# Is it possible to produce an example where Picard Lindelof existence and uniqueness theorem fails?

Taken from Picard–Lindelöf Theorem (Taken from Hirsch and Smale)

(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.

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.