Abstract
In this paper, we study the number of limit cycles emerging from the period annulus by perturbing the Hamiltonian system . The period annulus has a heteroclinic cycle connecting two hyperbolic saddles as the outer boundary. It is proved that there exist at most and at least limit cycles emerging from the period annulus, and limit cycles are near the boundaries.
1. Introduction
The maximal number of limit cycles of the -degree polynomial systemis the topic of second part of Hilbert’s 16th problem [1]. It is rather difficult even for finding the limit cycles of a concrete system of degree . There exist several weaker versions, one of them is studying the limit cycles of the following system, which has simple form and potential applications in natural science,Equation (2) includes many nonlinear oscillators such as the classical Van der Pol equation with and . When the degree of (2) is larger, it is usually called strongly nonlinear oscillator, which has rich application in material science; see a systematic study in the relatively new monograph [2]. System (2) has the following planar form after introducing and taking new damping effect as , showing it is very weak.In fact, studying the maximal number of limit cycles of (2) is the intersection of weak Hilbert’s 16th problem [3] and Smale’s 13th problem [4]. The latter version is studying the limit cycles of classical Liénard system , and the former one is studying the limit cycles of perturbed polynomial Hamiltonian system:where , defines a family of closed curves , is a sufficiently small parameter, and is the coefficient vector of and . The problem is mainly restricted to first-order bifurcation and the train of thought is to investigate zeros of the integral; see Poincaré-Pontryagin Theorem [5].This integral is called Abelian integral; see the survey works [6, 7] for smooth systems and for nonsmooth cases see relatively new papers [8, 9].
System (3) will be called being of type when , . Let denote the exactly maximal number of zeros of for (3) of type in this paper. It is still very difficult even for the simpler systems (3) to get the exact bound of . We recommend the work [10] and its introduction part for the related results with different .
One more interesting restriction is that the period annulus is unique and bounded by a asymmetric heteroclinic cycle. For this case, it was proved that and in [11–13]. For , the general form isKazemi et al. [14] and Sun [15] studied system (6) of the case . They proved that corresponding system (6). However, it is still open that whether or is the sharp bound. Sun and Zhao [16] studied the case that , under which the unperturbed system has a heteroclinic cycle connecting two nilpotent cusps surrounding a nilpotent center. They prove a same result as [15]. For a period annulus of with a heteroclinic cycle, which connects two hyperbolic saddles and surrounds an elementary center, Sun [10] proved that it has at most limit cycles and there indeed exist limit cycles for some possible parameters; in other words, is the sharp bound. It was proved that for this case. It is interesting that the sharp bound is obtained for systems (6) without nilpotent singularities. Researchers usually believe that the perturbed systems has the same results on the number of limit cycles by first-order bifurcation when the unperturbed systems have the same topological portraits. One question is that does the nonexistence of the nilpotent singularity lead to the sharp bound , or is there any systems of the same kind may not has the sharp bound ?
2. Main Result
The main aim of this paper is to report a different result of . We pay attention mainly on (6) with fixed , to studyunder the symmetric perturbations of ; that is,where and . The Hamiltonian is defines a family of ovals , which are closed clockwise orbits of (7). The outer boundary of has the heteroclinic loop as its outer boundary, connecting the hyperbolic saddles and , at which , and the elementary center at the origin, , is the inner boundary; see Figure 1.
Correspondingly, we have the Abelian integral defined on ,where ,
We will mainly pay attention to the least upper and lower bounds of number of zeros of . It should be noted that there is no , and in the perturbation, their integration on each closed orbit will vanish. Symbolic computation will be the main tools to assist our analysis. We do not use the parameter in our analysis because it induces rather complex computation. We fix in our analysis and the same dynamical portraits are kept. However, we note that the same results are obtained when taking several values for near via the completely same analysis.
Theorem 1. Considering the Liénard system (8), (i) there exist at most four limit cycles emerging from the period annulus for all , and (ii) there exist some parameters such that system (8) can have exact limit cycles bifurcated out near the boundary of the annulus.
In Section 3, the parameter space is splitted into 4 parts, on each of which the upper bound of will be analyzed via an algebraic criteria. In Section 4, 3 zeros of will be detected via asymptotic analysis. Last, we present a discussion to compare the previous results as ours to stress our main purpose of this work.
3. Proof of Main Result (i)
First, we divide the parameter space into the following four parts:Defining . First, we have the following.
Lemma 2. for , where and is a polynomial.
Proof. On each closed curves , ThenBy Lemma 4.1 of [17], taking and , we obtainwhereThereforeApplying Lemma 4.1 of [17] again, thenwhere with Inserting (18) into (17),Final applying Lemma 4.1 of [17] giveswhere , and is some polynomial of degree .
Combining (20) and (21) gives where .
Further, we have the following.
Lemma 3. Let and then where .
