ON THE NUMBER OF LIMIT CYCLES FOR A GENERALIZATION OF LIÉNARD POLYNOMIAL DIFFERENTIAL SYSTEMS

where g1(x) = εg11(x)+ε g12(x)+ε g13(x), g2(x) = εg21(x) + ε g22(x) + ε g23(x) and f(x) = εf1(x) + εf2(x) + ε f3(x) where g1i, g2i, f2i have degree k, m and n respectively for each i = 1, 2, 3, and ε is a small parameter. Note that when g1(x) = 0 we obtain the generalized Liénard polynomial differential systems. We provide an upper bound of the maximum number of limit cycles that the previous differential system can have bifurcating from the periodic orbits of the linear center ẋ = y, ẏ = −x using the averaging theory of third order.


Introduction and statement of the main results
The second part of the 16th Hilbert's problem wants to find an upper bound for the maximum number of limit cycles that a polynomial vector field of a fixed degree can have.In this paper we will try to give a partial answer to this problem for the class of polynomial differential systems ẋ = y − g 1 (x), ẏ = −x − g 2 (x) − f (x)y, (1) where g 1 (x) = εg 11 (x) + ε 2 g 12 (x) + ε 3 g 13 (x), g 2 (x) = εg 21 (x) + ε 2 g 22 (x) + ε 3 g 23 (x), where g 1i , g 2i , f i have degree k, m and n respectively for each i = 1, 2, 3, and ε is a small parameter.When g 1 (x) = 0 the differential system (1) coincides with the generalized Liénard polynomial differential systems.The classical Liénard polynomial differential systems are ẋ = y, ẏ = −x − f (x)y, (2) where f (x) is a polynomial in the variable x of degree n.For these systems [Lins et al., 1977] stated the conjecture that if f (x) has degree n ≥ 1 then system (2) has at most [n/2] limit cycles.Here [x] denotes the integer part function of x ∈ R.They proved this conjecture for n = 1, 2. The conjecture for n = 3 has been proved recently by [Li & Llibre, 2012].For n ≥ 5 the conjecture is not true, see [De Maesschalck & Dumortier, 2011] and [Dumortier et al., 2007].So it remains to know if the conjecture is true or not for n = 4.Many of the results on the limit cycles of polynomial differential systems have been obtained by considering limit cycles which bifurcate from a single degenerate singular point (i.e., from a Hopf bifurcation), that are called small amplitude limit cycles, see for instance [Lloyd, 1988].There are partial results concerning the maximum number of small amplitude limit cycles for Liénard polynomial differential systems.The number of small amplitude limit cycles gives a lower bound for the maximum number of limit cycles that a polynomial differential system can have.
There are many results concerning the existence of small amplitude limit cycles for the following generalization of the classical Liénard polynomial differential system (2) where g(x) and f (x) are polynomials in the variable x of degrees m and n, respectively.We denote by H(m, n) the maximum number of limit cycles that systems (3) can have.This number is usually called the Hilbert number for systems (3).
(i) [Liénard, 1928] proved that if m = 1 and F (x) = ∫ x 0 f (s)ds is a continuous odd function, which has a unique root at x = a and is monotone increasing for x ≥ a, then equation (3) has a unique limit cycle.
(ii) [Rychkov, 1975] proved that if m = 1 and F (x) is an odd polynomial of degree five, then equation (3) has at most two limit cycles.
Up to now and as far as we know only for these four cases ((iii)-(vii)) the Hilbert number for systems (3) has been determined.
The maximum number of small amplitude limit cycles for systems (3) is denoted by Ĥ(m, n).[Blows & Lloyd, 1984], [Lloyd & Lynch, 1988] and [Lynch, 1995] have used inductive arguments in order to prove the following results.
[ Llibre et al., 2009] compute the maximum number of limit cycles Hk (m, n) of systems (3) which bifurcate from the periodic orbits of the linear center ẋ = y, ẏ = −x, using the averaging theory of order k, for k = 1, 2, 3.
In [Llibre & Valls, 2011] the authors studied using the averaging theory of first and second order the more general system where g 1i , f 1i , g 2i , f 2i have degree k, l, m and n respectively for each i = 1, 2, and ε is a small parameter.Using the averaging method of first and second order they proved the following result.
Theorem 1.1.For |ε| sufficiently small the maximum number of limit cycles of the generalized Liénard polynomial differential systems (4) bifurcating from the periodic orbits of the linear center ẋ = y, ẏ = −x using the averaging theory of second order is: [ Alavez-Ramirez et al., 2012] studied system (1) with g 13 (x) = g 23 (x) = f 23 (x) = 0 and they proved the following result.
Theorem 1.2.For |ε| sufficiently small the maximum number of limit cycles of the generalized Liénard polynomial differential systems (4) with f 11 (x) = f 12 (x) = 0 bifurcating from the periodic orbits of the linear center ẋ = y, ẏ = −x using the averaging theory of third order is where O(l) is the largest odd integer ≤ l, and E(l) is the largest even integer ≤ l.
In the present paper we study system (1), i.e. we extend the results of [Alavez-Ramirez et al., 2012] because in [Alavez-Ramirez et al., 2012] first g 13 (x) = g 23 (x) = f 3 (x) = 0, and additionally the study of the limit cycles coming from averaging of second and third order is made under some restrictive conditions.Using the averaging method of third order we will show our main result: Theorem 1.3.For |ε| sufficiently small the maximum number of limit cycles of the generalized Liénard polynomial differential systems (1) bifurcating from the periodic orbits of the linear center ẋ = y, ẏ = −x using the averaging theory of third order is λ 1 equal to Note that λ 0 < λ 1 .The proof of Theorem 1.3 is given in Section 3.
The results that we shall use from the averaging theory of third order for computing limit cycles are presented in Section 2.

