Stochastic parametric resonance of a fractional oscillator

Underdamped motion of a fractional oscillator with fluctuating eigenfrequency subjected to an internal additive fractional noise is investigated. A friction kernel of the viscoelastic type, with memory, is assumed as a power-law function of time. The exact formula for the variance of the output signal in the long-time limit is derived. It is established that of intermediate values of the memory exponent, the energetic stability of the oscillator is enhanced in comparison with the cases of strong and low memory. Dependence of such an anomalous property of the oscillator on other system parameters is also discussed.