The Center Conditions and Bifurcation of Limit Cycles at the Degenerate Singularity of a Three-Dimensional System

Shugang Song,Qinlong Wang,Jingjing Feng

doi:10.1155/2014/546243

Shugang Song, Qinlong Wang + Show 1 more

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https://doi.org/10.1155/2014/546243

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Abstract

We investigate multiple limit cycles bifurcation and center-focus problem of the degenerate equilibrium for a three-dimensional system. By applying the method of symbolic computation, we obtain the first four quasi-Lyapunov constants. It is proved that the system can generate 3 small limit cycles from nilpotent critical point on center manifold. Furthermore, the center conditions are found and as weak foci the highest order is proved to be the fourth; thus we obtain at most 3 small limit cycles from the origin via local bifurcation. To our knowledge, it is the first example of multiple limit cycles bifurcating from a nilpotent singularity for the flow of a high-dimensional system restricted to the center manifold.

Highlights

IntroductionAbout dynamical behavior of the trajectories of threedimensional system, bifurcation of limit cycles is one of major concerns; for Hopf bifurcation of a nondegenerate equilibrium with a pair of pure imaginary roots and a negative one, many investigations have been carried out in the past decades, for example, [1,2,3,4] for the three-dimensional chaotic systems, [5,6,7,8] for the three-dimensional Lotka-Volterra systems, and [9] for general three-dimensional systems
We discuss the centerfocus problem for the flow restricted to the center manifold, which closely relates to the maximum number of limit cycles bifurcating from the origin
For system (5), the origin is a center on the local center manifold if and only if the following condition is satisfied: d1 = 0 or a0 = b0 = 0

Summary

Introduction

About dynamical behavior of the trajectories of threedimensional system, bifurcation of limit cycles is one of major concerns; for Hopf bifurcation of a nondegenerate equilibrium with a pair of pure imaginary roots and a negative one, many investigations have been carried out in the past decades, for example, [1,2,3,4] for the three-dimensional chaotic systems, [5,6,7,8] for the three-dimensional Lotka-Volterra systems, and [9] for general three-dimensional systems. We discuss the centerfocus problem for the flow restricted to the center manifold, which closely relates to the maximum number of limit cycles bifurcating from the origin. In [17, 18], the authors gave an inverse integral factor method of calculating the quasi-Lyapunov constants of the three-order nilpotent critical point; it is convenient to compute the higher order focal values and solve the center-focus problem of the equilibrium. We extend this method’s application to the three-dimensional system (1) and consider its specific example as follows: a0ux. The system and problem are all considered for the first time

Quasi-Lyapunov Constants and Center Conditions

Multiple Limit Cycle Bifurcation of the System

Conclusions