The averaging theory of third order
The averaging theory for studying specifically limit cycles up to third order in ε was developed in [Buicȃ & J. Llibre, 2004].It is summarized as follows.

Consider the differential system
where are locally Lipschitz with respect to x, and R is twice differentiable with respect to ε.
We define where (ii) For V ⊂ D an open and bounded set and for each ε Then for |ε| > 0 sufficiently small there exists a T -periodic solution ϕ(•, ε) of the system such that ϕ(0, a ε ) → a ε when ε → 0.
The expression d B (F 10 + εF 20 + ε 2 F 30 , V, a ε ) ̸ = 0 means that the Brouwer degree of the function F 10 + εF 20 + ε 2 F 30 : V → R n at the fixed point a ε is not zero.A sufficient condition in order that this inequality is true is that the Jacobian of the function F 10 + εF 20 + ε 2 F 30 at a ε is not zero.
If F 10 is not identically zero, then the zeros of F 10 + εF 20 + ε 2 F 30 are mainly the zeros of F 10 for ε sufficiently small.In this case the previous result provides the averaging theory of first order.
If F 10 is identically zero and F 20 is not identically zero, then the zeros of F 10 + εF 20 + ε 2 F 30 are mainly the zeros of F 20 for ε sufficiently small.In this case the previous result provides the averaging theory of second order.
If F 10 and F 20 are identically zero and F 30 is not identically zero, then the zeros of F 10 + εF 20 + ε 2 F 30 are mainly the zeros of F 30 for ε sufficiently small.In this case the previous result provides the averaging theory of third order.

Proof of Theorem 1.3
We shall need the third order averaging theory to prove Theorem 1.3.We write system (1) in polar coordinates (r, θ) where x = r cos θ, y = r sin θ, r > 0.
We shall work only with the necessary conditions ( 11) and ( 12), and consequently we improve the results of [Alavez-Ramirez et al., 2012] in two ways.First because we work with this necessary conditions and second because we consider in the differential system (1) terms up to order ε 3 while in [Alavez-Ramirez et al., 2012] they consider only up to order ε 2 .
In order to apply the third order averaging method we need to compute the corresponding function F 30 (r) that we rewrite as with It was proved in [Alavez-Ramirez et al., 2012] that using the integrals of the Appendix, or in [Llibre & Valls, 2011] that y 1 = y 1 (θ, r) is equal to where and We also note that (16) The proof of Theorem 1.3 will be a direct consequence of the next auxiliary lemmas.
Lemma 3.1.The integral F 1 30 (r) is a polynomial in the variable r of degree (For an explicit expression of the polynomial F 1 30 (r) we refer the reader to the proof of Lemma 3.1).
Hence, using the formulae in the appendix and the explicit formula of F 1 30 (r) given in ( 13) we obtain that F 1 30 (r) is equal to where δ i for i = 1, . . ., 6 are constants depending on t, i and j.
Lemma 3.2.The integral F 3 30 (r) is a polynomial in the variable r of degree λ 3 equal to (For an explicit expression of the polynomial F 3 30 (r) we refer the reader to the proof of Lemma 3.2).
Proof.We first note that ∂F 2 (r, θ)/∂r is equal to Hence, using the formulae in the appendix and the explicit formula of F 3 30 (r) given in ( 13) we obtain that F 3 30 (r) is equal to where ρ i for i = 1, . . ., 12 are constants depending on t, i and j.
Lemma 3.3.The integral F 4 30 (r) is a polynomial in the variable r of degree λ 4 equal to (For an explicit expression of the polynomial F 4 30 (r) we refer the reader to the proof of Lemma 3.3).
Proof.We will first compute in F 3 all the terms that have non-zero integral from 0 to 2π.To do it, we note that in view of (10) F 3 is equal to Then where the constants ζ 1 , ζ 2 depend on i.Furthermore, using the formulae of the Appendix we conclude that 1 2πr where the constants ζ l for l = 3, . . ., 9 depend on i, j.Finally, since A 2 1 /r 2 is equal to using the formulae of the Appendix we conclude that where the constants ζ l for l = 10, . . ., 13 depend on t, i, j.
Lemma 3.4.The integral F 2 30 (r) is a polynomial in the variable r of degree λ 5 equal to (For an explicit expression of the polynomial F 2 30 (r) we refer the reader to the proof of Lemma 3.4).
Proof.We note that F 2 30 (r) is equal to We will first compute the terms in For this we note that Now using the non-zero formulae in the appendix we conclude that F 2 30 (r) is equal to where the constants µ i for i = 1, . . ., 12 depend on t, i, j.

Now we compute
Hence using the formulae in the appendix we deduce that where ∆ i,r,s , Γ i,r,s and Υ i,r,s are non-zero constants.