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

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 e (coupling strength) and decreasing alpha (fractional order) can both accelerate the synchronization speed. SR mainly occurs in the high-alpha or high-sigma (noise amplitude) region, which means that SR emergence can be controlled by adjusting alpha or sigma. The damping force, coupling force and frequency fluctuations compete with each other; thus, the SR intensity should be maximized by adjusting alpha, e and sigma simultaneously.