove analytically the existence of such transversal intersection point (z5∗ , z6∗ , z7∗ ). In order to apply Lemmas 4.4 and 4.5, we make the following change of variables, 102563793961 93673471843 5228323783 1104348344 z5 = − u1 + u2 − u3 − , 75692301 117235838 13687494949 4231608813 114951879798 66846520379 1728113446 859801297 z6 = − u1 + u2 + u3 + , 118751113 116131808 4218432187 7705547304 131538341646 23870722389 1387092713 2790022856 z7 = u1 − u2 + u3 − . 188147809 57947275 5464980203 3302284149 Then, fe12 = ξ1 (u1 , u2 , u3 ), fe14 = ξ2 (u1 , u2 , u3 ), fe15 = ξ3 (u1 , u2 , u3 ), fe16 = ξ4 (u1 , u2 , u3 ), η = ξ5 (u1 , u2 , u3 ), Jf = ξ6 (u1 , u2 , u3 ). Now, the problem is to check that the varieties defined by ξ1 = 0, ξ2 = 0, and ξ3 = 0 intersect transversally near the origin (see Figure 5). u3 0 u2 u1 Figure 5. Plot of ξ1 = ξ2 = ξ3 = 0 in a neighborhood of the origin. The varieties are drawn in green, blue, and red, respectively. The length of the edges of the 3d-cube is 10−10 . We notice that the functions ξi , i = 1, . . . , 6, are polynomials in (u1 , u2 , u3 ). More- over, ξi , i = 1, . . . , 4 have degree 10, ξ5 has degree 4, and ξ6 has degree 27. We see that √ the coefficients of the previous polynomials depend on the irrational numbers π and 3. More specifically, the coefficients of ξi , i = 1, . . . , 4, depend on π up to power 12, √ ξ5 depends on π up to power 5, and ξ6 depends on π up to power 37. The number 3 appears in these coefficients with no exponent. √ For each function ξi , let µi denote the degree with respect to π. Then, substituting 3, π, π , . . . , πiµ by p1 , p2 , . . . , pµi +1 , 2 respectively, all the coefficients of the polynomial ξi (p1 , p2 , . . . , pµi +1 , u1 , u2 , u3 ) are HILBERT NUMBER IN PIECEWISE QUADRATIC SYSTEMS 23 now rational numbers defined as the quotient of two big integers, around 1000 figures each. Moreover, they have 6292, 6292, 6006, 6292, 220 and 234668 monomials. Finally, we can apply Theorem 2.4 to (ξ1 , ξ2 , ξ3 ) in B = [−h, h]3 , with h = 10−10 , because ξ1 (B1− ) ⊂ [a− − 1 , b1 ] ≈ [−3.58842524 · 10 −32 , −3.20226086 · 10−32 ], −32 ξ1 (B1+ ) ⊂ [a+ + 1 , b1 ] ≈ [3.28009013 · 10 , 3.66625367 · 10−32 ], ξ2 (B2− ) ⊂ [a− − 2 , b2 ] ≈ [−3.70185379 · 10 −32 , −3.04215854 · 10−32 ], −32 ξ2 (B2+ ) ⊂ [a+ + 2 , b2 ] ≈ [3.16666084 · 10 , 3.82635712 · 10−32 ], ξ3 (B3− ) ⊂ [a− − 3 , b3 ] ≈ [−4.72369496 · 10 −32 , −1.41476503 · 10−32 ], −32 ξ3 (B3+ ) ⊂ [a+ + 3 , b3 ] ≈ [2.14481860 · 10 , 5.45375151 · 10−32 ]. We notice that the values for a± i and bi , for i = 1, 2, 3, are all rational numbers ± explicitly computed using Lemmas 4.4, 4.5, and     21053343141 1783366216531 √ 716035 978122 π∈ , , 3∈ , . 