The proposed research deals with questions of stability, equilibrium, and bifurcation for random dynamical systems generated by a stochastic differential equation. Attention will be paid to the development of stochastic bifurcation theory, namely the study of the qualitative changes in long term behavior which can occur when one or more parameters in the stochastic differential equation are varied. Typically bifurcation occurs when a Lyapunov exponent changes sign. Stochastic averaging methods will be developed to calculate Lyapunov exponents, and rigorous results will relate the behavior of the linearized process to that of the underlying nonlinear system. The concepts of random sinks (depending on the past) and random sources (depending on the future) will be studied. The random sinks and sources may be single points, or sets of some (non-integer) Hausdorff dimension. A phase portrait for a random dynamical system will then consist of a mixture of past and future information. Two physical models will be studied in detail: the stability of the vibrations of a flexible beam under stochastic forcing, and the effects of noise on the phase-locking and bifurcation behavior of the leaky integrate-and-fire oscillators which occur as models of neural activity.
The project will study mathematical models used to describe the behavior of certain physical systems as they evolve in time under the influence of random perturbations. These models are used widely in physics (e.g. random vibrations of mechanical systems), chemistry (e.g. rates of chemical reactions), biology (e.g. generation of nerve impulses), finance (e.g. option pricing) and elsewhere. Particular attention will be paid to descriptions of the long term behavior of such systems, and in particular to situations where a gradual change in some underlying constants can cause a sudden change in the long term behavior of the system. The project will develop a general theory for such systems. It will also study in detail two special cases: the vibrations of a thin flexible beam, and the integrate-and-fire model for activity within a single neuron in the brain.
