Smooth dynamical system

Summary

A smooth dynamical system is a dynamical system whose phase space is a smooth manifold and whose time evolution is differentiable. In discrete time it is a diffeomorphism iterated forward and backward; in continuous time it is the flow of a smooth vector field. Differentiability allows one to linearise the motion along orbits, which leads to hyperbolicity, invariant manifolds and Lyapunov exponents. Most dynamical systems of mechanics and of the sciences are smooth.