6701487259 567663097408 413403 564719 Hence, we conclude that there exists a point (ξ1∗ , ξ2∗ , ξ3∗ ) in B such that ξ1 = ξ2 = ξ3 = 0. Additionally, ξ4 , ξ5 , ξ6 do not vanish on B. Indeed, ξ4 (B) ⊂ [a− − 4 , b4 ] ≈ [8.87669600 · 10 −26 , 8.87671664 · 10−26 ], ξ5 (B) ⊂ [a− − 5 , b5 ] ≈ [1.85317477 · 10 −11 , 1.85317520 · 10−11 ], −67 ξ6 (B) ⊂ [a+ + 6 , b6 ] ≈ [−5.37983643 · 10 , −5.37983443 · 10−67 ], Again, the values for a±i and bi , i = 4, 5, 6, are rational numbers explicitly computed ± using Lemmas 4.4 and 4.5. So, we conclude that the solution provided by Theorem 2.4 is a transversal solution of S.  5. Appendix: Explicit computations of the integrals This section is devoted to provide explicit expressions for some of the integrals necessary to compute the averaged functions. We also introduce new special integral functions as well as some of their properties and relations. The proofs follow closely the results from [30]. For each pair of natural numbers k and `, we define the following functions: Z α+θ Z α+θ α sin(kψ) α cos(kψ) Sk,` (r, θ) = ` dψ, Ck,` (r, θ) = dψ, α (1 + r cos ψ) α (1 + r cos ψ)` Z α+π Z α+π α sin(kθ) α cos(kθ) sk,` (r) = ` dθ, ck,` (r) = dθ, α (1 + r cos θ) α (1 + r cos θ)` Z π Z π sin(kθ) λ(r, θ) cos(kθ) λ(r, θ) λ sk,` (r) = ` dθ, λ ck,` (r) = dθ. (25) 0 (1 + r cos θ) 0 (1 + r cos θ)` Z π Z π φ sin(kθ) φ(r, θ) φ cos(kθ) φ(r, θ) sk,` (r) = dθ, c k,` (r) = dθ, 0 (1 + r cos θ)` 0 (1 + r cos θ)` Z π Z π θ θ sin(kθ) θ θ cos(kθ) sk,` (r) = ` dθ, ck,` (r) = ` dθ. 0 (1 + r cos θ) 0 (1 + r cos θ) 24 L. P. C. DA CRUZ, D. NOVAES, AND J. TORREGROSA Here, r ∈ (−1, 1), θ ∈ [−π, π], and φ, λ are the two periodic functions r  !! 1 1−r θ φ(r, θ) = √ θ − 2 arctan tan , 1−r 2 1+r 2 (26) λ(r, θ) = log(1 + r cos θ). Lemma 5.1. Let φ, λ be the functions defined by (26). Then, φ(r, 0) = φ(−r, 0) = 0, φ(r, π) = φ(r, −π) = 0, and φ(−r, t + π) = φ(r, t). Moreover, ∂ rφ(r, θ) sin θ φ(r, θ) = 2 + 2 , ∂r 1−r (1 − r )(1 + r cos θ) ∂ cos θ λ(r, θ) = . ∂r 1 + r cos θ Proof. The first properties follow simply by substituting θ = 0, π in the definition of φ. The last is satisfied because F (r, 0) = 0 and the derivative, with respect to θ, of F (r, θ) = φ(−r, θ + π) − φ(r, θ) vanishes identically. The expressions of their derivatives are easily to be checked.  