Faculty of Mathematics
Collaborative Research Centre 701
Spectral Structures and Topological Methods in Mathematics

# Project A3

## Stochastic dynamics and bifurcations

Principal Investigator(s) Other Investigators

## Summary:

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This project addresses central questions about nonlinear stochastic dynamics described by stochastic differential equations. The focus lies on noise-induced phenomena in systems undergoing bifurcations. Earlier studies of randomly perturbed slow--fast systems will be continued with detailed sample-path analysis of higher-dimensional bifurcations and an extension to partial differential equations.

Going beyond local analysis, problems concerning global dynamics will be investigated. On the one hand, we will study random return maps with the goal of quantifying the effect of noise on global dynamics by a combination of precise local analysis near bifurcation points of the deterministic system and estimates on random return maps. On the other hand, two types of networks of randomly perturbed dynamical systems will be investigated: Networks of identical multistable systems with nearest-neighbour coupling, and small feed-forward networks with more complex local dynamics. In both cases the ultimate goal is the quantification of the combined effect of periodic forcing and random perturbations.

 13079 From random Poincaré maps to stochastic mixed-mode-oscillation patterns PDF | PS.GZ 13024 Uniform propagation of chaos for a class of inhomogeneous diffusions PDF | PS.GZ 12074 On the noise-induced passage through an unstable periodic orbit II: General case PDF | PS.GZ 12047 Estimates for the rate of strong approximation in Hilbert space PDF | PS.GZ 12008 Sharp estimates for metastable lifetimes in parabolic SPDEs: Kramers’ law and beyond PDF | PS.GZ 11092 Self-stabilizing processes in multi-wells landscape in $\mathbb{R}^d$ - Invariant probabilities PDF | PS.GZ 11091 Exit problem of McKean-Vlasov diffusions in convex landscape PDF | PS.GZ 11090 Self-stabilizing processes in multi-wells landscape in $\mathbb{R}^d$ - Convergence PDF | PS.GZ 11065 McKean-Vlasov diffusions: from the asynchronization to the synchronization PDF | PS.GZ 11063 On the functions counting walks with small steps in the quarter plane PDF | PS.GZ