Complex Dynamics In Duffing-Van Der Pol System

In this thesis,we investigate the complex dynamics of the Duffing-Van der Pol system with external and parametric excitations.By applying Melnikov method,the threshold values of existence of chaotic motion are obtained under the periodic perturbation.By applying the second-order averaging method and Melnikov method,we prove the criterion of existence of chaos in averaged system under quasi-periodic perturbation forω= nω₁ +∈ν,n = 1,2,4,6,and cannot prove the criterion of existence of chaos in second-order averaged system under quasi-periodic perturbation forω= nω₁ +∈ν,n = 3,5,7 - 15,whereνis irrational toω₁,but can show the occurrence of chaos in original system by numerical simulation.Numerical simulations including heteroclinic and homoclinic bifurcation surfaces,bifurcation diagrams,Lyapunov exponent,phase portraits and Poincare map,not only show the consistence with the theoretical analysis but also exhibit some new complex dynamics:the period-doubling bifurcation from period-1,and period-2 orbit leading chaos,and the interleaving period-doubling bifurcation and the interleaving reverse period-doubling bifurcation leading to chaos,the chaotic regions with periodic window and invariant torus,and the large region of invariant torus,and the " small degree " (L＞0,～0)of chaotic regions,and the onset of chaos,and the interior crisis occurring more than one.The investigation for the Duffing-Van der Pol system,which hasn't done much yet,is of fundamental and even practical interest.And the dynamical behaviors of these systems will enrich the content of nonlinear dynamical systems and will be useful in other subjects such as optics,physics.The paper consists of three chapters.Chapter 1 is the preparation knowledge. A brief review of second-order averaging methods,Melnikov theory and chaos theory for continual dynamical system is presented.In chapter 2,we briefly introduce the backgrounds and histories of Duffing, Van der Pol and Duffing-Van der Pol systems.In chapter 3,we study Duffing-Van der Pol system with external and parametric excitations by using second-order averaging methods and Melnikov methods. The threshold values of existence of chaotic motion are obtained under the periodic perturbation,and the criterion of existence of chaos in averaged system under quasi-periodic perturbation forω= nω₁ +∈ν,n = 1,2,4,6,and cannot prove the criterion of existence of chaos in second-order averaged system under quasi-periodic perturbation forω=nω₁ +∈ν,n = 3,5,7—15,whereνis irrational toω₁,but can show the occurrence of chaos in original system by numerical simulation.The numerical simulations not only show the consistence with the theoretical analysis but also exhibit some new complex dynamics.