A criterion to detect a nontrivial homology of an invariant set of a flow in $\mathbb{R}^3$

TL;DR

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Contribution

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Abstract

Consider a flow in $\mathbb{R}^3$ and let $K$ be the biggest invariant subset of some compact region of interest $N \subseteq \mathbb{R}^3$. The set $K$ is often not computable, but the way the flow crosses the boundary of $N$ can provide indirect information about it. For example, classical tools such as Wa\.{z}ewski's principle or the Poincar\'e-Hopf theorem can be used to detect whether $K$ is nonempty or contains rest points, respectively. We present a criterion that can establish whether $K$ has a nontrivial homology by looking at the subset of the boundary of $N$ along which the flow is tangent to $N$. We prove that the criterion is as sharp as possible with the information it uses as an input. We also show that it is algorithmically checkable.