Bifurcations and Chaos in the Duffing Equation with Parametric Excitation and Single External Forcing

We study the Duffing equation with parametric excitation and single external forcing and obtain abundant dynamical behaviors of bifurcations and chaos. The criteria of chaos of the Duffing equation under periodic perturbation are obtained through the Melnikov method. And the existence of chaos of the averaged system of the Duffing equation under the quasi-periodic perturbation Omega = n omega + epsilon nu (where nu is not rational relative to omega) and n = 1, 2, 4, 6 is shown, but the existence of chaos of averaged system of the Duffing equation cannot be proved when n = 3, 5, 7- 15, whereas the occurrence of chaos in the original system can be shown by numerical simulation. Numerical simulations not only show the correctness of the theoretical analysis but also exhibit some new complex dynamical behaviors, including homoclinic or heteroclinic bifurcation surfaces, bifurcation diagrams, Lyapunov exponent diagrams, phase portraits and Poincare maps. We find a large chaotic region with some solitary period parameter points, a large period and quasi-period region with some solitary chaotic parameter points, period-doubling to chaos and chaos to inverse period-doubling, nondense curvilinear chaotic attractor, nonattracting chaotic motion, nonchaotic attracting set, fragmental chaotic attractors. Almost chaotic motion and almost nonchaotic motion appear through adjusting the parameters of the Duffing system, which can be taken as a strategy of chaotic control or a strategy of chaotic motion to nonchaotic motion.