Effective Fokker-Planck Equation for Birhythmic Modified van der Pol Oscillator

TL;DR

This paper derives an analytical solution to the Fokker-Planck equation for a birhythmic modified van der Pol oscillator, revealing how noise influences attractor probabilities and system dynamics.

Contribution

It provides an explicit phase-amplitude approximation solution for the Fokker-Planck equation in a birhythmic oscillator, linking noise effects to attractor behavior and phase transition analogies.

Findings

Analytical probability distributions match numerical results when frequencies are equal.

Noise causes merging of attractors and reduces the birhythmic region.

Attractors exhibit phase-like properties with distinct probabilities.

Abstract

We present an explicit solution based on the phase-amplitude approximation of the Fokker-Planck equation associated with the Langevin equation of the birhythmic modified van der Pol system. The solution enables us to derive probability distributions analytically as well as the activation energies associated to switching between the coexisting different attractors that characterize the birhythmic system. Comparing analytical and numerical results we find good agreement when the frequencies of both attractors are equal, while the predictions of the analytic estimates deteriorate when the two frequencies depart. Under the effect of noise the two states that characterize the birhythmic system can merge, inasmuch as the parameter plane of the birhythmic solutions is found to shrink when the noise intensity increases. The solution of the Fokker-Planck equation shows that in the birhythmic region, the two attractors are characterized by very different probabilities of finding the system in such a state. The probability becomes comparable only for a narrow range of the control parameters, thus the two limit cycles have properties in close analogy with the thermodynamic phases.