Dynamical Systems

Faculty with research interests in dynamics are Sasa Kocic and Samuel Lisi.

A dynamical system is a rule that defines how the state of a system changes with time. Formally, it is an action of reals (continuous-time dynamical systems) or integers (discrete-time dynamical systems) on a manifold (a topological space that looks like Euclidean space in a neighborhood of each point).

Roughly, the study of continuous-time dynamical systems is the study of differential equations; the study of discrete-time dynamical systems is the study of iterations of functions. The theory of dynamical systems is a very broad field closely intertwined with many other areas of mathematics. In particular, it has close relations with ergodic theory, probability theory, number theory, geometry, topology and mathematical physics.

Dr. Kocic’s particular research interests involve renormalization, rigidity theory of dynamical systems and KAM (Kolmogorov-Arnold-Moser) theory. Renormalization is a tool that originated in physics (quantum field theory, statistical mechanics) and, in the last 40 years, has become a powerful tool of the modern theory of dynamical systems. So far, there are two fields medals awarded for works involving renormalization (Curtis McMullen, 1998; Artur Avila, 2014). It is potentially also a powerful tool beyond dynamics. In dynamics, it is often the only tool to prove some rigidity results in one-dimensional dynamics, for example. Rigidity refers to a property that some systems that are a priori equivalent in some weak (e.g., topological) sense are actually equivalent in much stronger (e.g., smooth) sense. It is a notion which exists in many areas of mathematics. KAM theory is a classic topic in dynamical systems that concerns the persistence of quasiperiodic solutions (stable motions) under small perturbations of an integrable system.

Dr. Lisi’s particular research interests involve Hamiltonian systems and symplectic topology, and notably phenomena of rigidity in this context. Hamiltonian systems are a special class of dynamical systems, providing a framework for the study of systems with a conservation of energy. A classical example of such a system is the gravitational n-body problem of Newton, where n point masses (idealized planets) interact according to Newton’s law of gravity. Attempts to understand the behavior of this and related systems have inspired a lot of great mathematics and continue to be a rich source of inspiration. Symplectic topology studies the differential topology of symplectic manifolds, which notably include the phase spaces of Hamiltonian systems. The remarkable fact is that symplectic topology, qualitative behavior of Hamiltonian systems and geometric topology have a lot to say about each other.