Local Branches Of Several Types Of High-dimensional Systems

The main contribution of the paper is to study the local bifurcations of some higher dimensional differential systems.Bifurcations and limit cycles of nonlinear dynamical systems play an important role in many fields and have broad theoretical and practical applications.At present,the bifurcation theory of planar systems is very rich and deep.But as for higher dimensional systems,the bifurcation theory frame is still incompletely and the methods are relatively weak.This paper study the Hopf bifurcation,zero-Hopf bifurcation and bifurcation of periodic orbits on some special higher dimensional system and obtain some new results.The first chapter introduces the background of the research.Some preliminary ac-knowledgements related with the basic concepts and theorems are ready on the second chapter for the following work,including the limit cycles,Poincare map,bifurcation theo-ries of the planar systems,structural stability and so on.The reduction method which is most commonly used in bifurcations of higher dimensional systems is the center manifold.We introduce some theorems about it at the end of this section.In chap.3 we study the Hopf bifurcation of a 3 dimensional system,which is introduce recently by Ovsyannikov I.I.and Turaev D.V.and been named as Extended Lorenz System(abbreviated ELS).This system has five parameters.We selected one among them as the Hopf bifurcate parameter.With the aid of center manifold theorem,we reduce the original 3D system to the restriction on the local 2D center manifold.Thus we obtain three limit cycles,each bifurcating from the equilibrium of the system by the Hopf bifurcation theory of the planar system.The zero-Hopf bifurcation of the higher dimensional systems is discussed in the 4th chapter.We elaborate the Lyapunov-Schmidt reduction method introduced by Cesari and Hale.The Lyapunov-Schmidt reduction can be applied to the local and global bifurcation of the higher dimensional systems,including the bifurcation of periodic orbits,Hopf bifur-cation,zero-Hopf bifurcation,bifurcation in resonance and sometimes the bifurcation of functional differential equations.We discuss the codimensional 2 zero-Hopf bifurcation of the system(ELS)and give the sufficient conditions of the existence of limit cycles arising from the zero-Hopf equilibria.In chap.5 we study the zero-Hopf bifurcation of system(ELS)by means of averag-ing theory,which is widely used in the bifurcation theory of higher dimensional system.According to the averaging theory,we pass the system(ELS)to so called average system,which is autonomous and periodic.The numbers of limit cycles of the original system near the equilibrium is one to one corresponds to the zero of the average function in the right hand side of the average system.The result obtained based on the averaging theory is compared with the one gained by the Lyapunov-Schmidt reduction.The last chapter summarize the main contributions of the article and look forward to the future study?...