On classifications of dynamical systems

Robert McLachlan, Massey University, New Zealand

Quite a lot of dynamical systems theory is concerned with studying general properties that are true for all systems in a certain class - all Hamiltonian systems, say, or all systems with a given symmetry group. So it is natural to ask in what ways one can classify systems by their algebraic structure. This leads to some old theory, e.g. Cartan's 1913 classification of the primitive infinite dimensional diffeomorphism groups, some questions old but still open, e.g. the classification of systems preserving foliations of phase space, and many new questions. This study arose in an attempt to say just what it is we mean by a "geometric" property in geometric numerical integration.