Bifurcations in tri-stable Duffing–Van der Pol oscillator with recycling noise

Abstract

Recycling noise is a kind of more common noise. The nonlinear dynamic system can be controlled by adjusting its parameters. However, so far, the effect of recycling noise on tri-stable dynamic system has not been reported. In this paper, stochastic P-bifurcations in tri-stable Duffing–Van der Pol oscillator induced by additive recycling noise are investigated. Firstly, the stationary probability density function is derived using stochastic averaging method. Then, the general expression of the critical parameter conditions of stochastic P-bifurcation is given by the singularity theory. The stationary probability density of response amplitude in different parameter areas are also shown, which is verified by Monte Carlo numerical simulation. Based on these results, the influence of related parameters of recycling noise and damping coefficient on the stochastic P-bifurcation is studied. The result shows that the critical parameter of bifurcation can be changed by adjusting the delay time and fraction coefficient of the recycling noise. It has also been found that the stationary probability density and stochastic bifurcation show a periodic dependence on the delay time.

Fractional stochastic vibration system under recycling noise

“…Setting τ 2, k 0.4, ε −0.1, α 1 1.51, α 2 2.85, α 3 1.693, α 4 0.312, and ω 1 in Eq. 13 as that in Refs [30,53], the stochastic P-bifurcation is studied hereby. Keeping D 1 0.005 constant, we draw the joint probability density function section and top view of Eq.…”

Response and Bifurcation of Fractional Duffing Oscillator under Combined Recycling Noise and Time-Delayed Feedback Control

“…First, the influence of time-delayed feedback control is not considered, set the system parameters as [18,19] , stochastic P-bifurcation refers to the change in the number of peaks of the probability density function curve. Keep D = 0.9 constant, and draw the joint probability density function diagram, section and top view of system (26) under the influence of different fractional orders.…”

“…Finally, the influence of time-delayed feedback control is considered, set the system parameters as c = 0.2 ω = 1 α = 1 ε = 0.2 k = 0.4 τ 1 = 2 a 1 = 0.1 p = 0.2 D = 0.9 a 2 = 0.1 [18,19] , drawing the joint probability density function diagram, section and top view of system (26) under the influence of different time-delay.…”