Bifurcations of safe basins and chaos in softening Duffing oscillator under multi-frequency harmonic and hounded noise excitation

The erosion of the safe basins and related chaotic motions of a softening Duffing oscillator under multi-frequency external periodic forces and bounded random noise are investigated. The system Melnikov integral and the parametric threshold for onset of chaos are obtained. The Melnikov global perturbation technique is therefore generalized to higher dimensional systems, and the erosion of safe basins is discussed. As an alternative definition, stochastic bifurcation may be defined as a sudden change in the character of stochastic safe basins when the bifurcation parameter of the system passes through a critical value, which applies successfully to either randomly perturbed motions, or purely deterministic motions. It is found that increasing the number of forcing frequencies or increasing the random noise may destroy the integrity of the safe basins to lead to the occurrence of the stochastic bifurcation and make the threshold for onset of chaos vary in a larger extent.