The Double Hopf Bifurcation Analysis And Periodic Solutions By Homotopy Analysis Method For Two Coupled Van Der Pol Oscillators With Delay Coupling

Along with thorough research and an urgent need for engineering application,almost all of practical engineering problems are described by nonlinear dynamical systems,especially coupled nonlinear systems and time-delay nonlinear systems.What is more,there exist some complex nonlinear dynamical systems,which have both coupling and time delay.And we call they coupled nonlinear systems with time delay.So in this paper,we study the dynamical behavior of a system of two coupled van der Pol oscillators with delay coupling,by the method of multiple scales(MMS)and homotopy analysis method(HAM).And the system can be described as follows:This article consists of three chapters.In the first chapter,the research background is briefly introduced,and the research status of double Hopf bifurcation and HAM is respectively discussed.In the second chapter,we first analyze the distribution of the eigenvalues of its linearized characteristic equation,and obtain the critical values for the occurrence of double Hopf bifurcation,which is caused by time delay and strength of coupling.And MMS is adopted to acquire the normal form equations.Then,the dynamical behavior around the 3:5 weakly resonant double Hopf bifurcation point is unfolded and classified.In the third chapter,HAM is applied to obtain analytical approximate solutions of the system with parameter values located in different regions.And to illustrate the accuracy of the present method,the solutions obtained in this paper are compared with those of Runge-Kutta method,which shows HAM is effective and feasible.