Complex Dynamics In Duffing System With Nonlinear Damping And External Excitations

In this thesis, by applying bifurcation theory, second order averaging method, Melnikov's method in applied dynamical systems, and some numerical simulations, Duffing system with nonlinear dissipation or damping and two external excitations is investigated in detail. The criterion of existence of chaos under the periodic perturbation is given by using Melnikov's method. By using second-order averaging method and Melnikov's method we proved the criterion of existence of chaos in averaged systems under quasi-periodic perturbation(the ratio of the two external forcings is approximately equal to 1:1, 1:3 or 1:5 and is irrational). Numerical simulations including heteroclinic and homoclinic bifurcation surfaces, bifurcation diagrams, maximum Lyapunov exponents and Poincarémap are given to illustrate the theoretical analysis, and to expose the more new complex dynamical behaviors such as symmetry breakings, reverse period-doubling bifurcations routes to chaos from period-2 orbit, period-doubling bifurcations routes to chaos, coexistence of multiple chaotic attractors, boundary crisis, suddenly disappearing of chaos, conversion of period-1 orbit to chaos through period-doubling bifurcations routes, suddenly increasing of chaotic scale, period six window, convergence of chaotic dynamic to period four orbit, the coexistence of periodic orbit and the chaotic attractors, alternate conversion of periodic orbits, quasi-periodic orbits and chaotic dynamics, chaotic dynamics suddenly disappearing and converting to period one orbits and converting to chaos again through forward period-doubling bifurcations routes, and so on. The theoretical and numerical results will enrich the research of bifurcation and chaos.