Topics in Finite and Infinite Dimensional Random Dynamical Systems

Michigan State University, East Lansing MI

Investigators

Abstract

Random dynamical systems arise in the modeling of many phenomena in physics, biology, economics, etc. when uncertainties or random influences, called noises, are taken intoaccount. These random effects are not only introduced to compensate for the defects in some deterministic models, but also are often rather intrinsic phenomena. A fundamental problem is to study the dynamical behavior of orbits of random dynamical systems. This project seeks to establish some of the basic geometric framework for infinite dimensional random dynamical systems. By proving existence and robustness of random invariant manifolds and foliations, results on structural stability of deterministic systems under random perturbations will be sought. Floquet theory containing the multiplicative ergodic theorem will be developed, statistical properties of random attractors, and pattern formation in random media will be explored. The theory of smooth conjugacy for finite dimensional random dynamical systems will also be developed. The broader impacts of this project include the training of graduate students and junior faculty members in this emerging area of dynamics. Applications to the many fields mentioned above will create opportunities to interact on many levels with students and faculty members from other disciplines and will lead to advances in technology. One such example are the possible applications to the study of nano-devices, for instance, random thermal fluctuations or quantum effects must be accounted for in order to accurately predict performance. That field is one of many that is completely open to analysis using tools of the type to be developed here.