Periodic Solutions Bifurcating From a Curve of Singularity of the Jerk System

Abstract

We investigate a periodic solution which bifurcates from a curve of the singularity of the jerk system in .More precisely, we give the explicit states for the existence of a periodic solution of the jerk system with a nonisolated singular point, where for each singular point has a simple pair of purely imaginary and one zero eigenvalues. We recall for this point of singularity as a zero-Hopf (z-H) singular point. The coefficients in the jerk system are described for which the z-H singularity occur at each point of that curve of singularity. We show that for each point at that curve of singularity there is only one family of parameters which exhibits such type of singular points. The method of averaging in the second order is utilized to determine one periodic solution which bifurcates from any point of that curve of singularity. As far as, we realize that this investigation is the study on bifurcations from a curve of nonisolated z-H singularity to provide a periodic solution via the method of averaging. Under a generic small perturbation at the parameters, we prove that a periodic solution will be bifurcated at any point that located on a curve of a singularity of the jerk system.

Keywords

• jerk system
• periodic orbit
• zero-hopf singularity
• the method of averaging