LIMIT CYCLES FOR A GENERALIZATION OF POLYNOMIAL LIÉNARD DIFFERENTIAL SYSTEMS

We study the number of limit cycles of the polynomial differential systems of the form ẋ = y − f1(x)y, ẏ = −x − g2(x) − f2(x)y, where f1(x) = εf11(x) + εf12(x) + εf13(x), g2(x) = εg21(x) + εg22(x) + εg23(x) and f2(x) = εf21(x)+εf22(x)+εf23(x) where f1i, f2i and g2i have degree l, n and m respectively for each i = 1, 2, 3, and ε is a small parameter. Note that when f1(x) = 0 we obtain the generalized polynomial Liénard differential systems. We provide an accurate upper bound of the maximum number of limit cycles that this 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
The second part of the 16th Hilbert's problem wants to find an upper bound on the maximum number of limit cycles that the class of all polynomial vector fields with 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 (1)ẋ = y − f 1 (x)y,ẏ = −x − g 2 (x) − f 2 (x)y, where f 1 (x) = εf 11 (x) + ε 2 f 12 (x) + ε 3 f 13 (x), g 2 (x) = εg 21 (x) + ε 2 g 22 (x) + ε 3 g 23 (x) and f 2 (x) = εf 21 (x) + ε 2 f 22 (x) + ε 3 f 23 (x) where f 1i , f 2i and g 2i have degree l, n and m respectively for each i = 1, 2, 3, and ε is a small parameter. When f 1 (x) = 0 these systems coincide with the generalized polynomial Liénard differential systems (2)ẋ = y,ẏ = −g(x) − f (x)y, where f (x) and g(x) are polynomials in the variable x of degrees n and m, respectively. The classical polynomial Liénard differential systems are where f (x) is a polynomial in the variable x of degree n. For these systems in 1977 Lins, de Melo and Pugh [15] stated the conjecture that if f (x) has degree n ≥ 1 then system (3) has at most [n/2] limit cycles. They prove this conjecture for n = 1, 2. The conjecture for n = 3 has been proved recently by Chengzi Li and Llibre in [16]. For n ≥ 5 the conjecture is not true, see De Maesschalck and Dumortier [7] and Dumortier, Panazzolo and Roussarie [8]. 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 [20]. There are partial results concerning the maximum number of small amplitude limit cycles for Liénard polynomial differential systems. Of course, 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 generalized Liénard polynomial differential system (2). We denote by H(m, n) the number of limit cycles that systems (2) can have. This number is usually called the Hilbert number for systems (2).
Up to now and as far as we know only for these five cases ((iii)-(vii)) the Hilbert number for systems (2) has been determined.
LIMIT CYCLES OF A CLASS OF POLYNOMIAL DIFFERENTIAL SYSTEMS. 3 In 2010 Llibre, Mereu and Teixeira [18] compute the maximum number of limit cyclesH k (m, n) of systems (2) which bifurcate from the periodic orbits of the linear centerẋ = y,ẏ = −x, using the averaging theory of order k, for k = 1, 2, 3.
In [19] the authors studied using the averaging theory of first and second order the more general systeṁ where g 1i , f 1i , g 2i , f 2i have degree l, k, 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.
In Alavez-Ramirez et al. in [2] they studied the polynomial differential systeṁ where g 1i , g 2i , f 2i have degree l, m and n respectively for each i = 1, 2, and ε is a small parameter. They proved the following result.

Theorem 2.
For |ε| sufficiently small the maximum number of limit cycles of the generalized Liénard polynomial differential systems (6) bifurcating from the periodic orbits of the linear centerẋ = y,ẏ = −x using the averaging theory of third order is where O(k) is the largest odd integer ≤ k, and E(k) is the largest even integer ≤ k.
In the present paper we study system (1). We define Λ by Using the averaging method of third order we will show the following result that is the main result of the paper.

Theorem 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 centerẋ = y,ẏ = −x using the averaging theory of third order is at most Λ.
The proof of Theorem 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 first and second order
The averaging theory for studying specifically limit cycles up to third order in ε was developed in [4]. It is summarized as follows. Consider the differential system T -periodic in the first variable, and D is an open subset of R. Assume that the following conditions hold.
Lipschitz with respect to x, and R is twice differentiable with respect to ε.
LIMIT CYCLES OF A CLASS OF POLYNOMIAL DIFFERENTIAL SYSTEMS. 5 We define F k0 : D → R for k = 1, 2 as where ) dt.
(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 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 3
We shall need the third order averaging theory to prove Theorem 3. We write system (1) in polar coordinates (r, θ) where In this way system (1) will become written in the standard form for applying the averaging theory. If we write Now taking θ as the new independent variable, system (9) becomes It was proved in [19] that where t varies from 0 to λ given in the statement of Theorem 1, and S s,i,j ≥ 0 for s = 1, . . . , 4.
In order to apply the third order averaging method we need to compute the corresponding function F 30 (r) that we rewrite as where it was proved in [19] that using the integrals of the appendix we obtain that y 1 = y 1 (θ, r) is equal to whereγ i,l are constant. Again from [19] we have that We also note that The proof of Theorem 3 will be a direct consequence of the next auxiliary lemmas.
For an explicit expression of the polynomial F 1 30 (r) we refer the reader to the proof of Lemma 4.
is a polynomial in the variable r of degree less than or equal to λ 1 given by Proof. We first note that and y 1 (r, θ) 2 is equal to Hence, using the integrals which are zero in the formulae in the appendix and the explicit formula of F 1 30 (r) given in (10) we obtain that 4πF 1 30 (r) is equal to Then, now using the integrals which are not zero in the formulae in the appendix we conclude that F 1 30 (r) is equal to for some constants ρ s for s = 1, . . . , 19 which depend on t, i, j.
For an explicit expression of the polynomial F 3 30 (r) we refer the reader to the proof of Lemma 5. Proof. Using the expression of F 2 (r, θ) in (12) and y 1 (θ, r) in (11) together with eliminating the integrals that are zero (see the formulae in the Appendix for those integrals) we have that 2πF 3 30 (r) is equal to (1 − cos t+2 θ) cos i+j θ(1 − 3 cos 2 θ + 2 cos 4 θ) dθ.
Then, now using the integrals in the appendix which are not zero we conclude that For an explicit expression of the polynomial F 4 30 (r) we refer the reader to the proof of Lemma 6. Lemma 6. The integral F 2 30 (r) is a polynomial in the variable r of degree less than or equal to of degree less than or equal to λ 3 given by Proof. We have that and we denotẽ Using the formulae in the Appendix it is easy to see that for some constant W . Now we note that Using the formulae in the Appendix we obtain for some constants γ κ,i,j for κ = 1, . . . , 6.

Appendix: Formulae
In this appendix we recall some formulae that will be used during the paper, see for more details [1]. For i ≥ 0 we have ∫ 2π 0 cos 2i+1 θ sin 2 θ dθ = where P i,s,r , Q i,s,r , R i,j , R is,r are non-zero constants.