Measure and Statistical Attractors for Nonautonomous Dynamical Systems

Abstract

Various inequivalent notions of attraction for autonomous dynamical systems have been proposed, each of them useful to understand specific aspects of attraction. Milnor’s notion of a measure attractor considers invariant sets with positive measure basin of attraction, while Ilyashenko’s weaker notion of a statistical attractor considers positive measure points that approach the invariant set in terms of averages. In this paper we propose generalisations of these notions to nonautonomous evolution processes in continuous time. We demonstrate that pullback/forward measure/statistical attractors can be defined in an analogous manner and relate these to the respective autonomous notions when an autonomous system is considered as nonautonomous. There are some subtleties even in this special case–we illustrate an example of a two-dimensional flow with a one-dimensional measure attractor containing a single point statistical attractor. We show that the single point can be a pullback measure attractor for this system. Finally, for the particular case of an asymptotically autonomous system (where there are autonomous future and past limit systems) we relate pullback (respectively, forward) attractors to the past (respectively, future) limit systems.