Saturated invariant sets and the boundary behaviour of differential systems (Q1260856)

The paper is related to persistence theory of dynamical systems in a domain with invariant boundary, i.e. the existence of trajectories converging to the boundary is considered. We present the results in the case of a Kolmogorov system \(\dot x^ i=x^ i f^ i(x)\), \(i=1,\dots,n\), \(x=(x^ 1,\dots,x^ n)\in \mathbb{R}^ n_ +\). It is pointed out that, given a compact isolated invariant set \(K\subset\partial\mathbb{R}^ n_ +\), sign conditions on \(f^ i\) plus the nontriviality of the Conley index of \(K\) with respect to the boundary flow imply the existence of a point \(x_ 0\in\text{Int }\mathbb{R}^ n_ +\) whose \(\omega\)-limit set is contained in \(K\). The main result of the paper is a ``boundary index'' theorem, the generalization of the above observation to a finite collection of compact isolated invariant subsets of \(\partial \mathbb{R}^ n_ +\).