Sharp Bound of the Number of Zeros for a Li´enard System with a Heteroclinic Loop

In the presented paper, the Abelian integral I ( h ) of a Li´enard system is investigated, with a heteroclinic loop passing through a nilpotent saddle. By using a new algebraic criterion, we try to ﬁnd the least upper bound of the number of limit cycles bifurcating from periodic annulus.


Introduction
A well-known analytic system with planar polynomial differential equation of degree n is of the form: _ x � P n (x, y), _ y � Q n (x, y).
In 1977, Arnold [1] proposed weak Hilbert's 16th problem and studied the number of zeros of the Abelian integral: where p and q are the polynomials of degree n ≥ 2 and Γ h are some closed ovals of corresponding Hamiltonian. More precisely, H(x, y) is the Hamiltonian function of special form of (1): where H(x, y), p(x, y), and q(x, y) are the polynomials of x and y, their degrees satisfy max degp, degq � n, deg(H) � n + 1, and ε is a positive and sufficiently small parameter.

Some Preliminaries
For system (3), some related definitions and significative results are introduced, it can be seen in [21][22][23] in detail. (i) e family of sets f 0 , f 1 , f 2 , . . . , f n− 1 is called a Chebyshev system (T-system for short) provided that any nontrivial linear combination k 0 f 0 (x) + k 1 f 1 (x) + · · · + k n− 1 f n− 1 (x) has at most n − 1 isolated zeros on J. (ii) An ordered set of n functions f 0 , f 1 , f 2 , . . . , f n− 1 is called a complete Chebyshev system (CT-system for short) provided any nontrivial linear combi- is called an extended complete Chebyshev system (ECT-system for short) if the multiplicities of zeros are taken into account.
. . , f n− 1 is an ECTsystem on J; therefore, it is an ECT-system on J and then a T-system on J; however, the inverse implications are not true.
Let H(x, y) � (1/2)y 2 + A(x) in (5) be an analytic function. e set of ovals Γ h � H(x, y) � h inside periodic annulus is defined by h ∈ (h 1 , h 2 ) � J. Supposed that P is a punctured neighborhood of the origin foliated by ovals Γ h , then the projection of P on the x-axis is an interval (x l , x r ) with x l < 0 < x r . It is easy to know that xA ′ (x) > 0, has a zero of even multiplicity at x � 0, and there exists an analytic involution Lemma 1 (see [22]). On (x r , x l ), supposed that an analytic function f i (x) satisfies where h ∈ (h 1 , h 2 ), s ∈ N, and I h is the oval surrounding the origin inside the level curve A(x) + (1/2)y 2m � h . Setting

Discrete Dynamics in Nature and Society
If the following assumptions are satisfied then for all nontrivial linear combination of I 0 , I 1 , . . . , I n− 1 has at most n + k − 1 zeros on (h 1 , h 2 ) counting the multiplicities. Meantime, I 0 , I 1 , . . . , I n− 1 is called a T-system with accuracy k on (h 1 , h 2 However, the third condition above always not been satisfied, so we usually apply the next lemma to increase the power of y in I i . Lemma 2 (see [22]). Let Γ h be an oval inside the level curve A(x) + (1/2)y 2m � h , F(x) be a function which satisfies where G(

Lemma 3
where