Hopf-hopf Bifurcation Of Two Coupled Van Der Pol Oscillators With Delay Coupling

Time-delay double-coupled van der Pol oscillators are widely used in biological?chemical?physical and engineering fields.With different coupling strengths and time delays,it is able to a series of complex dynamical behaviors,such as equilibrium point bifurcation?Hopf bifurcation?Hopf-Hopf bifurcation?invariant tori,etc.In this thesis,we mainly study the existence of Hopf-Hopf bifurcation and quasi-periodic invariant tori of the double-coupled van der Pol oscillator model.Firstly,the research background and present situation of van der Pol oscillator model are introduced.Secondly,the coupling strength and time-delay are selected as the bifurcation parameters,and the critical condition of the Hopf-Hopf bifurcation of the system is obtained,by using the normal form theory of timedelay differential equations and the central manifold theorem,the canonical form of the system near the critical point is calculated to the fifth order.Finally,we obtain the parameter condition on the existence of invariant tori of the truncated normal form.Because the Hopf-Hopf bifurcation is two-co-dimensional and the truncated system may not be equivalent to the original system,that is,the existence of the invariant tori of the original system can not be obtained by the existence of the invariant tori of the truncated system.We use a KAM theorem to discuss the existence of quasi-periodic invariant tori after the truncation system is added with higher order terms.