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

We present an explicit solution based on the phase-amplitude approximation of theFokker-Planckequation associated with theLangevinequation of the birhythmic modifiedvan der Polsystem. Thesolution enables us to derive probability distributions analytically as well as the activation energiesassociated with switching between the coexisting different attractors that characterize the birhythmicsystem. Comparing analytical and numerical results we find good agreement when the frequencies ofboth attractors are equal, while the predictions of the analytic estimates deteriorate when the twofrequencies depart. Under the effect of noise, the two states that characterize the birhythmic systemcan merge, inasmuch as the parameter plane of the birhythmic solutions is found to shrink when thenoise intensity increases. The solution of theFokker-Planckequation shows that in the birhythmicregion, the two attractors are characterized by very different probabilities of finding the system insuch 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.