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Given an vector field $v:U (\subset R^n)\to{R^n}$. Consider the differential equation $\dot{x} = v(x)$. It's given that this differential equation with an intial condition $x(0) = x_0$ has unique solution for all $x_0$ in U.

Let $\Phi_{x_0}$ be the solution with initial value $x_0$. Is it true that $\Phi_{x_0+h} = \Phi_{x_0}+h$ ? If not, is there a way to prove that the function $\Psi(x,t) = \Phi_{x}(t)$ is smooth, given that $v$ is smooth vector field and U has no singular points of $v$, i.e $v(x) \neq 0$ for all x in U, using rectification theorem ?

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  • $\begingroup$ The answer to the first question is no and a point where the vector field vanishes is called a critical point, not singular point. See this article: en.wikipedia.org/wiki/… $\endgroup$ – jobe Feb 27 at 15:49

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