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Taken from Picard–Lindelöf Theorem (Taken from Hirsch and Smale) enter image description here

(Pretty sure $F$ can be relaxed to locally Lipschitz)

Note that the solution is unique on some unspecified interval $(t_0 - a, t_0 + a)$ around the starting time $t_0$.

I am wondering if there exists a system such that it would exihibit non-uniqueness outside of this interval.

I am thinking about bifurcative systems but I am not exactly familiar with that.

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Non-uniqueness can occur when a solution leaves a region where $F$ is locally Lipschitz. But your $F$ is assumed to be locally Lipschitz on all of $\mathbb R^n$. So the thing that can fail is existence, i.e. the solution to the initial value problem goes off to infinity at some finite value of $t$, and so on an interval containing this value there does not exist a solution.

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