AMC-SS: Noise-Induced Transitions in Multiscale Systems

University Of Illinois At Urbana-Champaign, Urbana IL

Investigators

Abstract

The proposed research is an attempt to understand the effect of small noise on bifurcations in dynamical systems. While bifurcations involve sharp changes in behavior, noise has a smoothing effect (the probability density function solves a parabolic PDE, rather than a transport equation). It is this competition between dominant bifurcative behavior and small noise which is of interest to us. We seek to illuminate this competition by considering a number of specific problems. We in particular seek to study canard phenomena (which is a combination of bifurcation and multiple scales), two-point motion near a homoclinic orbit, and behavior of certain higher-dimensional systems. One of the central themes in applied mathematics is the problem of modelling evolving behavior. One of the main tools for doing so is differential equations; the theoretical underpinnings of differential equations are mathematically attractive, and it is fairly simple to convert qualitative effects into the vector fields which govern ODE's. An honest assessment of any behavior must, of course, also include noise. Usually this noise is small enough to have negligible effect, but in certain situations, even small noise may dominate the behavior of a system. Our interest is in these situations where noise causes macroscopically visible random transitions in a dynamical system. By means of certain carefully-chosen examples, we seek to understand a number of different phenomena where such random transitions occur. The specific equations we study appear in a number of areas: biology, the study of ocean currents, and materials science.