A dynamical system is a space (often a bare set or a manifold) together with a “law of motion” expressed by the action of a monoid : To model continuous time-evolution one may take to be the additive group of real numbers. For discrete time evolution is the additive group of integers (or just the monoid of natural numbers). In this case the law of motion is often given by an initial value problem for a differential equation “of evolution type”. In this case, the dynamical system is equivalently.