flow
A flow on a set $X$ is a group action of $(\mathbb{R},+)$ on $X$.
More explicitly, a flow is a function $\phi :X\times \mathbb{R}\to X$ satisfying the following properties:

1.
$\phi (x,0)=x$

2.
$\phi (\phi (x,t),s)=\phi (x,s+t)$
for all $s,t$ in $\mathbb{R}$ and $x\in X$.
The set $\mathcal{O}(x,\phi )=\{\phi (x,t):t\in \mathbb{R}\}$ is called the orbit of $x$ by $\phi $.
Flows are usually required to be continuous^{} or differentiable^{}, when the space $X$ has some additional structure (e.g. when $X$ 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^{}
$${y}^{\prime}=f(y),y(0)=x$$  (1) 
as a function of the initial condition^{} $x$, when the equation has existence and uniqueness of solutions. That is, if (1) has a unique solution ${\psi}_{x}:\mathbb{R}\to X$ for each $x\in X$, then $\phi (x,t)={\psi}_{x}(t)$ defines a flow.
Title  flow 

Canonical name  Flow1 
Date of creation  20130322 13:12:34 
Last modified on  20130322 13:12:34 
Owner  Koro (127) 
Last modified by  Koro (127) 
Numerical id  8 
Author  Koro (127) 
Entry type  Definition 
Classification  msc 37C10 