Poincaré Bifurcation of Limit Cycles from a Liénard System with a Homoclinic Loop Passing through a Nilpotent Saddle
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.
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 , 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  and Shi  proved that independently. In 1985, Li and Li  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.
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 , , [18, 19], and . 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 , .
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
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  (Theorem A) can be stated as follows.
Lemma 2 (see ). 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 ). 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
Lemma 5. where
Therefore, is a T-system with accuracy if and only if is the same as well.
Taking the following function where is an analytic involution, defined by on . Factoring yields where + + + + + , which defined on . Hence, with Suppose that , and then , where is the projection of on -axis and ; in other words,
Lemma 6. The function tuple is an T-system with accuracy 3 for .
Proof. Taking (23) into consideration, with the aid of Maple 16, we can obtain the 4 following Wronskians: where = + + + + + , and , ,, and are polynomials in of degrees 24, 50, 79, and 110, respectively. On the following, calculating the resultant with respect to between and gives From Sturm’s Theorem, we know that has two roots . Thus we will check if and have any common roots on by using the program with Maple 16 to find all the possible intervals: ; ; ; ; ; ; [regular _ chain] ; It is obvious that all the roots of the five regular chains do not satisfy (24), so we conclude that and have no common root on .
(i) Calculating the resultant with respect to between and , that is, eliminating from and gives , where is a polynomial of degree 122 in . Applying Sturm’s theorem to , there is a point, denoted by such that , with . Thus we will check if and have any common roots on by using the program with Maple 16 to find all the possible intervals: ; ; ; ; ; ; [regular _ chain] ; where , and , , and are polynomials in of degrees 108, 110, and 122, respectively. It is obvious that all the roots of the regular chains and do not satisfy (24), and the regular chains is square-free and zero-dimensional (because the number of variables equals the number of polynomials). and represent and in Maple; we use the following program to check their common roots: ; [regular _ chain] ; ; It means that there are 2 pairs of common roots of and in the listed intervals, respectively. However, there is not any pair of listed common root satisfing (24), so we conclude that on .
(ii) Calculating the resultant with respect to between and , that is, eliminating from and , gives , where is a polynomial of degree 254 in . Applying Sturm’s theorem to , there are five points, denoted by and , such that , in which , and Thus we will check if and have any common roots on by using the program with Maple 16 to find all the possible intervals: ; ; ; ; ; ; [regular _ chain] ; where , and , , and are polynomials in of degrees 228, 229, and 254, respectively. It is obvious that all the roots of the regular chains and do not satisfy (24); the regular chains are square-free and zero-dimensional (because the number of variables equals the number of polynomials). and represent and in Maple, we use the following program to check their common roots: ; [regular _ chain] ; ;