On simulation of a bistable system with fractional damping in the presence of stochastic coherence resonance

Grzegorz Litak,Marek Borowiec

doi:10.1007/s11071-014-1330-4

Abstract

A bistable dynamical system with the Duffing potential, fractional damping, and random excitation has been modelled. To excite the system, we used a stochastic force defined by Wiener random process of Gaussian distribution. As expected, stochastic resonance appeared for sufficiently high noise intensity. We estimated the critical value of the noise level as a function of derivative order $$q$$ . For smaller order $$q$$ , damping enhancement was reported.

Highlights

Fractional-order systems have been intensively studied in various contexts [1]
Cao et al [19] investigated the fractionally damped system response using phase diagrams, bifurcation diagrams and Poincare maps in a wide range of the fractional order that changes from 0.1 to 2.0. Their analysis results show that the fractional-order- damped Duffing system could be treated as a bifurcation parameter
In contrast to Ref. [21], we terminate the harmonic component and study the nonlinear Duffing system with a fractional derivative subjected to a random excitation force defined as generated with an additive white Gaussian noise term

Summary

Introduction

Fractional-order systems have been intensively studied in various contexts [1]. In mechanical engineering, there were suggestions to apply it for various complex nonviscous, memory- effected damping effects like rubbing or composite material response including natural wooden composites. Randomly excited nonlinear systems show a number of interesting features such as stochastic Hopf bifurcation [10], period-doubling bifurcations [11] and a stochastic resonance phenomenon [12] This resonance is characterized by the flow over the potential barrier. Their analysis results show that the fractional-order- damped Duffing system could be treated as a bifurcation parameter By continuing these studies, Chen et al [20] and Hu et al [21] analysed such a system with a bounded noise excitation term composed of harmonic excitation with an additional random phase. In contrast to Ref. [21], we terminate the harmonic component and study the nonlinear Duffing system with a fractional derivative subjected to a random excitation force defined as generated with an additive white Gaussian noise term

The model and equations of motion

Simulation results

Conclusion