Stochastic Bifurcations of a Fractional-Order Vibro-Impact System Driven by Additive and Multiplicative Gaussian White Noises

Yong-Ge Yang,Ya-Hui Sun,Wei Xu

doi:10.1155/2019/6737139

Abstract

Stochastic fractional-order systems or stochastic vibro-impact systems can present rich dynamical behaviors, and lots of studies dealing with stochastic fractional-order systems or stochastic vibro-impact systems are available now, while the discussion on the stochastic systems with both vibro-impact factors and fractional derivative element is rare. This paper is concerned with the stochastic bifurcation of a fractional-order vibro-impact system driven by additive and multiplicative Gaussian white noises. Firstly, we can remove the discontinuity of the original system with the help of nonsmooth transformation and obtain the equivalent stochastic system. Then, we adopt the stochastic averaging method to get the approximately analytical solutions. At last, an example is discussed in detail to assess the reliability of the developed approach. We also find that the coefficient of restitution factor, fractional derivative coefficient, and fractional derivative order can induce the stochastic bifurcation.

Highlights

As the fractional-order models can more accurately describe the complex systems than the integer-order models, the investigation on the fractional-order systems attracts more and more attention
Wang et al [35] put forward a new procedure based on the generalized cell mapping (GCM) method to explore the stochastic response of vibro-impact systems numerically
In this paper, we will explore the response of a fractional-order vibro-impact oscillator driven by additive and multiplicative Gaussian white noises

Summary

Introduction

As the fractional-order models can more accurately describe the complex systems than the integer-order models, the investigation on the fractional-order systems attracts more and more attention. Li et al [28] considered the bifurcation control of a Van der Pol oscillator using the fractional-order PID controller These studies focused on the dynamical behaviors of smooth systems instead of nonsmooth systems. Wang et al [35] put forward a new procedure based on the generalized cell mapping (GCM) method to explore the stochastic response of vibro-impact systems numerically. Much attention was devoted to the study of vibroimpact systems, little work focused on the investigation of vibro-impact systems with the fractional derivative damping under random excitation. In this paper, we will explore the response of a fractional-order vibro-impact oscillator driven by additive and multiplicative Gaussian white noises. According to Refs. [32, 37, 39], we have (y_− − y_+)δ(t − t∗) ≈ (1 − r)y_|y_|δ(y), and we can get the following equivalent oscillator without impact term: y€ + εβ sgn(y)Dα(|y|) + εf(|y|, y_ sgn(y))y_

Stochastic Averaging Method

Conclusions