The Semicontinuity of Attractors for Closed Relations on Compact Hausdorff Spaces

TL;DR

This paper proves that attractors for closed relations on compact Hausdorff spaces are semicontinuous, ensuring stability of attractor structures under small system perturbations, extending known results from flows and maps.

Contribution

It establishes semicontinuity of attractors for closed relations on compact Hausdorff spaces, generalizing previous results beyond flows and maps.

Findings

Attractors are semicontinuous for closed relations on compact Hausdorff spaces.

Semicontinuity guarantees preservation of attractor structure under small perturbations.

Extends known semicontinuity results from flows and maps to relations.

Abstract

We show that attractors are semicontinuous for closed relations on compact Hausdorff spaces. Semicontinuity is what guarantees that small changes to a system do not result in massive growth of certain features, notably attractors. That is, there is a certain preservation of structure. When it comes to flows, semiflows, and maps, it is well established that attractors are semicontinuous. In [2], relations were established as a way to generalize maps, and a formal definition of attractors was established. Relations (in the dynamical systems sense) represent discrete time systems, which may lack uniqueness (or existence) in forward time.