Limit Cycle Bifurcations Near a Cuspidal Loop

Pan Liu,Maoan Han

doi:10.3390/sym12091425

Abstract

In this paper, we study limit cycle bifurcation near a cuspidal loop for a general near-Hamiltonian system by using expansions of the first order Melnikov functions. We give a method to compute more coefficients of the expansions to find more limit cycles near the cuspidal loop. As an application example, we considered a polynomial near-Hamiltonian system and found 12 limit cycles near the cuspidal loop and the center.

Highlights

As we know, in the qualitative theory of planar differential systems, one of the most important problems is to study the number of limit cycles for a near-Hamiltonian system, which is closely related to the Hilbert’s 16th problem
In this paper we develop the idea for computing coefficients in the the expansion of M near a homoclinic loop passing through a hyperbolic saddle used in [18] to the case of a cuspidal loop
This paper considered the number of limit cycles bifurcating from a cuspidal loop with order 1

Summary

Introduction

In the qualitative theory of planar differential systems, one of the most important problems is to study the number of limit cycles for a near-Hamiltonian system, which is closely related to the Hilbert’s 16th problem. Many studies focused on the number of limit cycles for the following near-Hamiltonian system:. When ε = 0, system (1) becomes the following Hamiltonian system: ẋ = Hy , ẏ = − Hx . We suppose that the equation H ( x, y) = h defines a family periodic orbits Lh of system (2), where h ∈ J with J an open interval. The boundary of the family of periodic orbits Lh may be a center, a homoclinic loop or a heteroclinic loop, among other possibilities. To study limit cycle bifurcations of system (1), the following Melnikov function M(h, δ) =

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Results

Conclusion