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

The complex dynamics in Duffing-Van der Pol equation with linear restoring and external excitations are investigated by applying bifurcation theory,melnikov methods and second-order averaging method. We prove the threshold values of existence of chaotic motion under the periodic perturbation,and the criterionof existence of chaos in averaged system under quasi-periodic perturbation forω₂ = nω₁ +εν,n = 1,2,3,4,6, and cannot prove the criterion of existence of chaos in second-order averaged system under quasi-periodic perturbation forω₂ = nω₁ +εν, n = 5, 7, 9 - 15,whereνis not rational toω₁, but can show the occurrence of chaos in original system by numerical simulation. Numerical simulationsincluding heteroclinic and homoclinic bifurcation surfaces, bifurcation diagrams, Lyapunov exponent, phase portraits and Poincar(?) map, not only show the consistence with the theoretical analysis but also exhibit some new complex dynamics, including the periodic(revesed periodic) doubling bifurcation leading to chaos, and the interleaving occurrences of chaotic behaviors and periodic windows,onset of chaos, the chaos suddenly disappearing or suddenly converting to periodic one orbit, the large chaotic regions with period-windows and without period-windows, the almost symmetric chaos regions, the region of invariant torus without period-windows, and the interior crisis. The investigation for the Duffing-Vander Pol equations with linear-restoring and period external excitations has not done much yet,and will enrich the content of dynamic systems,and will be useful in other subject.The thesis consists of two chapters as the following. Chapter 1 is the preparationknowledge. A brief review of second-order averaging methods and Melnikov methods,chaos and some routes to chaos for continual dynamical system are presented,the backgrounds and histories of Duffing Duffing-Van der Pol equations. In chapter 2, we study Duffing-Van der Pol equation with linear-restoring and external excitations by using bifurcation theory second-order averaging methods and Melnikov methods, we prove that 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 +εν,n = 1.2,3,4,6,and cannot prove the criterion of existence of chaos under quasi-periodic perturbation forω₂ = nω₁ +εν, n = 5, 7, 9 - 15, but can show the occurrence of chaos in original system by numerical simulation.we also show the complex dynamics as the nine-systems parameters vating.