On asymptotically autonomous differential equations in the plane (Q1130201)

The qualitative behaviour of trajectories of solutions to perturbed autonomous differential equations in the plane is studied in the framework of an axiomatic theory of solution spaces for ordinary differential equations due to \textit{V. V. Filippov} [Russ. Math. Surv. 48, No. 1, 104-154 (1993); translation from Usp. Mat. Nauk. 48, No. 1(289), 103-154 (1993; Zbl 0808.34002)]. This theory provides a unified approach to the study of solutions of ordinary differential equations, including those with singularities, and of differential inclusions, by establishing different series of axioms which reflect fundamental properties of solution sets of ODEs and by introducing topological structures that make it possible to deal with the given sets of solutions as with elements of a topological space. More specifically, the paper is based on previous studies by Filippov on autonomous and perturbed autonomous differential equations, in particular, on planar ones, in the framework of the mentioned axiomatic theory. Two theorems on \(\omega\)-limit sets are generalized and a theorem by L. Markus (a kind of Poincaré-Bendixson type theorem). This last extension, which parallels to some degree work by \textit{H. R. Thieme} [J. Math. Biol. 30, No. 7, 755-763 (1992; Zbl 0761.34039)] -- in the sense that it deals with the case where the \(\omega\)-limit set of the asymptotically autonomous equation may contain stationary points of its limiting autonomous equation -- constitutes the main result of the paper.