Non-triviality of the Conley index for an isolating neighborhood $N$ and a flow $\varphi$ can be used to prove non-emptyness of the related isolated invariant set. In particular, if $N$ doesn't deformation retract to the (closed) exit set, then there is a nonempty positively invariant subset $$P=\{x\in N:\,\varphi([0,\infty),x)\subseteq N\}.$$ Can we use some related topological techniques to show that a given (nice) set $Init \subset N$ is contained in $P$? In other words, that the "initial" states never exit a given set $N$ in future times. Thanks for possible hint or reference.
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