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

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 \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 () 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.