The Existence Of Quasi-Periodic Invariant Tori For Delay-Coupled Limit Cycle Oscillators

Nonlinear coupling limit ring oscillators are widely involved in many field-s,the coupled system can produce rich dynamic characteristics.With different coupling strength and delays,the coupled system with delays possesses a series of complicated phenomenon,such as double Hopf bifurcation,quasi-periodic mo-tions,three-dimensional invariant tori and so on.In this thesis,the bifurcation theory and numerical analysis are applied to studying the existence of quasi-periodic invariant tori for a delay coupled limit cy-cle oscillator model with double Hopf bifurcations.First of all,we introduce the delayed coupling limit ring oscillator model;second,select the coupling strength and delay as bifurcation parameters,by analysing eigenvalues of the linearized system,the critical and transverse conditions of 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.Finally,near the double Hopf bifurcation point,we verify specification parameter conditions on the existence of invariant tori for the time-delay coupling limit ring oscilla-tor model truncated system.By using theorems 1 and 2 in[21],the sufficient conditions on the existence of quasi-periodic invariant tori in the system adding higher-order terms are verified by matlab simulation.Moreover,since all transfor-mations are reversible in the normalization process,it can be concluded that the original system also has quasi-periodic invariant 2-dimensional and 3-dimensional tori for most of parameter values admitted,respectively.