Bifurcation Analysis Of A Three Dimensional System

In order to enrich the stability and bifurcation theory of the three dimensional chaotic systems,taking a quadratic truncate unfolding system with the triple singularity equilibrium as the research subject,the existence of the equilibrium,the stability and the bifurcation of the system near the equilibrium under different parametric conditions are studied.Using the method of mathematical analysis,the existence of the real roots of the corresponding characteristic equation under the different parametric conditions are analyzed,and the local manifolds of the equilibrium are got,then the possible bifurcations are guessed.The conditions of codimension-one Hopf bifucation and the prerequisites of the supercritical and subcritical Hopf bifurcation are found by computation and the parametric conditions under which the equilibrium is saddle-focus is analyzed carefully by the Cardan formula.Moreover,the existence of the homoclinic or heteroclinic loop connecting saddle-focus is proved.These results show that the system has the Shilnikov's chaos and abundant stability and bifurcation.This method can be extended to study higher nonlinear systems.