---
tag: 02903SSA48
kind: definition
name: Flow of a vector field
msc: [37C10]
defines: 06738GGWWK
uses: [07BNRFQ0PJ, 025PKW8AMR, 019VSXSFMQ]
sources:
  - url: https://en.wikipedia.org/wiki/Flow_(mathematics)
    locator: Autonomous systems of ordinary differential equations
  - url: https://en.wikipedia.org/wiki/Flow_(mathematics)
    locator: Formal definition
---

## 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|019VSXSFMQ]] of the [[autonomous system|07BNRFQ0PJ]] $x' = f(x)$ 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|025PKW8AMR]] and [[time translation|0WA7ZR95V4]].
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|0YC8JT513J]].

## 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 $s + t$.