Cycling

The distribution of first-exit locations through an unstable periodic orbit of a planar system, subjected to additive white noise. See here for more details.

The expression we use in the animations below is

where

- the transient factor
*f*_{trans}(*t*) is only present when the system does not start in its quasistationary distribution; it increases from 0 to 1 in a time of order |log σ|, where σ is the noise intensity; - the exponential term decays on a timescale
*T*_{K}(σ) = e^{H/σ2}related to Kramers' time, where*H*is the quasipotential; - the periodic part is a universal function (for a suitable parametrization θ of the periodic orbit) of period
equal to the orbit's period
*T*times its Lyapunov exponent λ. The function is the density of a type-1 Gumbel distribution with mode -log 2/2 and scale parameter 1/2.

The following animation shows how the exit distribution changes as the noise intensity σ decreases from 1 to 0.0001.
The system starts in its quasistationary distribution, in order to avoid transients which become longer as the noise intensity decreases.
Parameter values are: Lyapunov exponent λ=1, quasipotential *H*=1 and period *T*=4. The distribution has been normalised so that the maximum, not the area under the curve, is constant.

A full resolution version of the animation is here (AVI, 22Mb).

The following animation shows how the exit distribution changes as the orbit's period increases from 0.001 times Kramers' time to 5 times Kramers' time.
Parameter values are: Lyapunov exponent λ=1, quasipotential *H*=1 and noise intensity σ=0.4. The distribution has been normalised so that the maximum, not the area under the curve, is constant.

A full resolution version of the animation is here (AVI, 13Mb).

The link with residence-time distributions for stochastic resonance is explained here.