Toward a Theory of Chaos
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WHAT IS a CHAOTIC ATTRACTOR? 1. Introduction J. Yorke Coined the Word 'Chaos' As Applied to Deterministic Systems. R. Devane
WHAT IS A CHAOTIC ATTRACTOR? CLARK ROBINSON Abstract. Devaney gave a mathematical definition of the term chaos, which had earlier been introduced by Yorke. We discuss issues involved in choosing the properties that characterize chaos. We also discuss how this term can be combined with the definition of an attractor. 1. Introduction J. Yorke coined the word `chaos' as applied to deterministic systems. R. Devaney gave the first mathematical definition for a map to be chaotic on the whole space where a map is defined. Since that time, there have been several different definitions of chaos which emphasize different aspects of the map. Some of these are more computable and others are more mathematical. See [9] a comparison of many of these definitions. There is probably no one best or correct definition of chaos. In this paper, we discuss what we feel is one of better mathematical definition. (It may not be as computable as some of the other definitions, e.g., the one by Alligood, Sauer, and Yorke.) Our definition is very similar to the one given by Martelli in [8] and [9]. We also combine the concepts of chaos and attractors and discuss chaotic attractors. 2. Basic definitions We start by giving the basic definitions needed to define a chaotic attractor. We give the definitions for a diffeomorphism (or map), but those for a system of differential equations are similar. The orbit of a point x∗ by F is the set O(x∗; F) = f Fi(x∗) : i 2 Z g. An invariant set for a diffeomorphism F is an set A in the domain such that F(A) = A. -
Writing the History of Dynamical Systems and Chaos
Historia Mathematica 29 (2002), 273–339 doi:10.1006/hmat.2002.2351 Writing the History of Dynamical Systems and Chaos: View metadata, citation and similar papersLongue at core.ac.uk Dur´ee and Revolution, Disciplines and Cultures1 brought to you by CORE provided by Elsevier - Publisher Connector David Aubin Max-Planck Institut fur¨ Wissenschaftsgeschichte, Berlin, Germany E-mail: [email protected] and Amy Dahan Dalmedico Centre national de la recherche scientifique and Centre Alexandre-Koyre,´ Paris, France E-mail: [email protected] Between the late 1960s and the beginning of the 1980s, the wide recognition that simple dynamical laws could give rise to complex behaviors was sometimes hailed as a true scientific revolution impacting several disciplines, for which a striking label was coined—“chaos.” Mathematicians quickly pointed out that the purported revolution was relying on the abstract theory of dynamical systems founded in the late 19th century by Henri Poincar´e who had already reached a similar conclusion. In this paper, we flesh out the historiographical tensions arising from these confrontations: longue-duree´ history and revolution; abstract mathematics and the use of mathematical techniques in various other domains. After reviewing the historiography of dynamical systems theory from Poincar´e to the 1960s, we highlight the pioneering work of a few individuals (Steve Smale, Edward Lorenz, David Ruelle). We then go on to discuss the nature of the chaos phenomenon, which, we argue, was a conceptual reconfiguration as -
Attractors of Dynamical Systems in Locally Compact Spaces
Open Math. 2019; 17:465–471 Open Mathematics Research Article Gang Li* and Yuxia Gao Attractors of dynamical systems in locally compact spaces https://doi.org/10.1515/math-2019-0037 Received February 7, 2019; accepted March 4, 2019 Abstract: In this article the properties of attractors of dynamical systems in locally compact metric space are discussed. Existing conditions of attractors and related results are obtained by the near isolating block which we present. Keywords: Flow, attraction neighborhood, near isolating block MSC: 34C35 54H20 1 Introduction The dynamical system theory studies the rules of changes in the state which depends on time. In the investigation of dynamical systems, one of very interesting topics is the study of attractors (see [1 − 4] and the references therein). In [1 − 3], the authors gave some denitions of attractors and also made further investigations about the properties of them. In [5], the limit set of a neighborhood was used in the denition of an attractor, and in [6] Hale and Waterman also emphasized the importance of the limit set of a set in the analysis of the limiting behavior of a dynamical system. In [7 − 9], the authors dened intertwining attractor of dynamical systems in metric space and obtained some existing conditions of intertwining attractor. In [10], the author studied the properties of limit sets of subsets and attractors in a compact metric space. In [11], the author studied a positively bounded dynamical system in the plane, and obtained the conditions of compactness of the set of all bounded solutions. In [12], the uniform attractor was dened as the minimal compact uniformly pullback attracting random set, several existence criteria for uniform attractors were given and the relationship between uniform and co-cycle attractors was carefully studied. -
