Noise-mediated dynamics in a two-dimensional oscillator: Exact solutions and numerical results

We derive a Fokker-Planck equation (FPE) to analyze the oscillator equations describing a nonlinear amplifier, exemplified by a two-junction Superconducting Quantum Interference Device (SQUID), in the presence of thermal noise. We show that the FPE admits a unique stationary solution and obtain analytical results for several parameters ranges. To solve the FPE numerically, we develop an efficient spectral method which exploits the periodicity of the probability density. The numerical method, combined with the exact solutions, allow us to rapidly explore the noise-mediated dynamics as a function of the control parameters.