Different Stochastic Resonances Induced by Multiplicative Polynomial Trichotomous Noise in a Fractional Order Oscillator with Time Delay and Fractional Gaussian Noise

Abstract

A general investigation on the mechanism of stochastic resonance is reported in a time-delay fractional Langevin system, which endues a nonlinear form multiplicative colored noise and fractional Gaussian noise. In terms of theoretical analysis, both the expressions of output steady-state amplitude and that of the first moment of system response are obtained by utilizing stochastic averaging method, fractional Shapiro and Laplace methods. Due to the presence of trichotomous colored noise, the excitation frequency can induce fresh multimodal Bona fide stochastic resonance, exhibiting much more novel dynamical behaviors than the non-disturbance case. It is verified that multimodal pattern only appears with small noise switching rate and memory damping order. The explicit expressions of critical noise intensity corresponding to the generalized stochastic resonance are given for the first time, by which it is determined that nonlinear form colored noise induces much more of a comprehensive resonant phenomena than the linear form. In the case of slow transfer rate noise, a newfangled phenomenon of double hypersensitive response induced by a variation in noise intensity is discovered and verified for the first time, with the necessary range of parameters for this phenomenon given. In terms of numerical scheme, an efficient and feasible algorithm for generating trichotomous noise is proposed, by which an algorithm based on the Caputo fractional derivative are applied. The numerical results match well with the analytical ones.

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