AMC-SS: Noise-Induced Transitions in Multiscale Systems

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The proposed research is an attempt to understand the effectof small noise on bifurcations in dynamical systems.While bifurcations involve sharp changes in behavior, noise has a smoothingeffect (the probability density function solves a parabolic PDE,rather than a transport equation).It is this competition between dominant bifurcative behaviorand small noise which is of interest to us. We seek to illuminatethis competition by considering a number of specific problems.We in particular seek to study canard phenomena (which isa combination of bifurcation and multiple scales), two-pointmotion near a homoclinic orbit, and behavior of certain higher-dimensionalsystems.One of the central themes in applied mathematics is the problemof modelling evolving behavior. One of the maintools for doing so is differential equations; the theoreticalunderpinnings of differential equations are mathematicallyattractive, and it is fairly simple to convert qualitativeeffects into the vector fields which govern ODE's.An honest assessment of any behavior must, of course, also includenoise. Usually this noise is small enough to have negligible effect, but in certainsituations, even small noise may dominate the behavior of a system.Our interest is in these situations where noise causes macroscopicallyvisible random transitions in a dynamical system.By means of certain carefully-chosen examples, we seek tounderstand a number of different phenomena where suchrandom transitions occur. The specific equations we studyappear in a number of areas: biology, the study of ocean currents,and materials science.