Bursting Oscillation And Its Bifurcation Mechanism Of Duffing-van Der Pol System With Periodic Excitation

In recent decades,with the development of science and technology and the deepening of theoretical research,scholars at home and abroad have conducted in-depth studies on the dynamic behavior of coupling systems under different time scales from theoretical analysis and numerical simulation.In this paper,we mainly study the bursting oscillation behavior of a Duffing-van der Pol system with periodic excitation.The research contents are as follows:A periodic excitation is introduced into a Duffing-van der Pol system.By using the slow-fast dynamic analysis method,the bifurcation behaviors of the slow-fast system are analysed.By analysis,we can acquire the fold bifurcation set and Hopf bifurcation set.According to the Hopf bifurcation theorem,the existence of Hopf bifurcation points is proved.And the first Lyapunov coefficient method is used to judge the stability and bifurcation direction of Hopf bifurcation.Using u1 and F as bifurcation parameters,the bifurcation behavior in two typical cases is analyzed,and the equilibrium curves under different u1are obtained.In the above two typical cases,the bifurcation mechanism of the bursting oscillation is analyzed.Through numerical simulations,we get four kinds of typical bursting oscilla-tions,namely symmetric fold/fold bursting,symmetric fold/supHopf bursting,symmetric subHopf/fold cycle bursting and symmetric subHopf/subHopf bursting.The bifurcation mechanisms of the four kinds of bursting oscillations are further revealed by using the overlap portrait.The CY limit cycles under different u1 in the case ? are discussed.Taking the symmetric fold/fold bursting oscillation in the case I as an example,the ef-fects of the excitation frequency ?,the excitation amplitude f and the parameter u1 on the bursting oscillation are discussed by the numerical simulation.And the parameter regions in which different oscillations occur are obtained.