We notice that not all the above integrals can be explicitly obtained. So, next lemma introduce some new functions. They, together with their derivatives, are useful for the proofs of the results. Lemma 5.2. Let Φ00 (r), Λ00 (r), and Λ01 (r) be the functions Z π 0 Φ0 (r) = φ(r, θ)dθ, 0 Z π  √  2(1 − 1 − r2 ) 0 Λ0 (r) = λ(r, θ)dθ = −π log , (27) 0 r2 Z π 0 λ(r, θ) π log(1 − r2 ) − Λ00 (r) Λ1 (r) = dθ = √ . 0 1 + r cos θ 1 − r2 Then, their derivatives can be expressed explicitly as functions of (26) and (27), d 0 r2 Φ00 (r) + λ(r, 0) − λ(−r, 0) Φ0 (r) = dr r(1 − r2 ) √ d 0 π(1 − 1 − r2 ) Λ0 (r) = − √ , dr r 1 − r2  2 √  d 0 1 0 (1 + r − 1 − r2 )π Λ (r) = r Λ1 (r) − √ . dr 1 1 − r2 r 1 − r2 The following results give recurrent formulas in terms of k and ` for all the functions defined at the beginning of this section. Proposition 5.3. The functions Sk,` α and Ck,`α , defined in (25), write as    0 k = 0, ` ≥ 0,    −1   cos(kα) − cos(k(θ + α) k k ≥ 1, ` = 0,    −1  α Sk,` (r, θ) = λ(r, α) − λ(r, θ + α) r k = 1, ` = 1,   1−` (1 + r cos(θ + α)) − (1 + r cos α) 1−`     k = 1, ` ≥ 2,   r(` − 1)   α 2 Sk−1,`−1 α (r, θ) − Sk−1,` (r, θ) r−1 − Sk−2,` α (r, θ) k ≥ 2, ` ≥ 1, HILBERT NUMBER IN PIECEWISE QUADRATIC SYSTEMS 25    θ k = 0, ` = 0,      sin(k(θ + α)) − sin(kα) k −1 k ≥ 1, ` = 0,     1  φ(r, α) − φ(r, θ + α) + θ(1 − r2 )− 2 k = 0, ` = 1, α Ck,` (r, θ) = r ∂ α   α C0,`−1 (r, θ) + C (r, θ) k = 0, ` ≥ 2,   ` − 1 ∂r 0,`−1  α    α C0,`−1 (r, θ) − C0,` (r, θ) r−1 k = 1, ` ≥ 1,    2 C α α  −1 α k−1,`−1 (r, θ) − Ck−1,` (r, θ) r − Ck−2,` (r, θ) k ≥ 2, ` ≥ 1, when r 6= 0. Furthermore, Sk,` α α (0, θ) = Sk,0 (r, θ) and Ck,` α α (0, θ) = Ck,0 (r, θ). Proof. The expressions of Sk,` α (r, θ) for k = 0, 1 follow by direct integration. When k ≥ 2 and ` ≥ 2, from its definition and by using elementary transformations, we get Z α+θ Z α+θ α sin((k−1)ψ) sin((k−1)ψ)(1+cos ψ) Sk−1,`−1 (r, θ) = dψ = dψ α (1+r cos ψ)`−1 α (1+r cos ψ)` Zα+θ Z α+θ (28) sin((k−1)ψ) sin((k−1)ψ) cos ψ dψ + r dψ. α (1+r cos ψ)` α (1+r cos ψ)` Using the identity 2 sin((k −1)ψ) cos ψ = sin(kψ)+sin((k −2)ψ) the above expression writes α α 1 α 1 α Sk−1,`−1 (r, θ) = Sk−1,` (r, θ) + r Sk,` (r, θ) + r Sk−2,` (r, θ). 2 2 Then, solving Sk,` α (r, θ) in this expression we recover the one appearing in the state- ment. The expression for Ck,0 α (r, θ) follows by a direct integration, whereas C0,1 α follows from the definition of φ given in (26) and the change of variables tan(ψ/2) = ϕ. The expression of C0,`α (r, θ), for ` ≥ 2, follows deriving with respect to r. The expression for Ck,` (r, θ) can be obtained analogously to Sk,` α α (r, θ).  