Random dynamical system

In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, a random dynamical system is a <a href="/facts/Dynamical_system/782gREl5">dynamical system</a> in which the <a href="/facts/Equations_of_motion/25msr1vE">equations of motion</a> have an element of randomness to them. Random dynamical systems are characterized by a <a href="/facts/State_space/H1vjb3LL">state space</a> S, a <a href="/facts/Set_(mathematics)/BfucfHMq">set</a> of <a href="/facts/Map_(mathematics)/FABF1l2f">maps</a> 
 
 
 
 Γ
 
 
 {\displaystyle \Gamma }
 
 from S into itself that can be thought of as the set of all possible equations of motion, and a <a href="/facts/Probability_distribution/EpsKKVRu">probability distribution</a> Q on the set 
 
 
 
 Γ
 
 
 {\displaystyle \Gamma }
 
 that represents the random choice of map. Motion in a random dynamical system can be informally thought of as a state 
 
 
 
 X
 ∈
 S
 
 
 {\displaystyle X\in S}
 
 evolving according to a succession of maps randomly chosen according to the distribution Q.
An example of a random dynamical system is a <a href="/facts/Stochastic_differential_equation/ol9a9WNq">stochastic differential equation</a>; in this case the distribution Q is typically determined by noise terms. It consists of a <a href="/facts/Base_flow_(random_dynamical_systems)/bf8swbXr">base flow</a>, the "noise", and a <a href="/facts/Oseledec_theorem/s2Ylkrhw">cocycle</a> dynamical system on the "physical" <a href="/facts/Phase_space/qPe4Fq57">phase space</a>. Another example is discrete state random dynamical system; some elementary contradistinctions between Markov chain and random dynamical system descriptions of a stochastic dynamics are discussed.

Random dynamical system open-in-new

Random dynamical system