Resonant Behavior of a Fractional Oscillator with Random Damping

A fractional oscillator with a power‐law memory kernel subjected to an external periodic force is considered. The influence of the fluctuating viscoelastic environment is modeled by a multiplicative dichotomous noise (fluctuating damping) and an additive internal noise. The main purpose of this work is to provide exact formulas for the analytic treatment of the dependence of spectral amplification on system parameters: viz. the noise correlation time, noise amplitude, memory exponent, and driving frequency. Based on those exact expressions we demonstrate that stochastic resonance is manifested in the dependence of the spectral amplification upon the noise parameters. Moreover, a critical memory exponent is found which marks the transitions between different dynamical regimes of the oscillator.