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

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 with 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.

Coherence and stochastic resonance in a birhythmic van der Pol system

“…In a bistable system, two barriers characterize the exit times from the two stable states: ΔU 1 and ΔU 3 for the escape from the orbit of radius A 1 and A 3 , respectively. As an example, in Tables 1 and 2 we report the energy barriers [12,30] of the multi-limit-cycle associated to the frequencies and amplitudes of the model for some values of the physical parameters α and β. In the shaded region of the parameters plane (α, β) of Figure 1, where two global minima appear, the potential U is symmetric (Tab.…”

“…The escape process is depicted in Figure 2 as a jump over an activation barrier ΔU 1→3 (from the leftmost minimum A 1 ) and ΔU 3→1 (from the rightmost minimum A 3 ) [12,18]. The characteristics height can be controlled by the variation of the parameters α and β [30]. The noisy birhythmic van der Pol equation (1) can be characterized through the distribution P (T ) of the escape times (denoted by T i ) from the two wells of potential U .…”

“…The probability for the system to hop between the potential wells is defined through noise dependent Kramers rate [47,48], which is the inverse of the average escape time T . The quantities T 1,3 [18,30] for small noise intensity D < ΔU i are given by (here the primes refer to differentiation respect to the radius A):…”

“…The quantities (10) and (11) measure the relative stabilities pertaining to the attractors with limit cycle amplitude A 1 and A 3 through the resident time given by the relation [18,30]:…”

“…This analysis could also take advantage of the nonequilibrium potential constructed on the basis of an extremum principle [27][28][29]. In fact, it can be postulated that the behavior of the escape times, or more accurately of the mean first passage time through the separatrix between the two attractors, exponentially depends upon a quantity (the quasi-potential, or the effective energy barrier) and it is inversely proportional to the noise intensity [12,18,30]. The quasi-potential approach has also been extended to correlated noise case both driven [31] and undriven [32], that also exhibits stochastic bifurcations [33].…”