Convergence to cycles as a typical asymptotic behavior in smooth strongly monotone discrete-time dynamical systems. (English) Zbl 0755.58039

Poláčik, P.; Tereščák, I.

Arch. Ration. Mech. Anal. 116, No. 4, 339-360 (1991).

Let \(X\) be a strongly ordered Banach space and \(F : X \to X\) a compact \(C^{ 1,\alpha}\)-map such that for any \(x \in X\) the derivative \(F'(x)\) is a strongly positive operator. Assume that any compact invariant set admits continuous separation, a quite technical condition introduced in the paper. Let \(G\) be a bounded positively invariant set. Then, the set of points from \(G\) having their \(\omega\)-limit sets equal to a periodic orbit contains an open and dense set in \(G\).

Reviewer: J.Ombach (Kraków)

MSC:

37C70

Attractors and repellers of smooth dynamical systems and their topological structure

Keywords:

monotone dynamical systems; generic properties

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