A flow on a set X is a group action of (,+) on X.

More explicitly, a flow is a function φ:X×X satisfying the following properties:

  1. 1.


  2. 2.


for all s,t in and xX.

The set 𝒪(x,φ)={φ(x,t):t} is called the orbit of x by φ.

Flows are usually required to be continuousMathworldPlanetmath or differentiableMathworldPlanetmathPlanetmath, when the space X has some additional structure (e.g. when X is a topological spaceMathworldPlanetmath or when X=n.)

The most common examples of flows arise from describing the solutions of the autonomousMathworldPlanetmath ordinary differential equationMathworldPlanetmath

y=f(y),y(0)=x (1)

as a function of the initial conditionMathworldPlanetmath x, when the equation has existence and uniqueness of solutions. That is, if (1) has a unique solution ψx:X for each xX, then φ(x,t)=ψx(t) defines a flow.

Title flow
Canonical name Flow1
Date of creation 2013-03-22 13:12:34
Last modified on 2013-03-22 13:12:34
Owner Koro (127)
Last modified by Koro (127)
Numerical id 8
Author Koro (127)
Entry type Definition
Classification msc 37C10