Abstract
In this article, we study the limit cycles in a generalized 5-degree Liénard system. The undamped system has a polycycle composed of a homoclinic loop and a heteroclinic loop. It is proved that the system can have 9 limit cycles near the boundaries of the period annulus of the undamped system. The main methods are based on homoclinic bifurcation and heteroclinic bifurcation by asymptotic expansions of Melnikov function near the singular loops. The result gives a relative larger lower bound on the number of limit cycles by Poincaré bifurcation for the generalized Liénard systems of degree five.
1. Introduction
Consider the following perturbed Hamiltonian system:where is a polynomial of degree , and are polynomials of degree , is the sufficiently small perturbation parameter, and with compact. Taking , system (1) becomes a Hamiltonian system:and system (1) is usually called a near-Hamiltonian system. We assume that the unperturbed system (2) has a family of periodic orbits defined by , and the family forms a periodic annulus. The inner boundary of the periodic annulus is a center which may be an elementary one or a nilpotent one, and the outer boundary of the periodic annulus is usually a homoclinic loop or a heteroclinic loop or a polycycle. Most periodic orbits are broken when system (2) is perturbed and only a finite number of periodic orbits persist as limit cycles of system (1), which is the so-called Poincaré bifurcation. The most efficient tool to study Poincaré bifurcation of system (1) is the following first-order approximation of the Poincaré map:which is a continuous integral over the continuous ovals of and is usually called the first-order Melnikov function or Abelian integral. The zeros of correspond to the limit cycles by Poincaré bifurcation for system (1) (see [1–3]). It should be noted that studying the maximal number of zeros of is the topic of weak Hilbert’s 16th problem. We note that the original version of Hilbert’s 16th problem asks the maximal number of limit cycles of a general polynomial system:
Let denote the maximal number of limit cycles of system (4) and denote the maximal number of limit cycles of system (1). Most results on the lower bounds of are obtained in the literature as we know, by studying . However, it should be noted that it is also very difficult to determine for a given degree . We refer the works [4] for the relatively new results on the lower bounds of .
In order to reduce the difficulty, researchers usually study some special forms of system (1), such aswhich is called Liénard system of type having the degree , where and are polynomials of degrees and , respectively, and is positive and very small. Let denote the maximal number of limit cycles of system (5) and denote the maximal number of limit cycles of system (5) of degree . Dumortier and Li studied four Liénard systems with different portraits of type in a series of papers [5–8] and gave the corresponding sharp bound of number of limit cycles by Poincaré bifurcation. The exact bounds of and for some special Liénard systems were reported in [9–14] and references therein. For results on associated with symmetric system (5), the relatively new works are referred [15–17]. It is usually very difficult to give the exact bound; however, lots of lower bounds of have been obtained by studying the number of limit cycles near the center, homoclinic loops, and heteroclinic loops (see [18]). In particular, Xu and Li [19] investigated a Liénard system of type and proved , , , . In this paper, we study a type Liénard system which has a polycycle consisting of a homoclinic loop and a heteroclinic loop:with special elliptic Hamiltonian function:where , , () are real bounded parameters. System (6) has the degree . Let denote the maximum number of limit cycles of (6) and denote the maximum number of limit cycles of (6) of degree . Our main interest is focused on .
The level sets (i.e., ) of Hamiltonian function (7) are sketched in Figure 1. defines the periodic orbits of system (6). There are a heteroclinic loop and a homoclinic loop for system (6) defined by , denoted by and , respectively; let us in the following say is a hetero-homoclinic loop. connects the nilpotent cusp and the hyperbolic saddle of system (6), connects a the hyperbolic saddle of system (6). There are three families of clockwise periodic orbits, one family is inside surrounding a center with , one family is inside surrounding another center with , and the third family is surrounding and . Three families of periodic orbits form three periodic annuli, the boundaries of which are and , and , and .
For system (6), we get the following main result.
