PlanetMath: flow

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flow

(Definition)

A flow on a set is a group action of $(\mathbb{R},+)$ on .

More explicitly, a flow is a function $\varphi:X\times \mathbb{R}\rightarrow X$ satisfying the following properties:

$\varphi(x,0) = x$
$\varphi(\varphi(x,t),s) = \varphi(x,s+t)$

for all

in $\mathbb{R}$ and $x\in X$ .

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

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

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

$\displaystyle y' = f(y),\;\;\; y(0)=x$

(1)

as a function of the initial condition

, when the equation has existence and uniqueness of solutions. That is, if (1) has a unique solution $\psi_x:\mathbb{R}\rightarrow X$ for each $x\in X$ , then $\varphi(x,t) = \psi_x(t)$ defines a flow.

"flow" is owned by Koro.

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Cross-references: equation, initial condition, ordinary differential equation, autonomous, solutions, topological space, structure, differentiable, continuous, orbit, properties, function, group action
There are 20 references to this entry.

This is version 5 of flow, born on 2002-12-07, modified 2006-09-15.
Object id is 3673, canonical name is Flow2.
Accessed 4484 times total.

Classification:

AMS MSC:

37C10 (Dynamical systems and ergodic theory :: Smooth dynamical systems: general theory :: Vector fields, flows, ordinary differential equations)

Pending Errata and Addenda

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