Geometrical definition In the geometrical definition, a dynamical system is the tuple . is the domain for time – there are many choices, usually the reals or the integers, possibly restricted to be non-negative. is a manifold, i.e. locally a Banach space or Euclidean space, or in the discrete case a graph. f is an evolution rule t → f t (with ) such that f t is a diffeomorphism of the manifold to itself. So, f is a "smooth" mapping of the time-domain into the space of diffeomorphisms of the manifold to itself. In other terms, f(t) is a diffeomorphism, for every time t in the domain .
geometrical definition in the geometrical definition a dynamical system is the tuple  is the domain for time – there are many choices usually the reals or the integers possibly restricted to be non-negative is a manifold ie locally a banach space or euclidean space or in the discrete case a graph f is an evolution rule t → f t (with ) such that f t is a diffeomorphism of the manifold to itself so f is a "smooth" mapping of the time-domain into the space of diffeomorphisms of the manifold to itself in other terms f(t) is a diffeomorphism for every time t in the domain 