Lecture Notes on Limiting Systems and Chain Transitive Sets
نویسنده
چکیده
1 Motivations Example 1.1. Let D be a closed subset of R n. We consider the nonau-tonomous ordinary differential system: dx dt = f (t, x), t ≥ 0, x(0) = x 0 ∈ D. Assume that lim t→∞ f (t, x) = f 0 (x) uniformly for x in any bounded subset of D. Then we have a limiting autonomous system: dx dt = f 0 (x), t ≥ 0, x(0) = x 0 ∈ D. Problem. Under what conditions can we lift the long-time properties of solutions of the limiting system (1.2) to the nonautonomous system (1.1)? To solve the above problem, one may use the theory of asymptotically autonomous semiflows, see [3, 2] and the references therein.
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