Abstract

In present paper, the number of zeros of the Abelian integral is studied, which is for some perturbed Hamiltonian system of degree 6. We prove the generating elements of the Abelian integral from a Chebyshev system of accuracy of 3; therefore there are at most 6 zeros of the Abelian integral.

1. Introduction

In many branches of science, such as mechanics, electronics, fluid mechanics, biology, chemistry, and astrophysics, one often deals with families of special planar differential equations which can model different natural phenomena. The main open problem in the qualitative theory of planar polynomial differential systems is determining the maximum number of limit cycles, which is the well-known second part of Hilbert’s 16th problem.

Let denote the maximal number of limit cycles of polynomial systems of degree n of the form The problem is still open even for . As the introduction in [1], there are few studies on an upper bound of . However, there have been many interesting results on the lower bound of it for ; see [1–3]. What is more, Chen [4] and Shi [5] proved that independently. In 1985, Li and Li [6] found by using the method of detection function, and then Han et al. [7, 8] also obtained with new different distributions of limit cycles by using the method of stability-changing of homoclinic orbits.

Recently, had been proved in [9] by Li, and then , , and   , had been obtained, respectively; see [10–15].

The intersection form of Smale’s problem and weak Hilbert’s 16th problem is studying the number of zeros of Abelian integral corresponding to the following generalised Liénard system: It is called Liénard system of type if and . There are abundant results for weak Hilberts 16th problem restricted to Liénard systems of type , especially for , such as types [16], [17], [18, 19], and [20]. More details of the relative researches can be seen in [21–32].

In present paper, we consider the following system: with , and are constants. Equation (3) holds the hyperelliptic Hamiltonian function The level sets (i.e., ) of Hamiltonian function (4) are sketched in Figure 1. for , where and , .

Figure 1 

The portrait of system (3).

The outer boundary of is a homoclinic loop passing through a nilpotent saddles defined by , and the inner boundary is an elementary center at the origin defined by .

2. Some Preliminaries

Some related definitions and useful results are introduced in this section. For more details, refer to [33–35].

Definition 1. Assume that are analytic functions on a real open interval J.
(i) The family of sets is called a Chebyshev system (T-system for short) provided that any nontrivial linear combination has at most isolated zeros on .
(ii) An ordered set of functions is called a complete Chebyshev system (CT-system for short) provided any nontrivial linear combination has at most zeros for all . Moreover, it is called an extended complete Chebyshev system (ECT-system for short) if the multiplicities of zeros are taken into account.
(iii) The continuous Wronskian of at is where is the first-order derivative of and is the th order derivative of , . The definitions imply that the function tuple is an ECT-system on ; therefore it is a ECT-system on and then a T-system on ; however, the inverse implications are not true.

Let be an analytic function. Assume there exists a punctured neighborhood of the origin foliated by ovals , which corresponds to clockwise periodic orbits of system (2) and forms a period annulus denoted by . The set of ovals inside the period annulus is parameterized by the energy levels . The projection of on the x-axis is an interval with . Under the above assumptions it is easy to verify that for all ; has a zero of even multiplicity at and has an analytic involution defined by

For the number of isolated zeros of nontrivial linear combination of some integrals of special form, the algebraic criterion in [34] (Theorem A) can be stated as follows.

Lemma 2 (see [15]). On , suppose that an analytic function satisfies that where , , and is the oval surrounding the origin inside the level curve . Setting that

If the following assumptions are satisfied(i) is nonvanishing on for ,(ii) has zeros on counting with multiplicities,(iii).

Then for all nontrivial linear combination of has at most zeros on counting the multiplicities. Meantime, is called a T-system with accuracy on , where is Wronskian of

However, the third condition above has not always been satisfied, so we usually apply the next lemma to increase the power of in .

Lemma 3 (see [34]). Let be an oval inside the level curve , and let be a function which satisfies is analytic at . Hence, where .

Proposition 4. is an T-system with accuracy 3, and is the same. Therefore there are at most 6 zeros for on .

3. The Least Upper Bound of Number of Zeros of

Multiplying by , it is obtained Setting and , quoting Lemma 3 to yields where , and Substituting (13) into (12) and multiplying again, it changes to Quoting Lemma 3, setting and , it is obtained where , and

Substituting (15) into (14) and multiplying again, it arrives at Quoting Lemma 3 again, then where , and From the computation above, the following result easily can be obtained.