Minimal forward random point attractors need not exist

TL;DR

This paper constructs an example of a random dynamical system with multiple forward point attractors but no minimal one, highlighting a fundamental difference from pullback attractors.

Contribution

It provides the first known example of a system with multiple forward point attractors lacking a minimal attractor, addressing an open question in the theory.

Findings

Existence of multiple forward point attractors without a minimal one.

Contrast between properties of forward and pullback attractors.

Clarification of the structure of random attractors in dynamical systems.

Abstract

It is well-known that random attractors of a random dynamical system are generally not unique. It was shown in recent work by Hans Crauel and the author that if there exist more than one pullback or weak random attractor which attracts a given family of (possibly random) sets, then there exists a minimal (in the sense of smallest) one. This statement does not hold for forward random attractors. The same paper contains an example of a random dynamical system and a deterministic family of sets which has more than one forward attractor which attracts the given family but no minimal one. The question whether one can find an example which has multiple forward {\em point} attractors but no minimal one remained open. Here we provide such an example.