UTA Department of Mathematics

Mathematics and Statistics Colloquium

Date/Time/Room: Friday (3/2/2007) at 2:30pm in 304 Pickard Hall

Speaker: Dr. Barbara Gentz, Associate Professor
Faculty of Mathematics
University of Bielefeld, Germany

"The effect of noise on slow-fast systems"

Abstract: Dynamical systems perturbed by weak noise often display metastable behavior, i.e., typical sample paths spend exponentially long time spans near local equilibria of the underlying deterministic system. When combined with time-period forcing, this phenomenon can give rise to nearly periodic large-amplitude oscillations between attractors.

The overdamped motion of a particle in a double-well potential can serve as a simple example, the particle being subject to two different kinds of perturbations: Small deterministic periodic driving and additive noise. The periodic driving is assumed to have too small an amplitude to allow for transitions between the potential wells in the absence of noise. Without deterministic forcing, the particle would jump from one potential well to the other at random times. When both perturbations are combined, however, and their amplitudes suitably tuned, the particle will flip back and forth between the wells in a close to periodic way. This surprising effect is known as stochastic resonance or noise-induced synchronization. We show how to derive a mathematically rigorous description of the behavior of individual paths.

We conclude by discussing a general approach to concentration results for sample paths in fully coupled, multi-dimensional slow-fast systems with noise.

(Joint work with Nils Berglund, CPT-CNRS Marseille, France)