Infinite-Dimensional Dynamical Systems

A common sense definition of a dynamical system is any phenomenon of nature (or even any abstract construct) evolving in “time”. A particular class of dynamical systems described by partial differential equations is usually called infinite-dimensional dynamical systems. It contains a huge variety of problems from the celebrated Ricci flow in geometry to weather prediction analysis.  Their common feature is that they are governed by their own nonlinear PDEs.

Mathematically, these PDEs form the so-called semi-group of non-linear operators acting in Banach Spaces. The right choice of such spaces is related to the fundamental question of PDE theory: global well posedeness for underlying equations.  

Specific questions of the mathematical theory of dynamical systems are: long time behaviour of solutions, their global or local stability. Local stability analysis leads to deep problems about the spectrum of the linearized operators while the existence, dimension and “topological” structure of attractors are typical goals in the global stability setting.

Dynamical systems arising in physics (of solids and fluids) and in geometry are of particular importance to the Centre.