The Existence Of Quasi-periodic Invariant Tori And Double Hopf Bifurcation Of Van Der Pol Oscillator With Delay

Since the van der Pol oscillator was introduced into the mathematical model of differential equations,the van der Pol oscillator model has been widely used in circuit engineering science and many physics problems.The van der Pol oscillator model with the change of parameters has complex dynamic all behaviors,such as Takens-Bogdanov bifurcation,double Hopf bifurcation,invariant tori,etcThis thesis is devoted to study the existence of quasi-periodic invariant tori for a delay coupled van der Pol oscillator model with double Hopf bifurcations.Firstly,we introduce the delayed coupling van der Pol oscillator model;then choosing the coupling parameter and delay as bifurcation parameters,the sufficient conditions on the critical point of double Hopf bifurcation are obtained,respectively.Based on the normal form method and the center manifold theorem of delay differential equations,a normal form of the nonresonant double Hopf bifurcation up to the fifth order is derived.In order to analyze the existence of invariant tori of truncated system,the normal forml is rewritten in polar coordinates.Finally,we can prove that the original system still has quasi-periodic invariant tori for most parameter values at the end of the thesis.