Theory and applications of random Poincaré maps

Nils Berglund
Institut Denis Poisson
University of Orleans, France

For deterministic ODEs, Poincaré maps (also called return maps) are useful to find periodic orbits, determine their stability, and analyse their bifurcations. The concept of Poincaré map naturally extends to the case of stochastic  differential equations, where it takes the form of a discrete-time, continuous-space Markov chain.

The talk will first present a motivation for random Poincaré maps, through the problem of describing interspike interval statistics for the stochastic FitzHugh-Nagumo equation. Then we will discuss spectral-theoretic results that allow to quantify the metastable long-time dynamics of systems with several periodic orbits, as well as some directions for future research.

Based on joint works with Damien Landon and Manon Baudel.

References:
1. NB and Damien Landon, Mixed-mode oscillations and interspike interval statistics in the stochastic FitzHugh-Nagumo model, Nonlinearity 25:2303-2335 (2012)
https://dx.doi.org/10.1088/0951-7715/25/8/2303

2. NB and Manon Baudel, Spectral theory for random Poincaré maps, SIAM J. Math. Analysis 49(6): 4319-4375 (2017)
https://dx.doi.org/10.1137/16M1103816