Entropy In Dynamical Systems

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Entropy in Dynamical Systems

Tomasz Downarowicz
Entropy in Dynamical Systems Book PDF

Author : Tomasz Downarowicz
Publisher : Cambridge University Press
Page : 405 pages
File Size : 47,6 Mb
Release : 2011-05-12
Category : Mathematics
ISBN : 9781139500876

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Entropy in Dynamical Systems by Tomasz Downarowicz Pdf

This comprehensive text on entropy covers three major types of dynamics: measure preserving transformations; continuous maps on compact spaces; and operators on function spaces. Part I contains proofs of the Shannon–McMillan–Breiman Theorem, the Ornstein–Weiss Return Time Theorem, the Krieger Generator Theorem and, among the newest developments, the ergodic law of series. In Part II, after an expanded exposition of classical topological entropy, the book addresses symbolic extension entropy. It offers deep insight into the theory of entropy structure and explains the role of zero-dimensional dynamics as a bridge between measurable and topological dynamics. Part III explains how both measure-theoretic and topological entropy can be extended to operators on relevant function spaces. Intuitive explanations, examples, exercises and open problems make this an ideal text for a graduate course on entropy theory. More experienced researchers can also find inspiration for further research.

Entropy in Dynamic Systems

Jan Awrejcewicz,J. A. Tenreiro Machado
Entropy in Dynamic Systems Book PDF

Author : Jan Awrejcewicz,J. A. Tenreiro Machado
Publisher : MDPI
Page : 172 pages
File Size : 52,6 Mb
Release : 2019-10-16
Category : Science
ISBN : 9783039216161

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Entropy in Dynamic Systems by Jan Awrejcewicz,J. A. Tenreiro Machado Pdf

In order to measure and quantify the complex behavior of real-world systems, either novel mathematical approaches or modifications of classical ones are required to precisely predict, monitor, and control complicated chaotic and stochastic processes. Though the term of entropy comes from Greek and emphasizes its analogy to energy, today, it has wandered to different branches of pure and applied sciences and is understood in a rather rough way, with emphasis placed on the transition from regular to chaotic states, stochastic and deterministic disorder, and uniform and non-uniform distribution or decay of diversity. This collection of papers addresses the notion of entropy in a very broad sense. The presented manuscripts follow from different branches of mathematical/physical sciences, natural/social sciences, and engineering-oriented sciences with emphasis placed on the complexity of dynamical systems. Topics like timing chaos and spatiotemporal chaos, bifurcation, synchronization and anti-synchronization, stability, lumped mass and continuous mechanical systems modeling, novel nonlinear phenomena, and resonances are discussed.

Dynamical Entropy in Operator Algebras

Sergey Neshveyev,Erling Størmer
Dynamical Entropy in Operator Algebras Book PDF

Author : Sergey Neshveyev,Erling Størmer
Publisher : Springer Science & Business Media
Page : 294 pages
File Size : 55,6 Mb
Release : 2006-09-22
Category : Mathematics
ISBN : 9783540346739

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Dynamical Entropy in Operator Algebras by Sergey Neshveyev,Erling Størmer Pdf

The book addresses mathematicians and physicists, including graduate students, who are interested in quantum dynamical systems and applications of operator algebras and ergodic theory. It is the only monograph on this topic. Although the authors assume a basic knowledge of operator algebras, they give precise definitions of the notions and in most cases complete proofs of the results which are used.

Topological Entropy and Equivalence of Dynamical Systems

Roy L. Adler,Brian Marcus
Topological Entropy and Equivalence of Dynamical Systems Book PDF

Author : Roy L. Adler,Brian Marcus
Publisher : American Mathematical Soc.
Page : 90 pages
File Size : 41,9 Mb
Release : 1979
Category : Ergodic theory
ISBN : 9780821822197

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Topological Entropy and Equivalence of Dynamical Systems by Roy L. Adler,Brian Marcus Pdf

The purpose of this work is to prove a theorem for topological entropy analogous to Ornstein's result for measure entropy. For this a natural class of dynamical systems is needed to play the same role for topological entropy as the Bernoulli shifts do for measure entropy. Fortunately there is just such a class--the topological Markov shifts. The main result of this paper is that topological entropy along with another number, called the ergodic period, is a complete set of invariants under this new equivalence relation for the class of topological Markov shifts.

