Resonance behavior of fractional harmonic oscillator driven by exponentially correlated dichotomous noises

Abstract

This paper investigates the resonance behaviors of a fractional-order harmonic oscillator driven by two exponentially correlated dichotomous noises, where the Caputo fractional derivative operator is applied to describe the power-law memory of the system. By using the stochastic averaging method and the Shapiro-Loginov formula, we derive the analytical expression of the output amplitude gain of the system, from which the existence and the correlation of noises are found to be keys for the occurrence of resonance. When either of the noises is absent or they are uncorrelated, the output amplitude gain is zero, indicating that the system is dissipative in this case. The numerical simulation shows that the system has rich resonance behaviors when noises are exponentially correlated. Three types of resonance, that is, the bona fide resonance, the classic stochastic resonance and the generalized stochastic resonance, are observed. And the effects of system parameters on these resonance behaviors are well discussed. Specifically, double-peak resonance and damping-coefficient–induced resonance are observed only in the fractional-order system rather than integer-order system.