Theorem 1. There exist some , , , such that system (6) has 9 limit cycles for the type system (6). Therefore, , .
The rest of this paper is organized as follows. In Section 2, we present some preliminaries which will be used in the next section. In Section 3, we study the asymptotic expansions of the related Melnikov functions for system (6). The proof of the main result is given in Section 4. Conclusion and discussions are drawn in Section 5.
2. Preliminary Lemmas
Consider the Melnikov function for the near-Hamiltonian system (1); we suppose the Hamiltonian system (2) has a bounded periodic annulus denoted by , and the period orbits in the periodic annulus are clockwise. The boundary of can be a center, a homoclinic loop, and a heteroclinic loop. We suppose the inner boundary of is defined by and the outer boundary is defined by ; correspondingly, we haveand can be expanded near its boundary (see [20–23]).
When the inner boundary of is elementary center, we suppose it is located at with , and for near ,
For the expansion of near , we have the following.
Lemma 1 (see [20]). Under the condition we suppose above, for the expansion of near the elementary center , we have
Under (9), the formulas of can be obtained by using the programs in [20].
When the outer boundary of is homoclinic loop denoted by passing through a hyperbolic saddle satisfying , for the expansion of near the homoclinic loop, we have the following.
Lemma 2. (i)(see [21]). Under the condition we suppose above, the function has the following expansion: for , where and depend on the coefficients of , , and . for is divergence at , is a constant.(ii)(see [21]). In particular, for near the hyperbolic saddle ,Then,
Definition 1. The values and are, respectively, called the first and second local Melnikov coefficients at the saddle , denoted by , , respectively.
Note.(i)The local coefficients are presented in the expansion of a Melnikov function, which are local quantities at a singular point depending on the coefficients of the expansions of , , and at the singular point.(ii)When there is a nilpotent cusp connecting a homoclinic loop, there are similarly some local coefficients at the cusp presented in the expansion of the corresponding Melnikov function near the homoclinic loop; we denote these first three local coefficients by , , and (see [22]).
Lemma 3 (see [22]). Suppose there is a nilpotent cusp which is a limit point of a homoclinic loop or a heteroclinic loop, and for near the nilpotent cusp :
Then, the first three local coefficients of the corresponding Melnikov functions at the nilpotent cusp are as follows:
When the outer boundary of is heteroclinic loop denoted by passing through a hyperbolic saddle and a nilpotent cusp satisfying , for the expansion of near the heteroclinic loop , we have the following.
Lemma 4 (see [23]). The expansion of near the heteroclinic loop has the formfor , where , , and are constants, andand . In particular, if , we have
In many cases, the Hamiltonian function is not of the form supposed in the above lemmas. Then, to apply the lemmas, we need to first introduce suitable linear change of variables which will cause a change of the first-order Melnikov function. The following lemma gives the relationship between the old and new Melnikov functions.
Lemma 5 (see [2]). Under the linear change of variables of the formand time rescaling , where , system (31) becomeswhereLetwhich is the Melnikov function of system (21). Then,
Remark 1. Usually when we apply Lemma 5, we always take a linear transformation such that because under this condition, the local coefficients between the old and the new ones, respectively, are the same (if ) or only different from a symbol “” (if ).
Let us now suppose system (2) has a hetero-homoclinic loop , where is heteroclinic loop connecting a cusp and a hyperbolic saddle satisfying and is homoclinic loop passing through the previous hyperbolic saddle . There exist three families of orbits, the first family of periodic orbits surrounding a center with in , the second family of periodic orbits surrounding another center with in , and the third family of periodic orbits surrounding . The portrait of is just the same as Figure 1.
Correspondingly, we have three Melnikov functions:From Lemmas 2–4, we have the following.
Lemma 6. Under the condition supposed above, the three Melnikov functions have the following expansions:for .for .for , where and are the same as before, and , , , and are some constants. and , , , , , , , , , , , where , , and are local coefficients at the nilpotent cusp which can be obtained by Lemma 3 and and are local coefficients at the hyperbolic saddle which can be obtained by Lemma 2, whileand ; in particular, if , we haveand ; in particular, if , we haveand if .
