Critical Transitions, Oscillations and Continuation for SDEs

In this talk, I am going to discuss three recent connected ideas in the theory of stochastic dynamical systems. We start by defining critical transitions using stochastic fast-slow systems and derive early-warning signs in the near-hyperbolic regime. Then we proceed to study noisy canard-induced mixed-mode oscillations in a regime where hyperbolicity is lost. As a last topic, we propose a new numerical method that uses deterministic continuation algorithms to capture dynamics of stochastic systems.