Modern Dynamical Systems Theory

Abstract

In order to analyse generic or typical properties of dynamical systems we consider the space of all C1-vector fields on a fixed differentiable manifold M. In the C1-metric, assuming M is compact, is a complete metric space and a generic subset is an open dense subset or an intersection of a countable collection of such open dense subsets of . Some generic properties (i.e. specifying generic subsets) in are described. For instance, generic dynamic systems have isolated critical points and periodic orbits each of which is hyperbolic. If M is a symplectic manifold we can introduce the space of all Hamiltonian systems and study corresponding generic properties.