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I stumbled upon the notion of a Lipschitz continuous dynamical system, and was wondering what this exactly meant. I guess it means that all solutions to (in my case the dynamical system is a set of ODE's) are Lipschitz continuous, but I wanted to be sure.

Thanks

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It means that the right side in $\dot y=f(t,y)$ is Lipschitz-continuous in $y$. Then the uniqueness theorem applies (Picard-Lindelöf) and also numerical simulation will give meaningful results.

The solutions will be continuously differentiable, if the function $f$ is Lipschitz in all variables, then the solutions have (locally) Lipschitz continuous derivatives.

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