flow
Flow2
2002-12-07 07:08:39
2006-09-15 06:13:11
Definition
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A \emph{flow} on a set $X$ is a group action of $(\mathbb{R},+)$ on $X$.
More explicitly, a flow is a function
$\varphi:X\times \mathbb{R}\rightarrow X$
satisfying the following properties:
\begin{enumerate}
\item $\varphi(x,0) = x$
\item $\varphi(\varphi(x,t),s) = \varphi(x,s+t)$
\end{enumerate}
for all $s,t$ 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 $x$ by $\varphi$.
Flows are usually required to be continuous or \PMlinkescapetext{even} 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
\begin{equation}\label{eq1} y' = f(y),\;\;\; y(0)=x \end{equation}
as a function of the initial condition $x$, when the equation has existence and uniqueness of solutions.
That is, if (\ref{eq1}) 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.