  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.