The next corollary follows straightaway by evaluating θ = π in the last result. Corollary 5.4. The functions sαk,` and cαk,` defined in (25) write as    0 k = 0, ` ≥ 0,       (λ(r, α) − λ(r, α + π)) r−1 k = 1, ` = 1,    (1 − (−1)k ) α sk,` (r) = cos(kα) k ≥ 1, ` = 0,   k  (1 + r cos α)1−` − (1 − r cos α)1−`    k = 1, ` ≥ 2,   r(1 − `)    2 s α (r) − sα (r) r−1 − sα k ≥ 2, ` ≥ 1, k−1,`−1 k−1,` k−2,` (r) 26 L. P. C. DA CRUZ, D. NOVAES, AND J. TORREGROSA    π k = 0, ` = 0,     π   φ(r, α) − φ(r, α + π) + √ k = 0, ` = 1,   1 − r2   r d   cα0,`−1 (r) + cα0,`−1 (r) k = 0, ` ≥ 2, α ck,` (r) = ` − 1 dr   −1 + (−1)k   sin(kα) k ≥ 1, ` = 0,   k   α    c0,`−1 (r) − cα0,` (r) r−1 k = 1, ` ≥ 1,    −1  2 c α α k−1,`−1 (r) − ck−1,` (r) r − cαk−2,` (r) k ≥ 2, ` ≥ 1, when r 6= 0. Furthermore, sα0,` (0) = 0, cα0,` (0) = π for ` ≥ 0 and 1 − (−1)k −1 + (−1)k sαk,` (0) = cos(kα), cαk,` (0) = sin(kα), k k for k ≥ 1 and ` ≥ 0. Proposition 5.5. The functions sλk,` and cλk,` , defined in (25), write as    0 k = 0, ` ≥ 0,    k  λ(r, 0) − (−1) λ(−r, 0) r   + (sk−1,1 (r) − sk+1,1 (r)) k ≥ 1, ` = 0,   k 2k  2 2 sλk,` (r) = λ(r, 0) − λ(−r, 0) k = 1, ` = 1,   2r   1  λ(−r, 0) + (` − 1)−1 λ(r, 0) + (` − 1)−1      − k = 1, ` ≥ 2,   r(` − 1) (1 − r)`−1 (1 + r)`−1  λ 2 sk−1,`−1 (r) − sλk−1,` (r) r−1 − sλk−2,` (r) k ≥ 2, ` ≥ 1,  0  Λ` (r) k = 0, ` = 0, 1,       r d λ   λ  ` − 1 dr c0,`−1 (r) − c1,` (r) + c0,`−1 (r) k = 0, ` ≥ 2,  cλk,` (r) = r  (ck−1,1 (r) − ck+1,1 (r)) k ≥ 1, ` = 0,   2k      cλ0,`−1 (r) − cλ0,` (r) r−1 k = 1, ` ≥ 1,     2 cλk−1,`−1 (r) − cλk−1,` (r) r−1 − cλk−2,` (r) k ≥ 2, ` ≥ 1, when r 6= 0. Furthermore, sλk,` (0) = sk,` (0) and cλk,` (0) = 0 for k ≥ 0 and ` ≥ 0. Proof. The general expression for k ≥, ` ≥ 1 follows similarly to (28). The other cases follow straightforward. In some of them, the integration by parts rule is necessary and also the fact that cλk,` is an even function.  HILBERT NUMBER IN PIECEWISE QUADRATIC SYSTEMS 27 Proposition 5.6. The functions sφk,` and cφk,` , defined in (25), write as    0 k = 0, ` ≥ 0,     ck,1 (r)   − k ≥ 1, ` = 0,    Λ0k(r)  0 Λ (r) sφk,` (r) = √0 − 1 k = 1, ` = 1,   r 1 − r 2 r    1 c0,`−1    c 0,` − √ k = 1, ` ≥ 2,   r(`  − 1) 1 − r 2    2 s φ φ −1 − sφk−2,` (r) k ≥ 2, ` ≥ 1, k−1,`−1 (r) − sk−1,` (r) r  0   Φ0 (r) k = 0, ` = 0,    2   cθ0,1 (r) 1 π   √ √   − k = 0, ` = 1,   1 − r  2 2 1 − r2    r d s (r) `(1 − r2 ) − 1 φ   φ 1,` c0,`−1 (r) − + c (r) k = 0, ` ≥ 2, cφk,` (r) = ` − 1 dr 1 − r2 (` − 1)(1 − r2 ) 0,`−1   sk,1 (r) (−1)k − 1   + √ k ≥ 1, ` = 0,   k k 2 1 − r2        cφ0,`−1 (r) − cφ0,` (r) r−1 k = 1, ` ≥ 1,      2 c φ k−1,`−1 (r) − cφ (r) r−1 − cφ (r) k−1,` k−2,` k ≥ 2, ` ≥ 1, when r 6= 