Discrete Dynamics in Nature and Society

Volume 2019, Article ID 6943563, 12 pages

https://doi.org/10.1155/2019/6943563

## Poincaré Bifurcation of Limit Cycles from a Liénard System with a Homoclinic Loop Passing through a Nilpotent Saddle

Department of Applied Mathematics, Guangxi University of Finance and Economics, Nanning, Guangxi, 530003, China

Correspondence should be addressed to Minzhi Wei; moc.361@321gnoixnayoaix

Received 22 February 2019; Accepted 14 May 2019; Published 2 June 2019

Academic Editor: Douglas R. Anderson

Copyright © 2019 Minzhi Wei et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### 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 , .