Resonant behavior of a fractional oscillator with fluctuating frequency

Abstract

The long-time behavior of the first moment for the output signal of a fractional oscillator with fluctuating frequency subjected to an external periodic force is considered. Colored fluctuations of the oscillator eigenfrequency are modeled as a dichotomous noise. The viscoelastic type friction kernel with memory is assumed as a power-law function of time. Using the Shapiro-Loginov formula, exact expressions for the response to an external periodic field and for the complex susceptibility are presented. On the basis of the exact formulas it is demonstrated that interplay of colored noise and memory can generate a variety of cooperation effects, such as multiresonances versus the driving frequency and the friction coefficient as well as stochastic resonance versus noise parameters. The necessary and sufficient conditions for the cooperation effects are also discussed. Particularly, two different critical memory exponents have been found, which mark dynamical transitions in the behavior of the system.

Trichotomous Noise Induced Resonance Behavior for a Fractional Oscillator with Random Mass

“…We emphasize that the following inequality must hold for the sake of the stability of solutions [39]:…”

Friction-induced Resonance in a Noisy Fractional Oscillator

“…(11), (18), (19), and (21), the dependence of SPA, variance σ 2 , and SNR on the friction constant γ. We will not study all possible models of the additive noise ξ (t) in Eq.…”

“…It is seen from Eqs. (19) and (24) that the formation of the multiresonance versus γ can be explained as an interplay of two different effects: one caused by the energetic instability of the oscillator and another, which is due to the friction-induced resonance of A(γ). Thus, in the case of an internal noise the resonance of SNR versus γ exists for all physical values of the parameters in the dynamical equation of the fractional oscillator (1).…”

“…This is quite surprising in view of the fact that the importance of multiplicative fluctuations and viscoelasticity for biological systems, e.g., living cells, has been well recognized [17,18]. Thus motivated, the authors of [10,19] have recently considered fractional oscillators with fluctuating eigenfrequency subjected to an external periodic force and an additive noise. These models demonstrate that an interplay of noises and memory can generate a variety of cooperative effects, such as memory-enhanced energetic stability, stochastic resonance (SR) versus noise parameters, as well as frictioninduced resonance.…”

Stochastic resonance in a harmonic oscillator with fractional-order external and intrinsic dampings

“…For example, Soika et al [30][31][32] obtained the expression of the output amplitude and investigated the stochastic resonance in a fractional oscillator with multiplicative noise. Zhong et al [33] obtained an expression for the signal-to-noise (SNR) and studied the resonant behavior of an underdamped fractional oscillator.…”