Bifurcations and chaos in Duffing equation with damping and external excitations

Abstract

Duffing equation with damping and external excitations is investigated. By using Melnikov method and bifurcation theory, the criterions of existence of chaos under periodic perturbations are obtained. By using second-order averaging method, the criterions of existence of chaos in averaged system under quasi-periodic perturbations for Ω = nω + ɛσ, n = 2, 4,6 (where σ is not rational to ω) are investigated. However, the criterions of existence of chaos for n = 1, 3, 5, 7–20 can not be given. The numerical simulations verify the theoretical analysis, show the occurrence of chaos in the averaged system and original system under quasiperiodic perturbation for n = 1, 2, 3, 5, and expose some new complex dynamical behaviors which can not be given by theoretical analysis. In particular, the dynamical behaviors under quasi-periodic perturbations are different from that under periodic perturbations, and the period-doubling bifurcations to chaos has not been found under quasi-periodic perturbations.