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

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