Combinatorial Dynamics And Entropy In Dimension One

Alseda Luis,Llibre Jaume,Misiurewicz Michal
Combinatorial Dynamics And Entropy In Dimension One Book PDF

Author : Alseda Luis,Llibre Jaume,Misiurewicz Michal
Publisher : World Scientific Publishing Company
Page : 344 pages
File Size : 40,6 Mb
Release : 1993-06-04
Category : Mathematics
ISBN : 9789814553223

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Combinatorial Dynamics And Entropy In Dimension One by Alseda Luis,Llibre Jaume,Misiurewicz Michal Pdf

In last thirty years an explosion of interest in the study of nonlinear dynamical systems occured. The theory of one-dimensional dynamical systems has grown out in many directions. One of them has its roots in the Sharkovski0 Theorem. This beautiful theorem describes the possible sets of periods of all cycles of maps of an interval into itself. Another direction has its main objective in measuring the complexity of a system, or the amount of chaos present in it. A good way of doing this is to compute topological entropy of the system. The aim of this book is to provide graduate students and researchers with a unified and detailed exposition of these developments for interval and circle maps. Many comments are added referring to related problems, and historical remarks are made. Request Inspection Copy

Invariance Entropy for Deterministic Control Systems

Christoph Kawan
Invariance Entropy for Deterministic Control Systems Book PDF

Author : Christoph Kawan
Publisher : Springer
Page : 290 pages
File Size : 47,5 Mb
Release : 2013-10-02
Category : Mathematics
ISBN : 9783319012889

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Invariance Entropy for Deterministic Control Systems by Christoph Kawan Pdf

This monograph provides an introduction to the concept of invariance entropy, the central motivation of which lies in the need to deal with communication constraints in networked control systems. For the simplest possible network topology, consisting of one controller and one dynamical system connected by a digital channel, invariance entropy provides a measure for the smallest data rate above which it is possible to render a given subset of the state space invariant by means of a symbolic coder-controller pair. This concept is essentially equivalent to the notion of topological feedback entropy introduced by Nair, Evans, Mareels and Moran (Topological feedback entropy and nonlinear stabilization. IEEE Trans. Automat. Control 49 (2004), 1585–1597). The book presents the foundations of a theory which aims at finding expressions for invariance entropy in terms of dynamical quantities such as Lyapunov exponents. While both discrete-time and continuous-time systems are treated, the emphasis lies on systems given by differential equations.

Nonlinear Dynamics and Entropy of Complex Systems with Hidden and Self-excited Attractors

Christos Volos,Sajad Jafari,Jacques Kengne,Jesus M. Munoz-Pacheco,Karthikeyan Rajagopal
Nonlinear Dynamics and Entropy of Complex Systems with Hidden and Self excited Attractors Book PDF

Author : Christos Volos,Sajad Jafari,Jacques Kengne,Jesus M. Munoz-Pacheco,Karthikeyan Rajagopal
Publisher : MDPI
Page : 290 pages
File Size : 42,6 Mb
Release : 2019-05-03
Category : Technology & Engineering
ISBN : 9783038978985

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Nonlinear Dynamics and Entropy of Complex Systems with Hidden and Self-excited Attractors by Christos Volos,Sajad Jafari,Jacques Kengne,Jesus M. Munoz-Pacheco,Karthikeyan Rajagopal Pdf

In recent years, entropy has been used as a measure of the degree of chaos in dynamical systems. Thus, it is important to study entropy in nonlinear systems. Moreover, there has been increasing interest in the last few years regarding the novel classification of nonlinear dynamical systems including two kinds of attractors: self-excited attractors and hidden attractors. The localization of self-excited attractors by applying a standard computational procedure is straightforward. In systems with hidden attractors, however, a specific computational procedure must be developed, since equilibrium points do not help in the localization of hidden attractors. Some examples of this kind of system are chaotic dynamical systems with no equilibrium points; with only stable equilibria, curves of equilibria, and surfaces of equilibria; and with non-hyperbolic equilibria. There is evidence that hidden attractors play a vital role in various fields ranging from phase-locked loops, oscillators, describing convective fluid motion, drilling systems, information theory, cryptography, and multilevel DC/DC converters. This Special Issue is a collection of the latest scientific trends on the advanced topics of dynamics, entropy, fractional order calculus, and applications in complex systems with self-excited attractors and hidden attractors.

