Flow (mathematics)

Categories: PlanetMath sourced articles | Differential equations

In mathematics, flow refers to the group action of a one-parameter group on a set. Flows typically arise as the solutions of ordinary differential equations. The idea of a vector flow, that is, the flow determined by a vector field, occurs in the areas of differential topology, Riemannian geometry and Lie groups. Specific examples of vector flows include the geodesic flow, the Hamiltonian flow, the Ricci flow, and the Anosov flow.

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Formal definition

A flow on a set <math>X</math> is a group action of <math>(\mathbb{R},+)</math> on <math>X.</math> More explicitly, a flow is a function <math>\varphi:X\times \mathbb{R}\rightarrow X</math> with <math>\varphi(x,0) = x</math> and that is consistent with the structure of a one-parameter group:

<math>\varphi(\varphi(x,t),s) = \varphi(x,s+t)</math>

for all <math>s,t</math> in <math>\mathbb{R}</math> and <math>x\in X.</math>

The set <math>\mathcal{O}(x,\varphi) = \{\varphi(x,t):t\in\mathbb{R}\}</math> is called the orbit of <math>x</math> by <math>\varphi.</math>

Flows are usually required to be continuous or even differentiable, when the space <math>X</math> has some additional structure (e.g. when <math>X</math> is a topological space or when <math>X = \mathbb{R}^n.</math>)

The most common examples of flows arise from describing the solutions of the autonomous ordinary differential equation

<math> y' = f(y),\;\;\; y(0)=x</math>

as a function of the initial condition <math>x</math>, when the equation has existence and uniqueness of solutions. That is, if the equation above has a unique solution <math>\psi_x:\mathbb{R}\rightarrow X</math> for each <math>x\in X</math>, then <math>\varphi(x,t) = \psi_x(t)</math> defines a flow.

This article incorporates material from Flow on PlanetMath, which is licensed under the GFDL.