Hirsch The investigator studies several areas of dynamical systems and its applications: (1) Certain stochastic processes such as generalized Polya Urns are now known to have sample paths whose limit sets are almost surely Chain Recurrent sets for an associated differential equation. This allows the use of dynamical systems theory in a variety of fields. Applications are made to Stochastic Approximations, Game Theory, Mathematical Economics of sustainable resources, Numerical Algorithms, and Neural Learning Algorithms. (2) New methods for proving existence of Horseshoe Chaos are applied to generalized Henon maps. The Smale-Birkhoff existence theorem is extended to replace the assumption of transverse intersections between stable and unstable manifolds by a much more general topological condition of nonzero intersection number. (3) New models of neural computation are constructed, based on varying competition between oscillatory or chaotic attractors, with competition controlled by an adaptive clocking system. Mathematical dynamical systems theory studies the long-term behavior of complex systems. This project develops rigorous mathematics that can aid in the analysis of systems in several applications. (1) In Economic Theory, markets are often modeled as a game played repeatedly between competitors who continually adapt their strategies (e.g., prices, production) in the light of experience. It is of great importance to determine whether this leads to a stable, predictable situation. This project studies how questions of this kind can be solved by translating them into mathematical problems in the seemingly different but well understood field of dynamical systems theory. This application builds directly on work by the recent Nobel Laureates in Economics, Nash and Harsanyi. The investigator is also collaborating with economists on applications of the same mathematical results to models of sustainable resources. (2) In many areas of science en gineering it is important to whether particular systems are "chaotic," i.e. inherently unpredictable in the long run. This project provides new means for determining chaotic dynamics. (3) The newly emerging field of Neural Computation involves dynamics in many ways. Some of the same mathematical ideas used in the other parts of this project are directly relevant. In collaboration with biophysicists, the investigator applies dynamical systems theory to new neural network systems that can efficiently "learn" to do a variety of useful tasks, such as robot control.