Dynamical Systems

Jürgen Jost
Dynamical Systems Book PDF

Author : Jürgen Jost
Publisher : Springer Science & Business Media
Page : 199 pages
File Size : 44,5 Mb
Release : 2005-11-24
Category : Science
ISBN : 9783540288893

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Dynamical Systems by Jürgen Jost Pdf

Breadth of scope is unique Author is a widely-known and successful textbook author Unlike many recent textbooks on chaotic systems that have superficial treatment, this book provides explanations of the deep underlying mathematical ideas No technical proofs, but an introduction to the whole field that is based on the specific analysis of carefully selected examples Includes a section on cellular automata

Local Entropy Theory of a Random Dynamical System

Anthony H. Dooley, Guohua Zhang
Local Entropy Theory of a Random Dynamical System Book PDF

Author : Anthony H. Dooley, Guohua Zhang
Publisher : American Mathematical Soc.
Page : 106 pages
File Size : 52,6 Mb
Release : 2014-12-20
Category : Mathematics
ISBN : 9781470410551

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Local Entropy Theory of a Random Dynamical System by Anthony H. Dooley, Guohua Zhang Pdf

In this paper the authors extend the notion of a continuous bundle random dynamical system to the setting where the action of R or N is replaced by the action of an infinite countable discrete amenable group. Given such a system, and a monotone sub-additive invariant family of random continuous functions, they introduce the concept of local fiber topological pressure and establish an associated variational principle, relating it to measure-theoretic entropy. They also discuss some variants of this variational principle. The authors introduce both topological and measure-theoretic entropy tuples for continuous bundle random dynamical systems, and apply variational principles to obtain a relationship between these of entropy tuples. Finally, they give applications of these results to general topological dynamical systems, recovering and extending many recent results in local entropy theory.

Combinatorial Dynamics and Entropy in Dimension One

Ll Alsedà,Jaume Llibre,Michał Misiurewicz
Combinatorial Dynamics and Entropy in Dimension One Book PDF

Author : Ll Alsedà,Jaume Llibre,Michał Misiurewicz
Publisher : World Scientific Publishing Company Incorporated
Page : 415 pages
File Size : 47,7 Mb
Release : 2000
Category : Mathematics
ISBN : 9810240538

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Combinatorial Dynamics and Entropy in Dimension One by Ll Alsedà,Jaume Llibre,Michał Misiurewicz Pdf

This book introduces the reader to two of the main directions of one-dimensional dynamics. The first has its roots in the Sharkovskii theorem, which describes the possible sets of periods of all periodic orbits of a continuous map of an interval into itself. The whole theory, which was developed based on this theorem, deals mainly with combinatorial objects, permutations, graphs, etc.: it is called combinatorial dynamics. The second direction has its main objective in measuring the complexity of a system, or the degree of "chaos" present in it. A good way of doing this is to study the topological entropy of the system. The aim of this book is to provide graduate students and researchers with a unified and detailed exposition of these developments for interval and circle maps. The second edition contains two new appendices, where an extension of the theory to tree and graph maps is presented without technical proofs.

Smooth Ergodic Theory of Random Dynamical Systems

Pei-Dong Liu,Min Qian
Smooth Ergodic Theory of Random Dynamical Systems Book PDF

Author : Pei-Dong Liu,Min Qian
Publisher : Springer
Page : 233 pages
File Size : 44,6 Mb
Release : 2006-11-14
Category : Mathematics
ISBN : 9783540492917

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Smooth Ergodic Theory of Random Dynamical Systems by Pei-Dong Liu,Min Qian Pdf

This book studies ergodic-theoretic aspects of random dynam- ical systems, i.e. of deterministic systems with noise. It aims to present a systematic treatment of a series of recent results concerning invariant measures, entropy and Lyapunov exponents of such systems, and can be viewed as an update of Kifer's book. An entropy formula of Pesin's type occupies the central part. The introduction of relation numbers (ch.2) is original and most methods involved in the book are canonical in dynamical systems or measure theory. The book is intended for people interested in noise-perturbed dynam- ical systems, and can pave the way to further study of the subject. Reasonable knowledge of differential geometry, measure theory, ergodic theory, dynamical systems and preferably random processes is assumed.