0. Furthermore, sφk,` (0) = cφk,` (0) = 0. Proof. The expression of sφk,` follows from the fact that it is an even function and the results in [30]. For k = 1, ` ≥ 1 and k ≥ 2, ` ≥ 1, proceeding analogously to the previous proofs, we get cφk,` . Finally, for k = 0, ` ≥ 2, we compute directly the derivative, with respect to r, of cφ0,`−1 . The other cases follow using the integration by parts rule.  The last result follows similarly as all the previous results. Proposition 5.7. The functions sθk,` and cθk,` defined in (25) write as    0 k = 0, ` ≥ 0,     −(−1)k π   k ≥ 1, ` = 0,   k   θ sk,` (r) = − πλ(−r, 0) + Λ00 (r) r−1 k = 1, ` = 1,       1 π   − c0,`−1 (r) k = 1, ` ≥ 2,   r(` − 1) (1 − r)`−1     2 s θ (r) − s θ (r) r−1 − sθk−2,` (r) k ≥ 2, ` ≥ 1, k−1,`−1 k−1,` 28 L. P. C. DA CRUZ, D. NOVAES, AND J. TORREGROSA    π2   k = 0, ` = 0,   2     π2   Φ 0 (r) + √ k = 0, ` = 1,   0 2 1 − r2    r d θ θ cθk,` (r) = ` − 1 dr c0,`−1 (r) + c0,`−1 (r) k = 0, ` ≥ 2,   −1 + (−1)k     k ≥ 1, ` = 0,    k2      c0,`−1 (r) − c0,` (r) r−1 θ θ k = 1, ` ≥ 1,       2 c θ k−1,`−1 (r) − cθ (r) r−1 − cθ k−1,` k−2,` (r) k ≥ 2, ` ≥ 1, when r 6= 0. Furthermore, sθ0,` (0) = 0, cθ0,` (0) = π 2 /2 for ` ≥ 0 and −(−1)k π −1 + (−1)k sθk,` (0) = , cθk,` (0) = . k k2 for k ≥ 1 and ` ≥ 0. Acknowledgements This work has been realized thanks to the Spanish MINECO MTM2016-77278-P (FEDER) grant, Catalan AGAUR 2017 SGR 1617 grant, and the Brazilian FAPESP 2016/11471-2 and CAPES BEX 13473/13-1 grants. References [1] V. Acary, O. Bonnefon, and B. Brogliato. Nonsmooth modeling and simulation for switched circuits, volume 69 of Lecture Notes in Electrical Engineering. Springer, Dordrecht, 2011. [2] N. N. Bautin. 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A concrete example of the existence of four limit cycles for plane quadratic systems. Sci. Sinica, 23(2):153–158, 1980. [32] H. Żołądek. Eleven small limit cycles in a cubic vector field. Nonlinearity, 8(5):843–860, 1995. [33] H. Żołądek. The cd45 case revisited. In Mathematical Sciences with Multidisciplinary Applica- tions, pages 595–625, Cham, 2016. Springer International Publishing. Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bel- laterra, Barcelona, Catalonia, Spain Email address: leonardo@mat.uab.cat Departamento de Matemática, Universidade Estadual de Campinas, Rua Sérgio Buarque de Holanda, 651, Cidade Universitária Zeferino Vaz, 13083–859, Campinas, SP, Brazil Email address: ddnovaes@ime.com Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bel- laterra, Barcelona, Spain Email address: torre@mat.uab.cat