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Asymptotically autonomous differential equations in the plane. (English) Zbl 0811.34036

The author studies the following questions: Assume that the equilibria of the limit equation of an asymptotically autonomous ordinary differential equation are isolated and that any solution of the limit equation converges to one of them. Does any solution of the asymptotically stable equation converge to an equilibrium of the limit equation as well?
The results and several examples show that the question cannot be positively answered without further conditions, even in the plane. The answer is positive whenever the equilibria of the limit equation are isolated compact invariant sets and are cyclically chained to each other.
This question has implications for studies of certain chemostat/gradostat and epidemic models.

MSC:

34D05 Asymptotic properties of solutions to ordinary differential equations
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