Effective Fokker-Planck equation for birhythmic modified van der Pol oscillator

Area of Science:

• Nonlinear Dynamics
• Statistical Physics
• Stochastic Processes

Background:

• The birhythmic modified van der Pol system exhibits coexisting attractors, leading to complex dynamics.
• Understanding transitions between these attractors and the influence of noise is crucial for characterizing the system's behavior.

Purpose of the Study:

• To derive an explicit analytical solution for the Fokker-Planck equation of the birhythmic modified van der Pol system.
• To determine probability distributions and activation energies for switching between coexisting attractors.
• To investigate the impact of noise intensity on the system's parameter space and attractor probabilities.

Main Methods:

• Phase-amplitude approximation of the Fokker-Planck equation.
• Analytical derivation of probability distributions and activation energies.
• Comparison of analytical results with numerical simulations.

Main Results:

• Good agreement between analytical and numerical results was found when attractor frequencies are equal.
• Analytical predictions degrade as attractor frequencies diverge.
• Increased noise intensity causes the parameter plane for birhythmic solutions to shrink, merging the two states.

Conclusions:

• The analytical solution provides insights into the activation energies and probability distributions of attractor switching.
• The two attractors exhibit significantly different probabilities, analogous to thermodynamic phases, especially in the birhythmic region.
• Noise plays a critical role in modulating the system's dynamics and the coexistence of attractors.