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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 resonancelikeboundary crossings. In this paper, we analyze the stochastic periodic orbitscaused by SISR in fast-slow systems. By introducing the notion of the meanfirst passage velocity toward the boundary, a distance matching condition isestablished, through which the critical transition position of boundarycrossing can be calculated. The theoretical stochastic periodic orbit can beaccordingly obtained via gluing the dynamics along the slow manifolds. It isshown that the theoretical predictions are in excellent agreement with theresults of Monte Carlo simulations for a piecewise linear FitzHugh-Nagumosystem even for large noise. Furthermore, the proposed method is extended tothe original FitzHugh-Nagumo system and also found to exhibit consistentaccuracy. These results provide insights into the mechanisms of coherentbehaviors in fast-slow systems and will shed light on the coherent behaviors inmore complex systems and large networks.

Stochastic periodic orbits in fast-slow systems with self-induced stochastic resonance