Complex Dynamics In Duffing-Van Der Pol System With Nonlinear Restoring And External Excitations

The complex dynamical behaviours as the system's parametrics varing in Duffing-Van der Pol equation with nonlinear restoring and external excitations are investigated by using bifurcation theory,second-order averaging methods,Melnikov methods and chaotic theory.Wo prove that conditions of existence of harmoniz and its bifurcations,and The threshold values of existence of chaotic motion 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 forω₂ = nω₁ +∈ν,n=1,2,3,4,5,7,and cannot prove the criterion of existence of chaos in second-order averaged system under quasiperiodic perturbation forω₂ = nω₁ +∈ν,n = 6,8,9 - 15,whereω₁ is rational toν, 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,we found that periodic doubling bifurcation and the reversed periodic from period-one-three doubling bifurcation leading to chaos,and the interleaving occurrences of chaotic behaviors and periodic windows,the onset of chaos and chaos suddenly conventing to periodic-two ubits,the large chaotic regions with complex period-windows and without period-windows,the chaotic regious with complex invariant torus,the region of invariant torus without period-windows,and the interior crisis.The paper consists of three chapters,as the following,Chapter 1 is the preparation knowledge.A brief review of second-order averaging methods and Melnikov methods,chaos and some routes to chaos for continual dynamical system is presented.In chapter 2,we briefly introduce the backgrounds and histories of Duffing Duffing-Van der Pol equations.In chapter 3,we study Duffing-Van der Pol equation with nonlinear restoring and external excitations.we prove that the conditions of harmoniz and its bifurcation, and the threshold values of existence of chaotic motion under the periodic perturbation,and the criterion of existence of chaos in averaged system under quasi-periodic perturbation forω= nω₁ +∈ν,n = 1,2,3,4,5,7,and cannot prove the criterion of existence of chaos under quasi-periodic perturbation forω₂= nω₁ +∈ν,n = 6,8,9 - 15,but can show the occurrence of chaos in original system by numerical simulation.we find many new complex dynamics as the ten-system's parametries varing by numerical simulation.