WIAS Preprint No. 864, (2003)

On the noise-induced passage through an unstable periodic orbit I: Two-level model


  • Berglund, Nils
  • Gentz, Barbara

2010 Mathematics Subject Classification

  • 37H20 60H10 34F05


  • Stochastic exit problem, diffusion exit, first-exit time, large deviations, metastability, level-crossing problem, limit cycle, synchronization, phase slip, cycling, stochastic resonance




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.

Appeared in

  • J. Statist. Phys., 114 (2004), pp. 1577-1618

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