flow

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

More explicitly, a flow is a function$\phi :X\times ℝ\to X$satisfying the following properties:

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

2. $\phi (\phi (x,t),s)=\phi (x,s+t)$

for all $s,t$ in $ℝ$ and $x\in X$.

The set $𝒪(x,\phi )=\{\phi (x,t):t\in ℝ\}$ 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 $X={ℝ}^n$.)

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 ${\psi }_x:ℝ\to X$ for each $x\in X$, then $\phi (x,t)={\psi }_x(t)$ defines a flow.