Center-focus and Hopf bifurcation for a class of quartic Kukles-like systems

Yusen Wu,Changjin Xu

doi:10.1186/1687-1847-2014-245

Abstract

In the present article, we solve the center-focus problem for a class of quartic Kukles-like systems with third-order nilpotent singularities and prove the existence of five limit cycles in the neighborhood of the origin.MSC:34C05, 37G15.

Highlights

One of the most classical problems in the qualitative theory of planar analytic differential systems is to characterize the local phase portrait at an isolated singular point
Once we have made a distinction between a center and a focus, another problem is to find the number of limit cycles bifurcated from the focus
If a real analytic system has a nilpotent center at the origin, after a linear change of variables and a rescaling of time variable, it can be written in the following form: x = y + X(x, y), ( . )

Summary

Introduction

One of the most classical problems in the qualitative theory of planar analytic differential systems is to characterize the local phase portrait at an isolated singular point. Using the integral factor method, [ ] investigated center conditions and bifurcation of limit cycles at the nilpotent critical point in a class of septic polynomial differential systems.

Results

Conclusion