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.
Consider a random dynamical system
on a complete separable metric space
, where the noise is chosen from a probability space
A naïve definition of an attractor
for this random dynamical system would be to require that for any initial condition
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
However, we have not yet considered the effect of the noise
, which makes the system non-autonomous (i.e. it depends explicitly on time).
seconds into the "future", and considering the limit as
seconds into the "past", and evolves the system through
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.
-almost surely unique random set such that 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.
, then the random global attractor is given by where the union is taken over all bounded sets
Crauel (1999) proved that if the base flow