What kinds of dynamics are there? Lie pseudogroups, dynamical systems, and geometric integration

We classify dynamical systems according to the group of diffeomorphisms to which they belong, with application to geometric integrators for ODEs. This point of view unifies symplectic, Lie group, and volume-, integral-, and symmetry-preserving integrators. We review the Cartan classification of the primitive infinite-dimensional Lie pseudogroups (and hence of dynamical systems), and select the conformal pseudogroups for further study, i.e., those that contract volume or a symplectic structure at a constant rate. Their special properties are illustrated analytically (by a study of their behaviour with respect to symmetries) and numerically (by a geometric calculation of Lyapunov exponents). We also briefly discuss the nonprimitive pseudogroups.