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.

Set-up and motivation

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Consider a random dynamical system   on a complete separable metric space  , where the noise is chosen from a probability space   with base flow  .

A naïve definition of an attractor   for this random dynamical system would be to require that for any initial condition  ,   as  . This definition is far too limited, especially in dimensions higher than one. A more plausible definition, modelled on the idea of an omega-limit set, would be to say that a point   lies in the attractor   if and only if there exists an initial condition,  , and there is a sequence of times   such that

  as  .

This is not too far from a working definition. However, we have not yet considered the effect of the noise  , which makes the system non-autonomous (i.e. it depends explicitly on time). For technical reasons, it becomes necessary to do the following: instead of looking   seconds into the "future", and considering the limit as  , one "rewinds" the noise   seconds into the "past", and evolves the system through   seconds using the same initial condition. That is, one is interested in the pullback limit

 .

So, for example, in the pullback sense, the omega-limit set for a (possibly random) set   is the 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.

Several examples of pullback attractors of non-autonomous dynamical systems are presented analytically and numerically.[1]

Definition

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The pullback attractor (or random global attractor)   for a random dynamical system is a  -almost surely unique random set such that

  1.   is a random compact set:   is almost surely compact and   is a  -measurable function for every  ;
  2.   is invariant: for all   almost surely;
  3.   is attractive: for any deterministic bounded set  ,
  almost surely.

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.

Theorems relating omega-limit sets to attractors

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The attractor as a union of omega-limit sets

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If a random dynamical system has a compact random absorbing set  , then the random global attractor is given by

 

where the union is taken over all bounded sets  .

Bounding the attractor within a deterministic set

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Crauel (1999) proved that if the base flow   is ergodic and   is a deterministic compact set with

 

then    -almost surely.

References

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  1. ^ Li, Jeremiah H.; Ye, Felix X. -F.; Qian, Hong; Huang, Sui (2019-08-01). "Time-dependent saddle–node bifurcation: Breaking time and the point of no return in a non-autonomous model of critical transitions". Physica D: Nonlinear Phenomena. 395: 7–14. arXiv:1611.09542. doi:10.1016/j.physd.2019.02.005. ISSN 0167-2789. PMC 6836434. PMID 31700198.

Further reading

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