title: Noise-controlled dynamics through the averaging principle for stochastic slow-fast systems.
abstract: The effect of noise on nonlinear systems is analyzed, considering the case of slow-fast systems. It is known that small noise perturbations can induce a deterministic limit cycle in excitable systems when a specific scaling between the noise strength and the time-scale separation is achieved, a mechanism called self-induced stochastic resonance (SISR). The present study is focused on the impact of order 1 noise using the stochastic averaging principle. We introduce an elementary system of two coupled FitzHugh-Nagumo equations, which display the following nontrivial noise-induced behavior: (i) in the noise-free case, or for very small noise, the system fluctuates around its resting state; (ii) for small noise, oscillations appear due to SISR; (iii) for intermediate noise, the system fluctuates again around its resting state; (iv) for larger noise, new oscillations are observed and their explanation requires the application of the stochastic averaging principle. It is suggested that in the perspective of biological systems, time-scale separation may act as a "noise averager," enabling a noise-controlled dynamical behavior through the averaging principle.
authors.affiliation: Laboratoire Analyse, Géométrie et Applications (LAGA), Université Paris 13, Villetaneuse, France. wainrib@math.univ-paris13.fr
authors.forename: Gilles
authors.initials: G
authors.lastname: Wainrib
mesh_terms: D007700:Kinetics; D017711:Nonlinear Dynamics; D013269:Stochastic Processes