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Let $\varphi: [0,T] \rightarrow H$ be a Hilbert space valued $C^1$-function. Let $H = X \oplus X^{\perp}$ such that $\varphi(0) \in X$ and the implication $\varphi(t) \in X \Rightarrow \varphi'(t) \in X$ holds.

I ask: How can I show that $\varphi$ stays in $X$? It sounds natural and I guess it is true, but I fail to see how it can be shown.

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    $\begingroup$ On the positive side: assume there is a smooth foliation on $H$ such that $\phi(t)$ is tangent to the leaves. Then $\phi(t)$ belongs to the same leaf. $\endgroup$
    – Pietro Majer
    Apr 8, 2018 at 18:23

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Consider $\phi(t) = (1-e^{-1/t^2}, t)\in \mathbb R^2$. It satisfies your assumptions for $X= 0\times \mathbb R$, but it does not stay in $X$.

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    $\begingroup$ Or even $\phi(t) = (t^2, t)$. $\endgroup$
    – Nate Eldredge
    Apr 8, 2018 at 18:57
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    $\begingroup$ In fact even $\phi(t):=t^2$ in $H:=\mathbb{R}$ with $X:=(0)$. $\endgroup$
    – Pietro Majer
    Apr 8, 2018 at 20:23

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