Bifurcations of safe basins and chaos in Flickering oscillator under multi-frequency harmonic and bounded noise excitation

The erosion of the safe basins and related chaotic motions of a Flickering oscillator under multi-frequency external periodic forces and bounded random noise are studied. By the Melnikov method, the system's Melnikov integral is computed and the parametric threshold for the onset of chaos is obtained. The Melnikov's global perturbation technique is therefore generalized to higher dimensional systems. Using the Monte-Carlo and Runge-Kutta method, the erosion of safe basins is also 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. This definition applies equally well either to randomly perturbed motions or to 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, bring forward the occurrence of the stochastic bifurcation and make the threshold for onset of chaos vary in a larger extent, which hence makes the system less safe and chaotic motions easier to occur.