Stochastic Bifurcations of a Fractional-Order Vibro-Impact Oscillator Subjected to Colored Noise Excitation

A stochastic vibro-impact system has triggered a consistent body of research work aimed at understanding its complex dynamics involving noise and nonsmoothness. Among these works, most focus is on integer-order systems with Gaussian white noise. There is no report yet on response analysis for fractional-order vibro-impact systems subject to colored noise, which is presented in this paper. The biggest challenge for analyzing such systems is how to deal with the fractional derivative of absolute value functions after applying nonsmooth transformation. This problem is solved by introducing the Fourier transformation and deriving the approximate probabilistic solution of the fractional-order vibro-impact oscillator subject to colored noise. The reliability of the developed technique is assessed by numerical solutions. Based on the theoretical result, we also present the critical conditions of stochastic bifurcation induced by system parameters and show bifurcation diagrams in two-parameter planes. In addition, we provide a stochastic bifurcation with respect to joint probability density functions. We find that fractional order, coefficient of restitution factor and correlation time of colored noise excitation can induce stochastic bifurcations.

Stochastic Bifurcations of Group-Invariant Solutions for a Generalized Stochastic Zakharov–Kuznetsov Equation

In this paper, we introduce the concept of stochastic bifurcations of group-invariant solutions for stochastic nonlinear wave equations. The essence of this concept is to display bifurcation phenomena by investigating stochastic P-bifurcation and stochastic D-bifurcation of stochastic ordinary differential equations derived by Lie symmetry reductions of stochastic nonlinear wave equations. Stochastic bifurcations of group-invariant solutions can be considered as an indirect display of bifurcation phenomena of stochastic nonlinear wave equations. As a constructive example, we study stochastic bifurcations of group-invariant solutions for a generalized stochastic Zakharov–Kuznetsov equation.

Stochastic Bifurcations, Chaos and Phantom Attractors in the Langford System with Tori

The variability of stochastic dynamics for a three-dimensional dynamic model in a parametric zone with 2-tori is investigated. It is shown how weak Gaussian noise transforms deterministic quasiperiodic oscillations into noisy bursting. The phenomenon of stochastic generation of a phantom attractor and its shift with noise amplification is revealed. This phenomenon, accompanied by order-chaos transitions, is studied in terms of stochastic [Formula: see text]- and [Formula: see text]-bifurcations.

Bifurcation Analysis of a Self-Sustained Birhythmic Oscillator Under Two Delays and Colored Noises

In this manuscript, an investigation on bifurcations induced by two delays and additive and multiplicative colored noises in a self-sustained birhythmic oscillator is presented, both theoretically and numerically, which serves for the purpose of unveiling extremely complicated nonlinear dynamics in various spheres, especially in biology. By utilizing the multiple scale expansion approach and stochastic averaging technique, the stationary probability density function (SPDF) of the amplitude is obtained for discussing stochastic bifurcations. With time delays, intensities and correlation time of noises regarded as bifurcation parameters, rich bifurcation arises. In the case of additive noise, it is identified that the bifurcations induced by the two delays are entirely distinct and longer velocity delay can accelerate the conversion rate of excited enzyme molecules. A novel type of P-bifurcation emerges from the process in the case of multiplicative colored noise, with the SPDF qualitatively changing between crater-like and bimodal distributions, while it cannot be generated when the multiplicative colored noise is coupled with additive noise. The feasibility and effectiveness of analytical methods are confirmed by the good consistency between theoretical and numerical solutions. This investigation may have practical applications in governing dynamical behaviors of birhythmic systems.