A 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:

1. 1.

   $\varphi(x,0)=x$

2. 2.

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

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 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

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 $\psi_{x}:\mathbb{R}\rightarrow X$ for each $x\in X$, then $\varphi(x,t)=\psi_{x}(t)$ defines a flow.