On the Noise-Induced Passage Through an Unstable Periodic Orbit I: Two-Level Model

Abstract

We consider the problem of stochastic exit from a planar domain, whose boundary is an unstable periodic orbit, and which contains a stable periodic orbit. This problem arises when investigating the distribution of noise-induced phase slips between synchronized oscillators, or when studying stochastic resonance far from the adiabatic limit. We introduce a simple, piecewise linear model equation, for which the distribution of first-passage times can be precisely computed. In particular, we obtain a quantitative description of the phenomenon of cycling: The distribution of first-passage times rotates around the unstable orbit, periodically in the logarithm of the noise intensity, and thus does not converge in the zero-noise limit. We compute explicitly the cycling profile, which is universal in the sense that in depends only on the product of the period of the unstable orbit with its Lyapunov exponent.

Study of noise-induced transitions in the Lorenz system using the minimum action method

“…figure 4.3c). It should be distinguished from the most probable exit point on a limit cycle when a small but finite amount of noise is present, which has been examined both analytically [43,11] and numerically [44] for a planar system, the inverted van de Pol system with the noise amplitude ε. The authors there discovered an oscillatory behavior or cycling effect, namely, that for each given noise amplitude ε, there is a most probable exit location A ε on the limit cycle and this location A ε (the peak of the exit distribution on the limit cycle) slowly oscillates along the limit cycle with the period ∼ |logε| as ε ↓ 0.…”

“…Examples of these systems range from Josephson junctions [5] and switching in lasers [6] to protein folding [7] and electronic circuits [8]. Some of the most interesting findings include a preexponential factor of the Kramers rate and the transition state of an unstable point [9], a symmetry breaking bifurcation of the optimal escape path [10], and the phenomenon of cycling of exit point distribution from a planar unstable limit cycle [11].…”

“…Even though many non-gradient systems still possess saddle points (with one positive eigenvalue) as transition states, they are not the only possible transition states, even in a simple planar system without a limit cycle [13], where the MPEP asymptotically approaches an unstable fixed point on the separatrix. In the case of a limit cycle, which itself is the basin boundary in a planar system, the MPEP spirals toward the limit cycle asymptotically and its ω-limit set is the complete limit cycle [11].…”

Stochastic Dynamic Bifurcations and Excitability

“…Unlike the case of exit from a potential well, we have to deal here with the problem of noise-induced escape through a characteristic boundary (Day, 1990a;Day, 1992), which does not necessarily follow an exponential law. If for instance the boundary is a periodic orbit, cycling occurs and the exit location depends logarithmically on the noise intensity (Day, 1990b;Day, 1994;Day, 1996;Berglund and Gentz, 2004). This is related to the fact that a characteristic boundary is not crossed instantanously the moment a small neighbourhood is reached.…”

Thermally activated escape far from equilibrium: A unified path-integral approach

“…In the most common case, a potential barrier DU much larger than the thermal energy kT yields an escape rate exponentially small in DU/kT, complemented by a T-independent pre-exponential factor (''attempt frequency"). First major generalizations, of importance for conceptual reasons as well as due to numerous applications, are periodically modulated potentials [43][44][45][46][47]15,16,48,49,19], resulting in a renormalization of DU and highly non-trivial pre-factors, both depending in a very complicated manner on many details of the considered model. Due to the notorious technical difficulties of the problem, all these previous works are focused on one or the other of the following specific regimes: weak, slow, fast, or moderately fast and moderately strong modulations.…”