Control of Chaos: Methods and Applications
Automation and Remote Control, Vol. 64, No. 5, 2003, pp. 673{713. Translated from Avtomatika i Telemekhanika, No. 5, 2003, pp. 3{45. Original Russian Text Copyright c 2003 by Andrievskii, Fradkov. REVIEWS Control of Chaos: Methods and Applications. I. Methods1 B. R. Andrievskii and A. L. Fradkov Institute of Mechanical Engineering Problems, Russian Academy of Sciences, St. Petersburg, Russia Received October 15, 2002 Abstract|The problems and methods of control of chaos, which in the last decade was the subject of intensive studies, were reviewed. The three historically earliest and most actively developing directions of research such as the open-loop control based on periodic system ex- citation, the method of Poincar´e map linearization (OGY method), and the method of time- delayed feedback (Pyragas method) were discussed in detail. The basic results obtained within the framework of the traditional linear, nonlinear, and adaptive control, as well as the neural network systems and fuzzy systems were presented. The open problems concerned mostly with support of the methods were formulated. The second part of the review will be devoted to the most interesting applications. 1. INTRODUCTION The term control of chaos is used mostly to denote the area of studies lying at the interfaces between the control theory and the theory of dynamic systems studying the methods of control of deterministic systems with nonregular, chaotic behavior. In the ancient mythology and philosophy, the word \χαωσ" (chaos) meant the disordered state of unformed matter supposed to have existed before the ordered universe. The combination \control of chaos" assumes a paradoxical sense arousing additional interest in the subject. -
ONE HUNDRED YEARS of COMPLEX DYNAMICS the Subject of Complex Dynamics, That Is, the Behaviour of Orbits of Holomorphic Functions
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by University of Liverpool Repository ONE HUNDRED YEARS OF COMPLEX DYNAMICS MARY REES The subject of Complex Dynamics, that is, the behaviour of orbits of holomorphic functions, emerged in the papers produced, independently, by Fatou and Julia, almost 100 years ago. Although the subject of Dynami- cal Systems did not then have a name, the dynamical properties found for holomorphic systems, even in these early researches, were so striking, so unusually comprehensive, and yet so varied, that these systems still attract widespread fascination, 100 years later. The first distinctive feature of iter- ation of a single holomorphic map f is the partition of either the complex plane or the Riemann sphere into two sets which are totally invariant under f: the Julia set | closed, nonempty, perfect, with dynamics which might loosely be called chaotic | and its complement | open, possibly empty, but, if non-empty, then with dynamics which were completely classified by the two pioneering researchers, modulo a few simply stated open questions. Before the subject re-emerged into prominence in the 1980's, the Julia set was alternately called the Fatou set, but Paul Blanchard introduced the idea of calling its complement the Fatou set, and this was immediately universally accepted. Probably the main reason for the remarkable rise in interest in complex dynamics, about thirty-five years ago, was the parallel with the subject of Kleinian groups, and hence with the whole subject of hyperbolic geome- try. A Kleinian group acting on the Riemann sphere is a dynamical system, with the sphere splitting into two disjoint invariant subsets, with the limit set and its complement, the domain of discontinuity, having exactly similar properties to the Julia and Fatou sets. -
Maps, Chaos, and Fractals
MATH305 Summer Research Project 2006-2007 Maps, Chaos, and Fractals Phillips Williams Department of Mathematics and Statistics University of Canterbury Maps, Chaos, and Fractals Phillipa Williams* MATH305 Mathematics Project University of Canterbury 9 February 2007 Abstract The behaviour and properties of one-dimensional discrete mappings are explored by writing Matlab code to iterate mappings and draw graphs. Fixed points, periodic orbits, and bifurcations are described and chaos is introduced using the logistic map. Symbolic dynamics are used to show that the doubling map and the logistic map have the properties of chaos. The significance of a period-3 orbit is examined and the concept of universality is introduced. Finally the Cantor Set provides a brief example of the use of iterative processes to generate fractals. *supervised by Dr. Alex James, University of Canterbury. 1 Introduction Devaney [1992] describes dynamical systems as "the branch of mathematics that attempts to describe processes in motion)) . Dynamical systems are mathematical models of systems that change with time and can be used to model either discrete or continuous processes. Contin uous dynamical systems e.g. mechanical systems, chemical kinetics, or electric circuits can be modeled by differential equations. Discrete dynamical systems are physical systems that involve discrete time intervals, e.g. certain types of population growth, daily fluctuations in the stock market, the spread of cases of infectious diseases, and loans (or deposits) where interest is compounded at fixed intervals. Discrete dynamical systems can be modeled by iterative maps. This project considers one-dimensional discrete dynamical systems. In the first section, the behaviour and properties of one-dimensional maps are examined using both analytical and graphical methods. -
