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Let $\dot x = f(x)$ be a differential equation (assume $f$ is smooth enough). Let $x_{0}$ be an initial condition and the limit set $\omega$ of $x_{0}$ contains only fixed points (non-hyperbolic for sure). Let $\phi(x,t)$ be the flow and $p \in \omega$. Can I argue that there exists a $n_{0}$ such that for $n \geq n_{0}$ we have that $\phi(x_{0},n)$ $\in W_{c\;s}$ the center-stable manifold at fixed point $p$ ($n$ is natural).

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