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Suppose $\begin{cases}x'(t)=f(x(t)) \\ x(0)=x_0\end{cases}$ where $f:\mathbb{R}^n\to\mathbb{R}^n$ is continuous and suppose $\phi(t,x_0)$ is a solution. Can someone provide an example of when $\phi(t+s,x_0)\neq\phi(t,\phi(s,x_0))$? I would think the example would have to violate the uniqueness of the IVP. I was thinking $f:\mathbb{R}\to\mathbb{R}$ being $f(x)=x^{1/3}$, with $\phi(s,x_0)=0$, but I don't think this works.

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    $\begingroup$ I think that more plausible example here is that your solution might not be defined for all real $t$ and $s$. For example, if $f(x) = x^2$ then all solutions of this system experience finite time blow-up : all solutions have the maximal interval of existence that is not $\mathbb{R}$. $\endgroup$ – Evgeny Aug 22 '16 at 19:11

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