Forward Omega Limit Sets of Nonautonomous Dynamical Systems
DISCRETE AND CONTINUOUS doi:10.3934/dcdss.2020065 DYNAMICAL SYSTEMS SERIES S Volume 13, Number 4, April 2020 pp. 1103{1114 FORWARD OMEGA LIMIT SETS OF NONAUTONOMOUS DYNAMICAL SYSTEMS Hongyong Cui, Peter E. Kloeden and Meihua Yang School of Mathematics and Statistics, and Hubei Key Laboratory of Engineering Modeling and Scientific Computing Huazhong University of Science and Technology Wuhan 430074, China Dedicated to J¨urgen Scheurle on his 65th birthday. Abstract. The forward !-limit set !B of a nonautonomous dynamical system ' with a positively invariant absorbing family B = fB(t); t 2 Rg of closed and bounded subsets of a Banach space X which is asymptotically compact is shown to be asymptotically positive invariant in general and asymptotic negative invariant if ' is also strongly asymptotically compact and eventually continuous in its initial value uniformly on bounded time sets independently of the initial time. In addition, a necessary and sufficient condition for a '- invariant family A = fA(t); t 2 Rg in B of nonempty compact subsets of X to be a forward attractor is generalised to this context. 1. Introduction. Omega limit sets play a fundamental role in characterising the behaviour of dynamical systems. Indeed, the attractor of an autonomous dynamical system is an omega limit set. In such systems the omega limit sets are invariant. The situation is more complicated in nonautonomous dynamical systems because the behaviour now depends on the actual time and not on the the time that has elapsed since starting. This leads to new concepts of nonautonomous attractors that consist of families of compact subsets. -
Math Morphing Proximate and Evolutionary Mechanisms
Curriculum Units by Fellows of the Yale-New Haven Teachers Institute 2009 Volume V: Evolutionary Medicine Math Morphing Proximate and Evolutionary Mechanisms Curriculum Unit 09.05.09 by Kenneth William Spinka Introduction Background Essential Questions Lesson Plans Website Student Resources Glossary Of Terms Bibliography Appendix Introduction An important theoretical development was Nikolaas Tinbergen's distinction made originally in ethology between evolutionary and proximate mechanisms; Randolph M. Nesse and George C. Williams summarize its relevance to medicine: All biological traits need two kinds of explanation: proximate and evolutionary. The proximate explanation for a disease describes what is wrong in the bodily mechanism of individuals affected Curriculum Unit 09.05.09 1 of 27 by it. An evolutionary explanation is completely different. Instead of explaining why people are different, it explains why we are all the same in ways that leave us vulnerable to disease. Why do we all have wisdom teeth, an appendix, and cells that if triggered can rampantly multiply out of control? [1] A fractal is generally "a rough or fragmented geometric shape that can be split into parts, each of which is (at least approximately) a reduced-size copy of the whole," a property called self-similarity. The term was coined by Beno?t Mandelbrot in 1975 and was derived from the Latin fractus meaning "broken" or "fractured." A mathematical fractal is based on an equation that undergoes iteration, a form of feedback based on recursion. http://www.kwsi.com/ynhti2009/image01.html A fractal often has the following features: 1. It has a fine structure at arbitrarily small scales. -
Chaos Theory: the Essential for Military Applications
U.S. Naval War College U.S. Naval War College Digital Commons Newport Papers Special Collections 10-1996 Chaos Theory: The Essential for Military Applications James E. Glenn Follow this and additional works at: https://digital-commons.usnwc.edu/usnwc-newport-papers Recommended Citation Glenn, James E., "Chaos Theory: The Essential for Military Applications" (1996). Newport Papers. 10. https://digital-commons.usnwc.edu/usnwc-newport-papers/10 This Book is brought to you for free and open access by the Special Collections at U.S. Naval War College Digital Commons. It has been accepted for inclusion in Newport Papers by an authorized administrator of U.S. Naval War College Digital Commons. For more information, please contact [email protected]. The Newport Papers Tenth in the Series CHAOS ,J '.' 'l.I!I\'lt!' J.. ,\t, ,,1>.., Glenn E. James Major, U.S. Air Force NAVAL WAR COLLEGE Chaos Theory Naval War College Newport, Rhode Island Center for Naval Warfare Studies Newport Paper Number Ten October 1996 The Newport Papers are extended research projects that the editor, the Dean of Naval Warfare Studies, and the President of the Naval War CoJIege consider of particular in terest to policy makers, scholars, and analysts. Papers are drawn generally from manuscripts not scheduled for publication either as articles in the Naval War CollegeReview or as books from the Naval War College Press but that nonetheless merit extensive distribution. Candidates are considered by an edito rial board under the auspices of the Dean of Naval Warfare Studies. The views expressed in The Newport Papers are those of the authors and not necessarily those of the Naval War College or the Department of the Navy. -
