  Noise is ubiquitous in various systems. In systems with multiple timescales,
noise can induce various coherent behaviors. Self-induced stochastic resonance
(SISR) is a typical noise-induced phenomenon identified in such systems,
wherein noise acting on the fast subsystem causes stochastic resonance-like
boundary crossings. In this paper, we analyze the stochastic periodic orbits
caused by SISR in fast-slow systems. By introducing the notion of the mean
first passage velocity towards the boundary, a distance matching condition is
established, through which the critical transition position of boundary
crossing can be calculated. The theoretical stochastic periodic orbit can be
accordingly obtained via gluing the dynamics along the slow manifolds. It is
shown that the theoretical predictions are in excellent agreement with the
results of Monte Carlo simulations for a piecewise linear FitzHugh-Nagumo
system even for large noise. Furthermore, the proposed method is extended to
the original FitzHugh-Nagumo system and also found to exhibit consistent
accuracy. These results provide insights into the mechanisms of coherent
behaviors in fast-slow systems and will shed light on the coherent behaviors in
more complex systems and large networks.
