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Convergence results and a PoincarĂ©-Bendixson trichotomy for asymptotically autonomous differential equations. (English) Zbl 0761.34039
The title aptly reflects the contents of this paper. It is concerned with non-autonomous systems (N) \(x'=f(t,x)\), with the property that the limit, as \(t\) tends to infinity, of \(f(t,x)\) is \(g(x)\), uniformly on compact subsets of Euclidean space. One would like to compare the asymptotic behavior of (N) with that of the autonomous system (A) \(y'=g(y)\). The basic results are due to Markus (1956) and they are stated carefully in this paper since one of the author’s points is that they are often misquoted. The following question is addressed by the author. Question: Assuming that the equilibria of (A) are isolated and that every solution of (A) converges to one of these, does every solution of (N) converge to one of these equilibria. This question is not settled by the results of Markus. The author provides an illuminating example to show that the answer is no, in general, even for planar systems. He proves the following extension of a result of Markus: suppose that the \(\omega\)- limit set, \(\omega\), of a forward bounded solution of a planar system (N) is contained in a neighborhood possessing at most finitely many equilibria of (A). Then the following trichotomy holds: (i) \(\omega\) consists of an equilibrium of (A), or (ii) \(\omega\) is the union of periodic orbits of (A) and possibly centers of (A) that are surrounded by periodic orbits of (A) lying in \(\omega\), or (iii) \(\omega\) contains equilibria of (A) that are cyclically chained to each other in \(\omega\) by (hetero-/homo-clinic) orbits of (A). A related open problem is stated for planar systems. A second answer obtained by Thieme to the question posed above is not restricted to planar systems. The answer is yes, provided that the equilibria of (A) are isolated and not cyclically chained to each other. The methods of proof are as interesting as the results. A more detailed presentation and further applications will appear in subsequent publications.
Reviewer: Hal Smith (Tempe)

MSC:
34D05 Asymptotic properties of solutions to ordinary differential equations
37-XX Dynamical systems and ergodic theory
92D30 Epidemiology
34C05 Topological structure of integral curves, singular points, limit cycles of ordinary differential equations
34C37 Homoclinic and heteroclinic solutions to ordinary differential equations
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