A Poincaré-Bendixson Theorem for Hybrid Systems
A POINCARE-BENDIXSON´ THEOREM FOR HYBRID SYSTEMS WILLIAM CLARK, ANTHONY BLOCH, LEONARDO COLOMBO Abstract. The Poincar´e-Bendixsontheorem plays an important role in the study of the qualitative be- havior of dynamical systems on the plane; it describes the structure of limit sets in such systems. We prove a version of the Poincar´e-BendixsonTheorem for two dimensional hybrid dynamical systems and describe a method for computing the derivative of the Poincar´ereturn map, a useful object for the stability analysis of hybrid systems. We also prove a Poincar´e-BendixsonTheorem for a class of one dimensional hybrid dynamical systems. 1. Introduction H. Poincar´econsidered the problem of characterizing the structure of limit sets of trajectories of analytic vector fields on the plane in 1886 [17]. I. Bendixson improved the solution proposed by Poincar´ein 1903 by solving the problem under the weaker hypothesis of C1 vector fields [2]. Since then, the investigation of the asymptotic behavior of dynamical systems has been essential to understanding their behavior. The theory of Poincar´e-Bendixsonstudies so-called limit sets. The classic version of the Poincar´e-BendixsonTheorem states that if a trajectory is bounded and its limit set does not contain any fixed points, then the limt set is a periodic orbit [16]. Therefore, the problem of determining the existence of limit cycles in planar continuous dynamics is well understood. Hybrid systems [6] are non-smooth dynamical systems which exhibit a combination of smooth and discrete dynamics, where the flow evolves continuously on a state space, and a discrete transition occurs when the flow intersects a co-dimension one hypersurface. -
Free Agency, Determinism, and Chaos Theory
Free Agency, Determinism, and Chaos Theory David B. Timtnins THE DOCTRINE OF FREE AGENCY, while not unique to Mormonism, is per- haps more central to Mormon doctrine than it is to that of any other church or philosophy. Doctrine and Covenants 93:29 tells us, "Man also was in the beginning with God. Intelligence, or the light of truth, was not created or made, neither indeed can be. All truth is independent in that sphere in which God has placed it, to act for itself, as all intelligence also; otherwise there is no existence." Section 58, verses 26-28, adds, "[I]t is not meet that I command in all things, for he that is compelled in all things is a slothful and not a wise servant; wherefore he receiveth no reward. Verily I say, men should be anxiously engaged in a good cause and do many things of their own free will.. for the power is in them, wherefore they are agents unto themselves." On the concept of agency depends Mormonism's explanation of the nature of God, humankind, good and evil, and—since Mormons expect to be doing more than merely adoring God in the hereafter—the future of humanity and the universe. While agency appears self-evident to the simple believer and the un- instructed, it is not so to most physicists, mathematicians, and philoso- phers. Indeed, the contrary doctrine of determinism has ruled the realm of science at least since the days of Simon Laplace, the renowned French polymath of the seventeenth century, who maintained that given the one- time location, direction, and speed of every particle in the universe he could calculate the future with perfect accuracy for all time. -
PY2T20: CHAOS and COMPLEXITY (12 Lectures) Stefan Hutzler
PY2T20: CHAOS AND COMPLEXITY (12 lectures) Stefan Hutzler lecture notes at: http://www.tcd.ie/Physics/Foams/Lecture Notes/PY2T20ChaosAndComplexity February 7, 2014 Contents 1 A glossary 1 2 Examples of non-linear and chaotic behaviour 3 2.1 Population dynamics . 3 2.2 Non-linear electrical circuit . 6 2.3 Lorenz model of atmospheric convection . 7 2.4 Summary of observations . 7 3 Universal properties and self-similarity 8 3.1 Feigenbaum constants . 8 3.2 Measuring chaos . 9 3.3 Universality of chaos . 10 4 Determinism 12 5 Dynamics in phase space: Motion of the pendulum 13 5.1 Equation of motion for damped driven pendulum . 13 5.2 Phase space . 15 5.3 Damped driven pendulum: period doubling and chaos . 17 5.4 Properties of trajectories . 17 6 Some theory of chaotic dynamics 19 6.1 Long-term behaviour of dissipative systems . 19 6.2 Stability of fixed points . 20 6.2.1 one dimension . 20 6.2.2 two dimensions . 21 6.2.3 three dimensions . 23 6.3 Analysis of limit cycles . 23 i 6.4 Examples for damped driven pendulum . 23 6.5 Quasi-periodicity . 24 6.6 Different routes to chaos . 24 7 Iterated maps 26 7.1 Motivation . 26 7.2 Bernoulli shift . 26 8 Fractals 29 8.1 A mathematical monster: the Koch curve . 29 8.2 Fractal dimensions . 30 8.3 Examples of fractals . 30 9 Strange attractors 32 9.1 Definition . 32 9.2 Baker’s transformation . 33 9.3 Stretching and folding for the logistic map . 33 10 Advanced topics 35 10.1 Hamiltonian systems (motivation) .