---
tag: 59GFP54D78
kind: definition
name: Dynamical system
msc: [37B02, 37C10, 37-01]
defines: BBZP1R872N
uses: [0N6R902FG1, 0BSJ7RQ9BW, 0EPAT51DXQ]
sources:
  - url: https://en.wikipedia.org/wiki/Dynamical_system
    locator: Formal definition
---

## Definition

Let $\mathbb{T}$ be an additive [[monoid|0N6R902FG1]], the *time*, usually $\N$, $\Z$, $[0, \infty)$ or $\R$.
Let $X$ be a [[set|0BSJ7RQ9BW]], the *phase space* or *state space*.
A *dynamical system* with time $\mathbb{T}$ on $X$ is a [[map|0EPAT51DXQ]] $\Phi \colon \mathbb{T} \times X \to X$ such that:

1. $\Phi(0, x) = x$ for every $x \in X$;
2. $\Phi(s, \Phi(t, x)) = \Phi(s + t, x)$ for all $s, t \in \mathbb{T}$ and $x \in X$.

The map $\Phi^t = \Phi(t, \cdot) \colon X \to X$ is the *time-$t$ map*, and the set $\set{\Phi(t, x) : t \in \mathbb{T}}$ is the *orbit* of $x$.

## Remarks

Clauses 1 and 2 say that $t \mapsto \Phi^t$ is a monoid homomorphism from $\mathbb{T}$ to the maps $X \to X$ under composition, that is, an action of $\mathbb{T}$ on $X$.
In practice $X$ carries more structure, and $\Phi$ is required to respect it.
Examples are continuous maps on a topological space, smooth maps on a manifold, or measurable maps on a measure space.
With $\mathbb{T} = \R$ the system is a flow, with $\mathbb{T} = [0, \infty)$ a semiflow, and with $\mathbb{T} = \N$ or $\Z$ a discrete-time system.

Example: $\mathbb{T} = \R$, $X = \R$ and $\Phi(t, x) = e^{-t} x$, the solutions of $x' = -x$.
Clause 1 holds since $e^{0} x = x$, and clause 2 holds since $e^{-s}(e^{-t} x) = e^{-(s + t)} x$.

Non-example: $\Phi(t, x) = x + t^2$ on $\R$ with $\mathbb{T} = \R$.
Clause 1 holds, but $\Phi(1, \Phi(1, 0)) = 2$ while $\Phi(2, 0) = 4$, so clause 2 fails.