Dynamical System (definition) - General Definition

General Definition

In the most general sense, a dynamical system is a tuple (T, M, Φ) where T is a monoid, written additively, M is a set and Φ is a function

with

for

The function Φ(t,x) is called the evolution function of the dynamical system: it associates to every point in the set M a unique image, depending on the variable t, called the evolution parameter. M is called phase space or state space, while the variable x represents an initial state of the system.

We often write

if we take one of the variables as constant.

is called flow through x and its graph trajectory through x. The set

is called orbit through x.

A subset S of the state space M is called Φ-invariant if for all x in S and all t in T

In particular, for S to be Φ-invariant, we require that I(x) = T for all x in S. That is, the flow through x should be defined for all time for every element of S.