Stability and Instability in Conservative Dynamical Systems

University Of Maryland, College Park, College Park MD

Investigators

Abstract

The theory of dynamical systems seeks to describe the behavior of systems that evolve with time, such as the motion of planets or of gas particles in the air. Dynamical systems play an important role in mathematics, as well as having numerous applications in physics, biology, computer science and other sciences. Their range of application is growing, and will continue to grow given the widespread use of mathematical models by scientists and engineers. A modern view of dynamical systems contains three classes, called elliptic, parabolic and hyperbolic, depending on the increasing sensitivity to initial conditions of the system. The principal investigator will study this classification, with particular interest paid to the interconnections between the three classes and their interactions with various areas of mathematics and other sciences. The large span of the project provides rich training opportunities for graduate students. In addition, there are likely to be successful transfers of methodologies from one field to another in the theory of dynamical systems. Elliptic dynamics often refers to recurrent behavior in dynamical systems that is at the other end of the spectrum from chaotic dynamics. The systems with stable asymptotic behavior that are best represented by quasi-periodic motions on tori, and which appear for example in KAM theory, are within elliptic dynamics. But instability is also possible in elliptic dynamics, due to the so-called Liouville phenomena, where the existence of fast periodic approximations may be the source of very complex ergodic behavior. Since chaotic dynamics, that is best represented by hyperbolic dynamical systems, is associated with exponential growth of orbit complexity, one may consider slow growth of various characteristics of such complexity as a hallmark of elliptic dynamics. In between the elliptic and the hyperbolic world lie the so-called parabolic dynamical systems, that are best represented by unipotent actions on homogeneous spaces. The polynomial shear or rate of separation between orbits of these flows makes their ergodic theory very special. The goal of this project is to push forward the study of these three paradigms, as well as to explore the interconnections between them. Some main directions are: KAM stability results beyond the classical theory; robust unstable behavior in analytic Hamiltonian dynamics; approximation by the conjugation method beyond its usual limits; extension of Ratner theory to non-algebraic parabolic flows; developing a KAM rigidity theory for higher rank parabolic actions; developing limit laws for higher rank hyperbolic actions and exploiting them in a systematic approach to Diophantine approximation theory; counting problems from a statistical point of view. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.