Random dynamical systems

Exploring the Pullback Attractors of a Low-Order Quasigeostrophic Ocean Model: The Deterministic Case

“…We have studied this double-gyre model's PBAs (i.e., those timedependent invariant sets that attract all trajectories initialized in the remote past) along with the invariant sample measures that live on these sets. PBAs (Arnold 1998;Ghil et al 2008;Chekroun et al 2011) and their close cousins, snapshot attractors (Romeiras et al 1990;Bódai et al 2011Bódai et al , 2013Bódai and Tél 2012;Drótos et al 2015), have only recently been recognized as the natural tools for investigating basic features of a changing nonequilibrium climate, and the theory of nonautonomous dynamical systems as the proper mathematical framework for such investigations.…”

“…The limit t / 2' is the pullback limit and it provides the natural generalization of the forward asymptotics associated with autonomous systems (Arnold 1998;Rasmussen 2007;Kloeden and Rasmussen 2011;Carvalho et al 2012;Ghil et al 2008;Chekroun et al 2011); see the appendix herein. In practical terms, though, a question arises: How large should jtj be for A(t, t) to be virtually independent of t?…”

“…A global PBA is defined in the mathematical literature as a time-dependent set A(t) in the system's phase space X that is invariant under its governing equations-together with the equally time-dependent, invariant measure m(t) supported on this set-and to which all trajectories in X starting in the distant past converge (Arnold 1998;Rasmussen 2007;Kloeden and Rasmussen 2011;Carvalho et al 2012). As we shall see later, such a global PBA might include two or more local attractors, which only attract trajectories from certain subsets of X.…”

Attractors for stochastic lattice dynamical systems with a multiplicative noise

“…It is worth mentioning that, although it is known that finite dimensional stochastic differential equations generate random dynamical systems (see Arnold [2] Chapter 1), this is not true in general for infinite dimensional equations. However, for particular kinds of noise it is possible to transform the stochastic differential equation into a random one so that it generates a random dynamical system.…”

Quantum-like formalism for cognitive measurements

“…Let us denote the set of all operators representing mental observables participating in the creation of the contents of consciousness by the symbol L cons (H). Let us consider a random dynamical system (RDS: see, for example, [64] for general theory) that at each instant of (mental) time chooses randomly some set of commutative operators A 1 , ..., A m ∈ L cons (H). The contents of consciousness at this instant of time is created by the simultaneous measurement of…”