Bifurcations Induced in a Bistable Oscillator via Joint Noises and Time Delay

Abstract

In this paper, noise-induced and delay-induced bifurcations in a bistable Duffing-van der Pol (DVP) oscillator under time delay and joint noises are discussed theoretically and numerically. Based on the qualitative changes of the plane phase, delay-induced bifurcations are investigated in the deterministic case. However, in the stochastic case, the response of the system is a stochastic non-Markovian process owing to the existence of noise and time delay. Then, methods have been employed to derive the stationary probability density function (PDF) of the amplitude of the response. Accordingly, stochastic P-bifurcations can be observed with the variations in the qualitative behavior of the stationary PDF for amplitude. Furthermore, results from both theoretical analyses and numerical simulations best demonstrate the appearance of noise-induced and delay-induced bifurcations, which are in good agreement.

Hopf-pitchfork bifurcation of coupled van der Pol oscillator with delay

“…Kumar et al [20] carry out investigations on the bifurcation characteristics of a Duffing-van der Pol oscillator subjected to white noise excitations. Fu et al [13] discuss noise-induced and delay-induced bifurcations in a bistable Duffing-van der Pol oscillator under time delay and join noises theoretically and numerically. Dubkov and Litovsky [7] investigate that the exact Fokker-Planck equation for the joint probability distribution of amplitude and phase of a van der Pol oscillator perturbed by both additive and multiplicative noise sources with arbitrary nonlinear damping is first derived by the method of functional splitting of averages.…”