Complex Dynamics In Duffing System

Duffing equation with fifth nonlinear restoring force and one external forcing term, including parametric excitation terms is investigated in detail. 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 forω₂ = nω₁ +∈v,n = 1,2,3,4,6,8, and cannot prove the criterion of existence of chaos in second-order averaged system under quasi-periodic perturbation forω₂ = nω₁ +∈v,n = 5,7,9 - 15, 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 Poincáre map, not only show the consistence with the theoretical analysis but also exhibit some new complex dynamics. Including the reversed periodic doubling bifurcation leading to chaos, and the interleaving occurrences of chaotic behaviors and periodic windows, the chaos suddenly disappearing, 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. We also given the chaotic attractor and invariant torus for showing the different dynamical behaviors.The paper consists of three chapters. In chapter 1, we briefly introduce the backgrounds and histories of Duffing Duffing equations.Chapter 2 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 3, we study Duffing equation with external forcing and parametric excitation terms by using second-order averaging methods and Melnikov methods. The threshold values of existence of chaotic motion are obtained un- der the periodic perturbation, and the criterion of existence of chaos in averaged system under quasi-periodic perturbation forω= nω₁ +∈v,n = 1,2,3,4,6,8, and cannot prove the criterion of existence of chaos under quasi-periodic perturbation forω₂ = nω₁ +∈v, n = 5,7,9 - 15, but can show the occurrence of chaos in original system by numerical simulation.