Flow of a vector field

Definition

Let $U$ be an open subset of $\mathbb{R}^n$ and let $f : U \to \mathbb{R}^n$ be locally Lipschitz. For $x_0 \in U$, let $\gamma_{x_0} : I(x_0) \to U$ be the maximal solution of the autonomous system $x' = f(x)$ with $x(0) = x_0$. Let $\Omega = \{(t, x_0) : x_0 \in U,\ t \in I(x_0)\}$. The flow of $f$ is the map $\varphi : \Omega \to U$ given by $\varphi(t, x_0) = \gamma_{x_0}(t)$. Another notation is $\varphi_t(x_0) = \varphi(t, x_0)$.

Remarks

The flow has the group properties $\varphi_0(x) = x$ and $\varphi_s(\varphi_t(x)) = \varphi_{s + t}(x)$, whenever both sides are defined. The second property follows from uniqueness and time translation. The set $\Omega$ is open and $\varphi$ is continuous on $\Omega$. If $\Omega = \mathbb{R} \times U$, the flow is complete. Then $t \mapsto \varphi_t$ is an action of the additive group $\mathbb{R}$ on $U$. For a linear system $x' = A x$, the flow is $\varphi_t(x) = e^{tA} x$, the matrix exponential.

Explanations

L1 human

The flow moves each point of the phase space along the solution through it. The map $\varphi_t$ says where each point is after time $t$. Moving for time $t$ and then for time $s$ is the same as moving for time $s + t$.