Nine Limit Cycles in a 5-Degree Polynomials Liénard System

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.


Introduction
Consider the following perturbed Hamiltonian system: _ x � H y (x, y) + εp(x, y, δ), _ y � − H x (x, y) + εq(x, y, δ), (1) where H(x, y) is a polynomial of degree n + 1, p(x, y, δ) and q(x, y, δ) are polynomials of degree n, ε is the sufficiently small perturbation parameter, and δ ∈ D ⊂ R N with D compact. Taking ε � 0, 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 Γ h defined by (x, y) | H(x, y) � h , and the family forms a periodic annulus. e 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. e 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 Γ h of H(x, y) and is usually called the first-order Melnikov function or Abelian integral. e zeros of M(h, δ) correspond to the limit cycles by Poincaré bifurcation for system (1) (see [1][2][3]). It should be noted that studying the maximal number of zeros of M(h, δ) 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: In order to reduce the difficulty, researchers usually study some special forms of system (1), such as which is called Liénard system of type (m, n) having the degree n * � max n + 1, m { }, where g(x) and f(x) are polynomials of degrees m and n, respectively, and ε is positive and very small. Let H(n, m) denote the maximal number of limit cycles of system (5) and I(n * ) denote the maximal number of limit cycles of system (5) of degree n * . Dumortier and Li studied four Liénard systems with different portraits of type (3,2) in a series of papers [5][6][7][8] and gave the corresponding sharp bound of number of limit cycles by Poincaré bifurcation. e exact bounds of H (4,3) and H (5,4) for some special Liénard systems were reported in [9][10][11][12][13][14] and references therein. For results on H(7, 6) associated with symmetric system (5), the relatively new works are referred [15][16][17]. It is usually very difficult to give the exact bound; however, lots of lower bounds of H(m, n) 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 (5, n) and proved H(5, 2) ≥ 3, H(5, 4) ≥ 5, H(5, 6) ≥ 10, H(5, 8) ≥ 10. In this paper, we study a type (5, 4) Liénard system which has a polycycle consisting of a homoclinic loop and a heteroclinic loop: with special elliptic Hamiltonian function: where 0 < ε ≪ 1, f(x) � n i�1 a i x i , a i (i � 0, 1, . . . , n) are real bounded parameters. System (6) has the degree n * � max 5, n + 1 { }. Let H * (5, n) denote the maximum number of limit cycles of (6) and I * (n * ) denote the maximum number of limit cycles of (6) of degree n * . Our main interest is focused on H * (5, 4). e level sets (i.e., H(x, y) � h) of Hamiltonian function (7) are sketched in Figure 1. H(x, y) � h defines the periodic orbits of system (6).
For system (6), we get the following main result.

Theorem 1.
ere exist some a 0 , a 1 , a 2 , a 3 such that system (6) has 9 limit cycles for the type (5, 4) system (6). erefore, H(5, 4) ≥ H * (5, 4) ≥ 9, I(5) ≥ I * (5) ≥ 9. e 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). e proof of the main result is given in Section 4. Conclusion and discussions are drawn in Section 5.

