Abstract
Fractional-order calculus is more competent than integer-order one when modeling systems with properties of nonlocality and memory effect. And many real world problems related to uncertainties can be modeled with stochastic fractional-order systems with random parameters. Therefore, it is necessary to analyze the dynamical behaviors in those systems concerning both memory and uncertainties. The period-doubling bifurcation of stochastic fractional-order Duffing (SFOD for short) system with a bounded random parameter subject to harmonic excitation is studied in this paper. Firstly, Chebyshev polynomial approximation in conjunction with the predictor-corrector approach is used to numerically solve the SFOD system that can be reduced to the equivalent deterministic system. Then, the global and local analysis of period-doubling bifurcation are presented, respectively. It is shown that both the fractional-order and the intensity of the random parameter can be taken as bifurcation parameters, which are peculiar to the stochastic fractional-order system, comparing with the stochastic integer-order system or the deterministic fractional-order system. Moreover, the Chebyshev polynomial approximation is proved to be an effective approach for studying the period-doubling bifurcation of the SFOD system.
Highlights
Fractional calculus [1] had been proposed more than 300 years along with classical calculus, but really came to life over the last few decades
There have been several mathematical methods used to obtain the response of the system with random parameters numerically, for example, Monte Carlo method [9], stochastic perturbation method [10, 11], and orthogonal polynomial approximation method proposed by Spanos and Ghanem [12] and Jensen and Iwan [13] and developed by Li et al [14, 15] who presented a new scheme for the estimation of the mean values of parameters in multiple degree of freedom structural systems and a dynamic condensation algorithm for the orderexpanded equation
Inspired by the discussion above, we focus on analyzing the period-doubling bifurcation of the stochastic fractional-order Duffing (SFOD) system using Chebyshev polynomial approximation
Summary
Fractional calculus [1] had been proposed more than 300 years along with classical calculus, but really came to life over the last few decades. There have been several mathematical methods used to obtain the response of the system with random parameters numerically, for example, Monte Carlo method [9], stochastic perturbation method [10, 11], and orthogonal polynomial approximation method proposed by Spanos and Ghanem [12] and Jensen and Iwan [13] and developed by Li et al [14, 15] who presented a new scheme for the estimation of the mean values of parameters in multiple degree of freedom structural systems and a dynamic condensation algorithm for the orderexpanded equation.
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
