The Existence Of Quasi-Periodic Invariant Tori For Double Hopf Bifurcation Of Van Der Pol-Duffing Oscillator With Delay Feedback Control

In this thesis,we mainly discuss the existence of quasi-periodic invariant torus near the double Hopf bifurcation point for the Van der Pol-Duffing model.The thesis is divided into four chapters.The first chapter mainly introduces the research background of the Van der Pol-Duffing model with delay.The second chapter introduces a KAM theorem that is used to prove the existence of quasiperiodic invariant tori.The main content will be detailed in Chapters 3 and 4respectively.In Chapter 3,we analyze existence conditions on the double Hopf bifurcation in the Van der Pol-Duffing model and a normal form of the system.Firstly,the decay rate and the decay are used as bifurcation parameters.When the linearized system has two pairs of purely imaginary roots,we can obtain the critical condition generating the double Hopf bifurcation.Then,under the time scale transformation,we easily derive the fifth-order norm form of the system near the critical point by the central manifold theorem and the normal form method of delay differential equations.Finally,we write the norm form in polar coordinates in order to analyze the existence of invariant tori of the truncated system conveniently.In Chapter 4,we analyze the existence of invariant tori of the truncated system.The existence of invariant tori of the truncated system cannot imply the existence of quasi-periodic invariant tori of the original system.But,we can prove that the original system still has quasi-periodic invariant tori for most parameter values by KAM theory at the end of the thesis.