In mathematics, the attractor of a random dynamical system may be loosely thought of as a set to which the system evolves after a long enough time. The basic idea is the same as for a deterministic dynamical system, but requires careful treatment because random dynamical systems are necessarily non-autonomous. This requires one to consider the notion of a pullback attractor or attractor in the pullback sense.
Contents
Set-up and motivation
Consider a random dynamical system
A naïve definition of an attractor
This is not too far from a working definition. However, we have not yet considered the effect of the noise
So, for example, in the pullback sense, the omega-limit set for a (possibly random) set
Equivalently, this may be written as
Importantly, in the case of a deterministic dynamical system (one without noise), the pullback limit coincides with the deterministic forward limit, so it is meaningful to compare deterministic and random omega-limit sets, attractors, and so forth.
Definition
The pullback attractor (or random global attractor)
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A ( ω ) is a random compact set:A ( ω ) ⊆ X is almost surely compact andω ↦ d i s t ( x , A ( ω ) ) is a( F , B ( X ) ) -measurable function for everyx ∈ X ; -
A ( ω ) is invariant: for allφ ( t , ω ) ( A ( ω ) ) = A ( ϑ t ω ) almost surely; -
A ( ω ) is attractive: for any deterministic bounded setB ⊆ X ,
There is a slight abuse of notation in the above: the first use of "dist" refers to the Hausdorff semi-distance from a point to a set,
whereas the second use of "dist" refers to the Hausdorff semi-distance between two sets,
As noted in the previous section, in the absence of noise, this definition of attractor coincides with the deterministic definition of the attractor as the minimal compact invariant set that attracts all bounded deterministic sets.
The attractor as a union of omega-limit sets
If a random dynamical system has a compact random absorbing set
where the union is taken over all bounded sets
Bounding the attractor within a deterministic set
Crauel (1999) proved that if the base flow
then