Perturbation of a Period Annulus with a Unique Two-Saddle Cycle in Higher Order Hamiltonian

Hongying Zhu,Sumin Yang,Weihua Huang,Xiaochun Hu

doi:10.1155/2019/5813596

Abstract

In this paper, we study the number of limit cycles emerging from the period annulus by perturbing the Hamiltonian systemx˙=y,y˙=x(x2-1)(x2+1)(x2+2). The period annulus has a heteroclinic cycle connecting two hyperbolic saddles as the outer boundary. It is proved that there exist at most4and at least3limit cycles emerging from the period annulus, and3limit cycles are near the boundaries.

Highlights

The maximal number of limit cycles of the n-degree polynomial system ẋ = Pn (x, y), (1)y = Qn (x, y) is the topic of second part of Hilbert’s 16th problem [1]
We study the number of limit cycles emerging from the period annulus by perturbing the Hamiltonian system ẋ = y, ẏ = x(x2−1)(x2 +1)(x2+2)
It is proved that there exist at most 4 and at least 3 limit cycles emerging from the period annulus, and 3 limit cycles are near the boundaries

Summary

Introduction

We study the number of limit cycles emerging from the period annulus by perturbing the Hamiltonian system ẋ = y, ẏ = x(x2−1)(x2 +1)(x2+2). It is proved that there exist at most 4 and at least 3 limit cycles emerging from the period annulus, and 3 limit cycles are near the boundaries.

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Results

Conclusion