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

Abstract

In a complex environment, the long-time memory and coupling effect are two important system characteristics that can be described by the fractional damping force and coupling force. This paper investigates the collective behaviors of two coupled fractional harmonic oscillators driven by different frequency fluctuations, including stability, synchronization and stochastic resonance (SR). Theoretically, the ‘synchronization condition’ and ‘stability condition’ of the system are derived. Comparative analysis shows that the latter is stricter than the former. Based on this, an analytical expression of the output amplitude gain is obtained. The numerical results show that when the stability condition is met, the average trajectories of two particles are both bounded and synchronous. Otherwise, they will diverge to infinity. Increasing ɛ (coupling strength) and decreasing α (fractional order) can both accelerate the synchronization speed. SR mainly occurs in the high-α or high-σ (noise amplitude) region, which means that SR emergence can be controlled by adjusting α or σ. The damping force, coupling force and frequency fluctuations compete with each other; thus, the SR intensity should be maximized by adjusting α, ɛ and σ simultaneously.

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

“…In this section, we simulate the system equation and verify the theoretical results by numerical results. For numerical simulation of fractional differential equations, starting from the definition of fractional differential, which can be considered as an extension of the limit form of integer order calculus and difference definition [38]. For a k-order derivable function f (t), its first derivative is defined as:…”

Influence of colored cross-correlated noise on stochastic resonance for an underdamped bistable system subjected to multiplicative and additive noises

“…The phenomenon of stochastic resonance (SR) is an interesting nonlinear phenomenon occurring in a noisy system, which means that the addition of certain amount of noise can optimizing the system output performance thanks to the cooperation between the noise and the system [7][8][9][10][11][12][13][14][15][16]. The SR phenomenon for various kinds of systems with cross-correlated noise has received great attention [17][18][19][20][21][22][23][24][25][26][27][28].…”