Maximilian Engel, TU München
Local phenomena in random dynamical systems: bifurcations, quasi-stationary dynamics and isochronicity
Random dynamical systems theory focuses on dynamical properties of stochastic systems, comparing trajectories with different initial conditions but driven by the same noise. A central question is the asymptotic behaviour of typical trajetories, which is often characterized by a Lyapunov spectrum and its bifurcation behavior. One part of the talk will focus on a new description of stochastic bifurcations, introducing the notion of conditioned Lyapunov exponents for trajectories that stay within a bounded domain for asymptotically long times. We show that conditioned Lyapunov exponents uncover local bifurcations that are typically destroyed in the presence of unbounded noise. A crucial ingredient for determining conditioned Lyapunov exponents is the spectral analysis of hypoelliptic Kolmogorov operators, whose properties can be exploited in computer-assisted proofs. The second part discusses a new approach for describing isochrons, which are cross-sections of limit cycles with fixed return times, in the context of (small) stochastic oscillations. We introduce stochastic isochrons as random stable manifolds for random periodic solutions with noise-dependent period and discuss how this interpretation may be linked to recent physics approaches via an appropriate (S)PDE analysis.
The talk is based on joint work with Thai Son Doan (Vietnam Academy of Science and Technology), Christian Kuehn (TU Munich), Jeroen Lamb and Martin Rasmussen (Imperial College London).