Dynamical system

A set $Ω$ whose elements are called states, and an object, called dynamical law, that tells us how to go from one state to the next one, i.e., they have a time evolution.

In the case of a discrete dynamical system, the dynamical law is a map

F : N \times Ω \to Ω

In this case, $N$ is something like "discrete time". The function $F$ is something like a "semigroup action", similar to a group action but with $N$ .

The continuos dynamical systems, on the other hand, correspond to system of first order ODEs, whose solution is a 1-parameter local group of transformations, roughly speaking

F : R \times Ω \to Ω

They can be identified with vector fields of the form

A = \partial_{t} + \sum ϕ_{j} \partial x_{j}

The more important cases, for me, are the classical Hamiltonian systems.

They can be integrable systems.