Next, we will discuss the Chebyshev property of instead of . Lemma 3 reveals that the system can be deduced into with a parameter. The combination will simplify the computation and lead to a better result (see Remark in Section 4).
It is ready to set the determining functions and then checking if the integral system has Chebyshev property by the algebraic criteria [17, 18]. We set where the symmetry of reveals that defines . Then, if restricting . Therefore, and
Computing the related Wronskions obtains The long expression of the polynomials , , , , and has the degrees of 36, 66, 90, 90, and 92, respectively. In particular, , and and have the degrees 92 and 90, respectively.
Lemma 4. , , and do not vanish on , and has a uniqueroot in , which is simple one.
Proof. The existence and nonexistence of zeros are verified by Sturm’s Theorem to , , , and , respectively. The remainder is to prove the zero of is of multiplicity 1. We have where do not vanish in , and this reveals that is a simple zero.
Based on Lemma 4, the Chebyshev criteria in [17, 18] reveal the following result.
Proposition 5. Both of and are Chebyshev systems. forms a Chebyshev system with accuracy 1. Hence, has at most zero(s) when for , respectively.
The nonvanishing of in implies that is well defined, Further, we have the following.
Lemma 6. has , , and roots in , counting multiplicities, when , , and , respectively, where , , and , and and are given below.
Proof. We define a curveThe points on the curve satisfy .
Further, where and .
does not vanish in , has 2 roots: denoted by , and isolated in and , , respectively. (Numerical approximation shows and ).
Direct computation reveals that Then, the graph of the function is clear: it arises from , decreasing to and then increasing to , last decreasing to ; see Figure 2.
From the above discussion, we have(i)for each , there exist two points , , which reveals that has two zeros for in when fixing in this interval.(ii)For each , there exist one point , which reveals that has one zero for .(iii)For each , there exist no point , showing does not vanish in .Direct computation gives where has only two zeros at and as shown before. Further, has no roots on , or (the intervals exists). Hence, has two roots ( , ) with multiply multiplicities 2 if and only if taking , , respectively. And more, has two simple roots when , has unique simple root for .
When the parameters in , we fix and redenote . Then,
Based on the above discussion, we apply the Chebyshev criteria to the wronskians and and have the following.
Proposition 7. forms one Chebyshev system with accuracy . Hence, when , has, at most, four zeros in .
Propositions 5 and 7 prove the part (i) of the main result.
Remark 8. In the above discussion, we did not prove the four generating elements , , and form a Chebyshev system with some accuracy directly, by which one can verify that forms one Chebyshev system with accuracy 2, showing has at most five roots in , even changing any order of in . This stresses that the combination between and is rather crucial for a smaller upper bound compared the direct analysis.
4. Proof of Main Result (ii)
We will detect zeros of by the expansion theory of the Abelian integrals, see [19]. The Abelian integrals can be expanded near the outer and inner boundaries of the period annulus. One can verify the coefficients are linear dependent and take them as free parameters to detect zeros. According to the type of the outer boundary and inner one, we can apply the theories and computation methods in [20, 21] to establish the asymptotic expansions of near and . Here, the normal forms for application the methods in [20, 21] are obtained by the transforms and . We omit the computation and analysis for briefness and have for , andfor . The coefficients are as follows:andwhere and ; see the Appendix for their exact values in detail.
Note which implies Solving the equations revealsTaking and substituting it into and reveal Taking and positive and sufficiently small, and , and then This shows that has no zero between and .
By (41), we take , , and as free parameters and change them in turn satisfyingThen, there exist 2 zeros of near in and zero in near .
Let (45) , a subset of some small neighborhood of . Then, the following assertion is clear.
Theorem 9. has, for , zeros in and zero in , totally zeros.
Similarly, we can have the following.
Theorem 10. can have zero in and zeros in .
Theorem 11. can have zeros in .
Theorem 12. can have zeros in .
Theorems 9, 10, 11, and 12 prove part (ii) of the main result.
Remark 13. System (8) has the same dynamical portraits as that investigated in [10], however, different results can be obtained. Therefore, the non-existence of nilpotent singularity may not be the key condition of existence of sharp bound . The combination analysis of and is rather important to get a smaller upper bound (comparing as ), while it is only to prove that the upper bound is by studying four generating elements directly.
Appendix
In this Appendix, we present the exact values of coefficients in the expansions of . with
Data Availability
The data used to support the findings of this study are available from the corresponding author upon request.
Conflicts of Interest
The authors declare that there are no conflicts of interest.
Authors’ Contributions
All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.
Acknowledgments
The authors express their gratitude to Dr. Xianbo Sun for his suggestion. This work is supported by Natural Science Foundation of Guangxi Province (2018GXNSFAA138198, 2016GXNSFDA380031), Program for Innovative Team of GUFE (2018-2021), and The Project of Young Teacher’s Upgrading Program in Guangxi Province (2019KY1310).