  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.