Dynamical Systems

I?Akov Grigor?evich Sina?
Dynamical Systems Book PDF

Author : I?Akov Grigor?evich Sina?
Publisher : World Scientific
Page : 694 pages
File Size : 41,7 Mb
Release : 1991
Category : Science
ISBN : 981020437X

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Dynamical Systems by I?Akov Grigor?evich Sina? Pdf

This volume consists of very high quality articles which not only give a very good account of this field in the Soviet Union, but also provide stimulating materials for researchers working on this topic.

Entropy of Complex Processes and Systems

Eugene Barsky
Entropy of Complex Processes and Systems Book PDF

Author : Eugene Barsky
Publisher : Elsevier
Page : 308 pages
File Size : 43,7 Mb
Release : 2020-08-17
Category : Science
ISBN : 9780128216620

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Entropy of Complex Processes and Systems by Eugene Barsky Pdf

Entropy of Complex Processes and Systems formalizes our understanding of many complex processes, including the development of the methodology of analytical computation of complex processes as applied in many industries, such as ore processing, or more generally, in areas of natural sciences. The adequacy of the results of these calculations is confirmed by numerous experimental data obtained both on pilots and industrial facilities. The book also provides a thorough analysis of the underlying physical foundations of entropy performed from new standpoints that are of interest to theoreticians studying contemporary expositions. Provides methodologies for controlling and optimizing complex processes in branches of industry that involve transformation of materials or substances Describes entropy as the universal characteristic of a stochastic process independent of the system Introduces a new definition of entropy specifically related to dynamical phenomena

Ergodic Theory and Dynamical Systems

Yves Coudène
Ergodic Theory and Dynamical Systems Book PDF

Author : Yves Coudène
Publisher : Springer
Page : 190 pages
File Size : 42,7 Mb
Release : 2016-11-10
Category : Mathematics
ISBN : 9781447172871

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Ergodic Theory and Dynamical Systems by Yves Coudène Pdf

This textbook is a self-contained and easy-to-read introduction to ergodic theory and the theory of dynamical systems, with a particular emphasis on chaotic dynamics. This book contains a broad selection of topics and explores the fundamental ideas of the subject. Starting with basic notions such as ergodicity, mixing, and isomorphisms of dynamical systems, the book then focuses on several chaotic transformations with hyperbolic dynamics, before moving on to topics such as entropy, information theory, ergodic decomposition and measurable partitions. Detailed explanations are accompanied by numerous examples, including interval maps, Bernoulli shifts, toral endomorphisms, geodesic flow on negatively curved manifolds, Morse-Smale systems, rational maps on the Riemann sphere and strange attractors. Ergodic Theory and Dynamical Systems will appeal to graduate students as well as researchers looking for an introduction to the subject. While gentle on the beginning student, the book also contains a number of comments for the more advanced reader.

Dynamical Systems, Ergodic Theory and Applications

L.A. Bunimovich,S.G. Dani,R.L. Dobrushin,M.V. Jakobson,I.P. Kornfeld,N.B. Maslova,Ya.B. Pesin,J. Smillie,Yu.M. Sukhov,A.M. Vershik
Dynamical Systems Ergodic Theory and Applications Book PDF

Author : L.A. Bunimovich,S.G. Dani,R.L. Dobrushin,M.V. Jakobson,I.P. Kornfeld,N.B. Maslova,Ya.B. Pesin,J. Smillie,Yu.M. Sukhov,A.M. Vershik
Publisher : Springer Science & Business Media
Page : 476 pages
File Size : 44,7 Mb
Release : 2000-04-05
Category : Mathematics
ISBN : 3540663169

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Dynamical Systems, Ergodic Theory and Applications by L.A. Bunimovich,S.G. Dani,R.L. Dobrushin,M.V. Jakobson,I.P. Kornfeld,N.B. Maslova,Ya.B. Pesin,J. Smillie,Yu.M. Sukhov,A.M. Vershik Pdf

This EMS volume, the first edition of which was published as Dynamical Systems II, EMS 2, familiarizes the reader with the fundamental ideas and results of modern ergodic theory and its applications to dynamical systems and statistical mechanics. The enlarged and revised second edition adds two new contributions on ergodic theory of flows on homogeneous manifolds and on methods of algebraic geometry in the theory of interval exchange transformations.