Absorbing set (random dynamical systems)
- Absorbing set (random dynamical systems)
In mathematics, an absorbing set for a random dynamical system is a subset of the phase space that eventually contains the image of any bounded set under the cocycle ("flow") of the random dynamical system. As with many concepts related to random dynamical systems, it is defined in the pullback sense.
Definition
Consider a random dynamical system "φ" on a complete separable metric space ("X", "d"), where the noise is chosen from a probability space (Ω, Σ, P) with base flow "ϑ" : R × Ω → Ω. A random compact set "K" : Ω → 2"X" is said to be absorbing if, for all "d"-bounded deterministic sets "B" ⊆ "X", there exists a (finite) random time "τ""B" : Ω → [0, +∞) such that
:
This is a definition in the pullback sense, as indicated by the use of the negative time shift "ϑ"−"t".
References
* cite journal
last = Robinson
first = James C.
coauthors = Tearne, Oliver M.
title = Boundaries of attractors of omega limit sets
journal = Stoch. Dyn.
volume = 5
year = 2005
issue = 1
pages = 97–109
issn = 0219-4937
doi = 10.1142/S0219493705001304 MathSciNet|id=2118757 (See footnote (e) on p. 104)
Wikimedia Foundation.
2010.
Look at other dictionaries:
List of mathematics articles (A) — NOTOC A A Beautiful Mind A Beautiful Mind (book) A Beautiful Mind (film) A Brief History of Time (film) A Course of Pure Mathematics A curious identity involving binomial coefficients A derivation of the discrete Fourier transform A equivalence A … Wikipedia
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 … Wikipedia
Markov chain — A simple two state Markov chain. A Markov chain, named for Andrey Markov, is a mathematical system that undergoes transitions from one state to another, between a finite or countable number of possible states. It is a random process characterized … Wikipedia
Mathematics and Physical Sciences — ▪ 2003 Introduction Mathematics Mathematics in 2002 was marked by two discoveries in number theory. The first may have practical implications; the second satisfied a 150 year old curiosity. Computer scientist Manindra Agrawal of the… … Universalium
Entropy — This article is about entropy in thermodynamics. For entropy in information theory, see Entropy (information theory). For a comparison of entropy in information theory with entropy in thermodynamics, see Entropy in thermodynamics and information… … Wikipedia
Directed percolation — In statistical physics Directed Percolation (DP) refers to a class of models that mimic filtering of fluids through porous materials along a given direction. Varying the microscopic connectivity of the pores, these models display a phase… … Wikipedia
Photon — This article is about the elementary particle of light. For other uses, see Photon (disambiguation). Photon Photons emitted in a coherent beam from a laser Composition Elementary particle … Wikipedia
atom — /at euhm/, n. 1. Physics. a. the smallest component of an element having the chemical properties of the element, consisting of a nucleus containing combinations of neutrons and protons and one or more electrons bound to the nucleus by electrical… … Universalium
Hawking radiation — is a thermal radiation with a black body spectrum predicted to be emitted by black holes due to quantum effects. It is named after the physicist Stephen Hawking, who provided a theoretical argument for its existence in 1974, and sometimes also… … Wikipedia