3. The Expansions of Melnikov Functions of System (6)
Liénard system (6) is a special form of near-Hamiltonian system (1) with , . In this section, we take ; then, system (6) is a Liénard system of type . There are three Melnikov functions corresponding to three periodic annuli of system (6):
By Theorem 1, we obtainfor .for .for .
By Lemma 1, for ,and for ,
In the following, we use the preliminary lemmas to compute the coefficients of the above expansions for , , and . Firstly, we havewhere
Secondly,where
Therefore,where .
In order to get the local coefficients at the cusp , we introduce the transformation ; then, system (6) becomeswith Hamiltonian function , , , .
Applying the formula for , , in Lemma 3, in accordance with Lemma 6, we have
In order to get the local coefficient near the hyperbolic saddle , we introduce the transformation ; then, system (6) becomeswith Hamiltonian system , and . Applying the formula for and in Lemma 2, in accordance with Lemma 5, we have
Let ; we have
Under this case,
Taking (47) into (48), we have with the help of Maple 13,where
Let , i.e., ; we have
Direct computation giveswhere
Then, by (49) and (51) and Lemma 6, we havewhere .
Last we compute , , , and in (36) and (37). Taking a linear transformation , with time scaling and applying Lemma 5 and program in [20], we obtain
Taking a linear transformation , with time scaling and applying Lemma 5 and program in [20], we obtain
4. Proof of Theorem 1
In this section, we prove Theorem 1. Solving the equations gives
We take and ; then,
Therefore, there exist , , and , satisfying , , and , respectively, such thatwhile taking ,and thus there exists a between and , such that
It is easy to find
Therefore, , , , and can be taken as free parameters; we denote them by , , , and , respectively. Next, we take these free parameters to perturb 8 zeros.
We take , in turn satisfying
We take in which is a very small neighborhood of .
Therefore, , , , for by Lemma 6; then, there exist 3 zeros of , and of satisfying .
Under (63), , , for by Lemma 6; then, there exist 2 zeros , of satisfying .
Under (63), , , , for by Lemma 6; then, there exist 3 zeros , , and of satisfying .
Note thatand in accordance with (59), we obtain a near , such that ; therefore, there is another zero of near .
The above 9 zeros are isolated real zeros; therefore, they correspond to 9 limit cycles of system (6). The main theorem is proved.
5. Conclusion and Discussion
In the paper, we have applied the theories of Hopf bifurcation, homoclinic loop bifurcation, and heteroclinic loop bifurcation to detect the limit cycles near the center and polycycle locally. The main routine is to prove the independence of the coefficients of the asymptotic expansions of the Melnikov functions and then treat them as free perturbation parameters. The computational analysis shows that there exist at least 9 limit cycles in the suitably damped system. The result gives a relative larger lower bound on the number of limit cycles for the Liénard system of degree 5. It is interesting to show that the asymptotic expansions of the Melnikov functions not only are the efficient tools to detect limit cycles such as a complicated investigation in [24] but also have been successfully applied to study the existence of periodic traveling waves and the coexistence of periodic solitary traveling waves (see [25, 26]). It may be more interesting to investigate the periodic traveling waves in external perturbation considered in the model given in [27]. It should also be pointed that there exist more questions left to solve for system (5), for example, what is the maximal number of limit cycles, or limit cycles by Poincaré bifurcation, or limit cycles by Hopf , or limit cycles by Hopf bifurcation, how about the global dynamics, and what is the period function of the undamped system. However, it is not easy at all to solve these questions.
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 they have no conflicts of interest.
Acknowledgments
This study was supported by the Natural Science Foundation of Guangxi (2018GXNSFAA138198) and Program for Innovative Team of GUFE (2018–2021).