The resonant behavior of fractional harmonic oscillator with fluctuating mass

Abstract

When moving in viscous medium, the mass of a Brownian particle is fluctuant and its damping force depends on the past velocity history. Therefore, in order to investigate the characteristics of Brownian motion in viscous medium, fractional harmonic oscillator is proposed in this paper for the first time so for as we know. First, the Shapiro-Loginov formula is fractionized to solve fractional stochastic differential equation with exponential correlative stochastic coefficients. Then, by using stochastic averaging method and fractional Shapiro-Loginov formula, the analytical expression of a system’s steady response amplitude is presented and the system’s resonant behavior is discussed accordingly. Finally, the reliability of theoretical results is tested by simulation experiments. All the research shows that: (1) Stochastic resonant behavior can be induced by mass fluctuation noise. (2) Parameter-induced resonance can be induced by memory damping force. (3) Under different parameter conditions, the system’s resonant forms are diverse.

Collective behaviors of two coupled harmonic oscillators driven by different frequency fluctuations with fractional damping

“…In the solution process, we will use the following formulas [60], which can be easily derived from the well-known Shapiro-Loginov procedure [61]:…”

Stochastic resonance of double fractional-order coupled oscillator with mass and damping fluctuations

“…In this paper, we model multiplicative noise μ(t) as symmetric dichotomous noise, whose values jump between two symmetric values σ and −σ(σ > 0). Assuming that the noise intensity is σ 2 and the noise correlation rate is λ, the dichotomous noise ξ(t) has the following statistical properties [29]: 6), represents internal noise caused by random collisions of molecules in the environment of the system. Here we assume that it is a Gaussian white noise with zero mean, that is,…”