A criterion to detect a non-trivial homology of an invariant set of a flow in R3

Abstract

Consider a flow in R3 and let K be the biggest invariant subset of some compact region of interest N ⊆ R. 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 Wazewski’s principle or the Poincaré–Hopf theorem can be used to detect whether K is non-empty or contains rest points, respectively. We present a criterion that can establish whether K has a non-trivial 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.