Duffing-Van der pol系统的Hopf分岔

Hopf bifurcation of a Duffing-Van der pol system

将保守Duffing系统作为未扰系统,并对它分四种情形进行了严格求解。用Melnikov函数方法研究了Duffing-Vanderpol系统的次谐分岔,获得了Duffing-Vanderpol系统的Hopf分岔条件。根据这些条件,在参数空间中确定了Hopf分岔曲线。在分岔曲线上取参数进行了数值模拟,所获得的奇、偶阶Hopf分岔与理论分析的结果完全一致。

Duffing equation x··+Gx+ξx3=0 was taken as a non-perturbed Hamiltonian system in a Duffing-van der pol system x··+Gx+ξx3+εβ(1+ηx2)x·=εf cos ωt,its four conditions were solved rigorously.Subharmonic bifurcation of the Duffing-van der pol system was studied by using the sub-Melnikov method,Hopf bifurcation conditions were obtained for the Duffing-van der pol system.Hopf bifurcation curves were gained based on the bifurcation conditions in the parameter space.On the bifurcation curves,parameters were taken to do numerical simulations,odd number order and even number order Hopf bifurcations obtained were in correspondence with the theoretical analyses.