Much of the research in dynamical systems concerns nonlinear partial differential equations that model significant physical phenomena. For example, hyperbolic systems of conservation laws and shock waves in continuum mechanics, Vlasov-Boltzmann systems in kinetic theory, nonlinear dispersive equations such as Schrödinger and KdV equations and cyclic systems of differential-delay equations all play a part.
Another aspect of the research in dynamical systems takes a geometric approach. Issues related to asymptotic and transient behavior, chaotic dynamics, transport and diffusion are studied in both dissipative and conservative systems. These phenomena are studied both theoretically and in the context of applications to topics such as nonlinear optics, theoretical chemistry, rigid body dynamics, fluid mechanics, oceanography and celestial mechanics.
The group in stochastic control theory works on virtually every aspect of the field: optimization, numerical methods, stability, nonlinear filtering, PDE methods, weak convergence and related approximation methods, large deviations and applications, nonlinear robust control, stochastic approximation, adaptive control, controlled heavy traffic models for queueing and similar networks, approximation methods for complex systems, etc. There is a strong interest in applications, and recently there has been much interest in applications to modern high speed telecommunications systems.