Pullback attractor

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. 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 φ {displaystyle varphi } on a complete separable metric space ( X , d ) {displaystyle (X,d)} , where the noise is chosen from a probability space ( Ω , F , P ) {displaystyle (Omega ,{mathcal {F}},mathbb {P} )} with base flow ϑ : R × Ω → Ω {displaystyle vartheta :mathbb {R} imes Omega o Omega } . A naïve definition of an attractor A {displaystyle {mathcal {A}}} for this random dynamical system would be to require that for any initial condition x 0 ∈ X {displaystyle x_{0}in X} , φ ( t , ω ) x 0 → A {displaystyle varphi (t,omega )x_{0} o {mathcal {A}}} as t → + ∞ {displaystyle t o +infty } . 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 a ∈ X {displaystyle ain X} lies in the attractor A {displaystyle {mathcal {A}}} if and only if there exists an initial condition x 0 ∈ X {displaystyle x_{0}in X} , there is a sequence of times t n → + ∞ {displaystyle t_{n} o +infty } such that This is not too far from a working definition. However, we have not yet considered the effect of the noise ω {displaystyle omega } , 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 t {displaystyle t} seconds into the 'future', and considering the limit as t → + ∞ {displaystyle t o +infty } , one 'rewinds' the noise t {displaystyle t} seconds into the 'past', and evolves the system through t {displaystyle t} 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 B ( ω ) ⊆ X {displaystyle B(omega )subseteq X} is the random set