Global attractors and steady states for uniformly persistent dynamical systems. (English) Zbl 1128.37016
It is known that, one of the important problem is to obtain sufficient conditions for the existence of interior global attractors for uniformly persistent dynamical systems. This is a nontrivial problem since the phase space \({\mathcal M}_0\) is an open subset of a complete metric space \(({\mathcal M},d)\). The main goal of this paper is to establish the existence of the interior global attractor (that is, the global attractor for \(T:({\mathcal M}_0, d)\to({\mathcal M}_0, d))\) and a fixed-point in \({\mathcal M}_0\). The authors provide four examples to show the existence of discrete – and continuous – time dynamical systems that admit global attractors, but no strong global attractors. Moreover, a simple periodic age-structured model is also studied to illustrate applicability in the case of convex \(k\)-contracting maps.
Reviewer: Messoud A. Efendiev (Berlin)
MSC:
37C25 | Fixed points and periodic points of dynamical systems; fixed-point index theory; local dynamics |
37C70 | Attractors and repellers of smooth dynamical systems and their topological structure |
37L05 | General theory of infinite-dimensional dissipative dynamical systems, nonlinear semigroups, evolution equations |
37N25 | Dynamical systems in biology |