Preliminary Lemmas
Consider the Melnikov function M(h, δ) for the near-Hamiltonian system (1); we suppose the Hamiltonian system (2) has a bounded periodic annulus Γ h denoted by H(x, y) � h , and the period orbits in the periodic annulus are clockwise. e boundary of Γ h can be a center, a homoclinic loop, and a heteroclinic loop. We suppose the inner boundary of Γ h is defined by H(x, y) � α and the outer boundary is defined by H(x, y) � β; correspondingly, we have and M(h, δ) can be expanded near its boundary (see [20][21][22][23]). When the inner boundary of Γ h is elementary center, we suppose it is located at C(x c , y c ) with H(x c , y c ) � α, and for (x, y) near C(x c , y c ), 2 Complexity For the expansion of M(h, δ) near C(x c , y c ), we have the following.
Lemma 1 (see [20]). Under the condition we suppose above, for the expansion of M(h, δ) near the elementary center Under (9), the formulas of b j can be obtained by using the programs in [20].
When the outer boundary of Γ h is homoclinic loop denoted by Γ β passing through a hyperbolic saddle S(x s , y s ) satisfying H(x s , y s ) � β, for the expansion of M(h, δ) near the homoclinic loop, we have the following.
(i) e 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 H(x, y), p(x, y), and q(x, y) at the singular point. (ii) When there is a nilpotent cusp S(x n , y n ) connecting a homoclinic loop, there are similarly some local coefficients at the cusp S(x n , y n ) presented in the expansion of the corresponding Melnikov function near the homoclinic loop; we denote these first three local coefficients by c 1 (S cusp , δ), c 3 (S cusp , δ), and c 4 (S cusp , δ) (see [22]).
Lemma 3 (see [22]). Suppose there is a nilpotent cusp S(x n , y n ) which is a limit point of a homoclinic loop or a heteroclinic loop, and for (x, y) near the nilpotent cusp S(x n , y n ): en, the first three local coefficients of the corresponding Melnikov functions at the nilpotent cusp S(x n , y n )are as follows: Complexity When the outer boundary of Γ h is heteroclinic loop denoted by Γ β passing through a hyperbolic saddle S(x s , y s ) and a nilpotent cusp N(x n , y n ) satisfying H(x s , y s ) � H(x n , y n ) � β, for the expansion of M(h, δ) near the heteroclinic loop Γ β , we have the following.

Lemma 4 (see [23]). e expansion of M(h, δ) near the heteroclinic loop L β has the form
and c 3 (δ) ∈ R. In particular, if c 1 (δ) � c 2 (δ), 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.

Complexity
Lemma 5 (see [2]). Under the linear change of variables of the form and time rescaling τ � kt, where D � a d − bc ≠ 0, system (31) becomes where Let which is the Melnikov function of system (21). en, Remark 1. Usually when we apply Lemma 5, we always take a linear transformation such that D/k � 1 because under this condition, the local coefficients between the old and the new ones, respectively, are the same (if |k|/D � 1 ) or only different from a symbol "− " (if |k|/D � − 1 ). Let us now suppose system (2) has a hetero-homoclinic loop Γ * � Γ 1 ∪ Γ 2 , where Γ 1 is heteroclinic loop connecting a cusp S 1 (x 1 , y 1 ) and a hyperbolic saddle S 2 (x 2 , y 2 ) satisfying H(x 1 , y 1 ) � H(x 2 , y 2 ) � β and Γ 2 is homoclinic loop passing through the previous hyperbolic saddle S(x 2 , y 2 ).
ere exist three families of orbits, the first family of periodic orbits Γ 1 h surrounding a center C 1 (x 3 , y 3 ) with H(x 3 , y 3 ) � α 1 in Γ 1 , the second family of periodic orbits Γ 2 h surrounding another center C 2 (x 4 , y 4 ) with H(x 4 , y 4 ) � α 2 in Γ 2 , and the third family of periodic orbits Γ * h surrounding Γ * . e portrait of Γ * is just the same as Figure 1.
Correspondingly, we have three Melnikov functions: From Lemmas 2-4, we have the following.

The Expansions of Melnikov Functions of System (6)
Liénard system (6) is a special form of near-Hamiltonian In this section, we take n � 4; then, system (6) is a Liénard system of type (5,4). ere are three Melnikov functions corresponding to three periodic annuli of system (6)ε � 0: By eorem 1, we obtain for 0 < − h ≪ 1.
for 0 < h ≪ 1. By Lemma 1, for 0 < h + 25/2688 ≪ 1, and for 0 < h + 8/21 ≪ 1, In the following, we use the preliminary lemmas to compute the coefficients of the above expansions for and M 3 (h, δ). Firstly, we have and in accordance with (59), we obtain a z * 30 near z 30 , such that M 3 (z * 30 , δ 1 ) � 0; therefore, there is another zero of M 3 (h, δ 1 ) near z 30 . e above 9 zeros are isolated real zeros; therefore, they correspond to 9 limit cycles of system (6). e main theorem is proved.

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. e 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. e computational analysis shows that there exist at least 9 limit cycles in the suitably damped system. e 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
e data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest
e authors declare that they have no conflicts of interest.