Hopf Bifurcation For Several Classes Of Chaotic Systems

The study of chaotic systems has become an important aspect in nonlinear science. Since the chaotic systems not only contain abundant complex and interested dynamical behaviors such as strange attractors and bifurcations, but also have potential applications in engineering science, ecology and economics.In this paper, three new chaotic (hyperchaotic) systems are constructed based on some classic chaotic systems. The existence of chaos in the new chaotic systems is investigated by Silnikov theorem and Lyapunov exponents, and the complex structure of the chaotic attractor in a new hyperchaotic system as well as its formation mechanism have been revealed by a constant controller. By choosing an appropriate bifurcation parameter, the existence of Hopf bifurcation is studied and we study in detail Hopf bifurcation in the integer-order chaotic systems by means of the normal form theory. With the stability theory, Hopf bifurcation and the Hopf circles in the new systems are also controlled and anti-controlled via linear feedback control as well as a modified projective synchronization method. Finally, the corresponding numerical simulations verify the theoretical analysis effectively.This paper is organized as following four chapters:In the first chapter, some important results of chaotic systems and Hopf bifurcation theory given by domestic and foreign scholars are presented, and their theoretical and practical value is introduced as well.In the second chapter, a new Rossler-like system is proposed. The existence of chaos and Hopf bifurcation in this system is investigated, the direction and the stability of bifurcating periodic solutions are presented in detail by the normal form theory. Furthermore, the anti-control of the Hopf circles is achieved by modified projective synchronization.In the third chapter, a new hyperchaotic system is presented and the complex structure of the chaotic attractor and its formation mechanism has been investigated by adding a constant controller. In addition, the existence of Hopf bifurcation in the system, the direction and the stability of bifurcating periodic solutions are investigated in detail.In the fourth chapter, a new four-dimensional fractional-order chaotic system is cons- tructed, the existence of chaos and Hopf bifurcation in the new fractional order system has been investigated by detail theoretical and numerical analysis. Furthermore, the Hopf bifurcation in the new system is eliminated via linear feedback control.