Stability And Hopf Bifurcation Of Shimizu-Morioka Time Delayed Feedback Control System

Time delay is ubiquitous in systems control,physics,ecology and other fields.A large number of studies on the causes and control methods of nonlinear systems,which are either from actual industrial production processes or natural and social scientific phenomena,show that even a small time delay in complex control systems may cause great change on the dynamics of systems.By introducing time delay,the chaotic system can be stabilized from the chaotic state and bifurcation may occur as time delay varies.This thesis focuses on the study of time-delay feedback control of Shimizu-Morioka chaotic system.Stability and Hopf bifurcation are analyzed by introducing two types of time delayed feedback into Shimizu-Morioka chaotic system.The occurrence of Hopf bifurcation and the properties of bifurcating periodic solutions are derived accordingly.The detailed work is as follows:1.By introducing a single time delayed feedback term into the third equation of Shimizu-Morioka chaotic system,the local stability of this system is studied by means of root distribution of transcendental equation.Regarding the time delay as the bifurcating parameter,the occurrence of Hopf bifurcation is analyzed and the explicit formulae for determing properties of bifurcating periodic solutions are derived by applying the center manifold theorem and norm form theory.Numerical simulations are presented to illustrate the applicability of the theoretical results.2.Shimizu-Morioka chaotic system with distributed time delayed feedback is studied.Regarding the mean time delay as the bifurcating parameter,stability and the occurrence of Hopf bifurcation of this system are investigated under the strong and weak kernel cases,respectively.Explicit formulae for determing the properties of Hopf bifurcation are derived by applying the center manifold theorem and norm form theory.An illustration numerical example is provided to verify the theoretical results.