Let $U\subset \mathbb{R}^n$ be open, let $f:U\to \mathbb{R}^n$ be locally Lipschitz, and consider the autonomous [ordinary differential equation](/page/Ordinary%20Differential%20Equation)
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\begin{align*}
\dot{x}=f(x).
\end{align*}
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For each $x\in U$, let $J_x\subset \mathbb{R}$ be the maximal existence interval containing $0$ for the solution with initial value $x(0)=x$. Let
Let $\varphi:D\to U$ be the maximal flow map, meaning that for every $x\in U$ the curve $t\mapsto \varphi(t,x)$ from $J_x$ to $U$ is the unique maximal solution of $\dot{x}=f(x)$ with initial value $\varphi(0,x)=x$. If $x_0\in U$ and $s\in J_{x_0}$, then