DOCSLIB.ORG

Complexity: Life at the Edge of Chaos, by Roger Lewin

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Complexity: Life at the Edge of Chaos, by Roger Lewin Book Review By Jean Mdlews l11e interaction of the components at the The "emergenists" (as t11e seekers of bottom of tl1e diagram give rise to properties tl1csc rules have been called) seem to lmve Complexity: Lifeat the Edge ofChaos, by thartould not have been predicted from what reached a tentative definition of' 'progress'' Roger Lewin (Macmillan, 1992) is a spell­ you know of the component parts. And the in the evolution of systems: the ability to bindingjoumey through ll1c fields and labo­ emergent properties then feed back, to influ­ process more and more information. What ratories oft hose who are pushing toward a set cncetllc behavioroftl1e individual interactors should interest readers ofPark Science is the of rules that some day may provide a grand that produced them. possible role of tl1 is new tllCOI)' as a push Wlification ofthe life sciences. Their various Le\\~n t11enjumpsback to tl1ebeginningof toward a holistic view of nature. The Sant.1 approaches to these elusive rules have landed the story-in the early 1960s when a bright Fe Institute people talk of" self-<>rganization them at the core of the current struggle to young scholar, Stuart Kauffman (now of the in complex systems, the emergence of pat­ redefine evolution-or at least to rewrite the University of Pennsylvania) began playing terns in evolutionary models that mimic Danvinian version of it. around very seriously witl1 random Boolean patterns in nature, and the idea that livi ng Lewin's story beg ins in Chaco Canyon. networks. KauiTma.n 's ignorance of matlle­ systems. as complex dynamical systems, arc NM, thecentcralmosta milleniumagoofthe matics scn'ed him well; he accomplished driven to these same patterns. They arc complex, sophisticated Anasazi culture. A)­ something no knowledgable mathematician saying tl1ere is a deep tl1eory to the order of though it disappeared like steam from a would have attempted. By incredible luck, nature." boiling kettle, theAnasazis' economic, polit­ early in rus computer runs, his modest net­ When they are accused of straying from ical, and religious web, which covered more work stumbled into an emergence oforder of mechanics and " looking for tl1e meaning of than a hundred thousand square miles, is a sort. His first thought was " Oh my God. li fe," they reply (in the words of Goodwin): referred to by today's archeologists as the J'vefoundsometl1ingprofound,'' and he told " We' re not looking for the meaning oflife. Chaco phenomenon. Le·win' ' I still tlUnk so. It'sthecrystaJiization more the meaning in life, the generation of From that bleak, arid terrain on the Colo­ oford er out of massively disordered systems. order. U1e generation ofpatt ern. the quality of rado Plateau, the story moves to consider­ It 's order for free." llus "accident," born t11c organism.' ' ation of how such complex systems as the of intuition and nurtured by diligence and Kauffman adds:' 'Pure Danvinismlcwcs Anasazi culture mjght have arisen from a luck, is one of the first building blocks in an you without an explanation of! he generation simple set oforga ni zational rules. For some­ edifice that hns arisen from similarly seren­ of biological form. In the Danvinian view. one who has read Jan1es Gleick's absorbing dipitous starting points in a scientific land­ organisms arejus t cobbled-! ogethcr products best-seller, Chaos, it maybe hard to imagine scape ranging from geology and biology to of random mutation and natural selection. a more entlm1llingjourney through the fron­ <trchcology and evolution. TI1e names or mindlessly rollowing adaptation first in one tiers of scientific discovery, but Lewin has contributors to this new scientific adventure direction, tl1cn the other. I find that deeply provided a woriliy sequel. inc lude Murray Geii-Mann, Warren w1satisfying and l don't think that's because ll1e theory of chaos is described early on McCulloch, John Maynard Srni U1, Per Bok. I want there to be some purpose in evolu­ in Lewin's book by Chris Langton of the James LO\·clock, Stuart Pimm, Richard tion.'' K<1ulTma n would refonnulatc Dar­ Santa Fe Institute as a subset of complexity Dawbns, John Cowan, Edward 0 . Wilson, winiCin theory to include sclf-organi/.ation. " in that you are dealing witl1 nonlinear Stephen Jay Gould, and Brian Goodwin. " We have no tllCOI)' in chemist!)'. physics. dynarnical systems." In thecaseof chaos, he Lewin describes tl1c debate between Gould biology, or beyond, that marries sclf-<>rgani­ explains, a few tlungs arc interacting, pro­ and Goodwin as to whetl1er complexity and zation and selection. To do so. CIS I think we ducing tremendously divergent behavior­ the edge ofc haos reveal a sort of progress in must, brings a ncwviewoflifc.'' lncffeet. he what he calls' 'deterministicchaos." It looks the random flow of Darwinian selection. says, it extends sclf-<>rg<~nization from the random, he says, but it's not, because it When Goodwin is challenged about lus def­ realm of physics. where it's accepted. into results from "oflcn quite simple equations inition oftl1e idea of"quality" in an organ­ biology. where it is sti ll viewed as mystic1l at that you can specify." In the case of com­ ism, he replies that by "quality" he means best and heretical at worst. plexity. Langton continues. " interactions in " the organism ns the cause a11d effect of Lewin is a Ph.D. in biochemistry from the a dynamjcaJ system give you an emergent itself, its own intrinsic order and organiza­ University of Liverpool. I lis most recent global order, witl1 a whole set of fascinating tion." Good\vin asks us to tl1ink of organ­ book. Bones ofCon tention. has been named properties" leading to what the Complexity isms as the result of a biological attractor-a the U.K.'s top science book for <1 general theorists call "emergence." sort of whirlpool in tl1e sea of a complex nudicncc. besting both Stephen Hnwking'sA Langton's view ofemergence in complex dynamical system. Then, he says, "you /Jriefllutory of Time, and James Gleiek 's systems looks like tllis: begin to approach what I mean by quality.'' Chaos. lnMay 1989,Lcwinrcccivcdthcfirst In addition to the gripping story of how Lewis Thomas Awnrd for Excellence in complexity tl~eory has grown, by leaps of Communicating Life Science. faitl1 and intuition simultaneously in differ­ From Comp/e:ri~l' ent disciplines and farfl unggcographic locn­ " ... if the concept of the edge of c hn os tions, I found most compelling the idea of does indeed trnnslatc from computer models Danvinian adaptation being only tl1e surface to tl1c real world. <IS Stu Knuffman. Chris manifestation of evolution, riding on the Langton, and others firmly believe it wi ll. deeper structure of rules that seem to govern then there will be nothing trivial about it at nonlinear dynamical systems of all bnds. all. Stu ·s cocvlution<~ry model systems get tluoughout the universe. Conrmrted on pa?,e 22 LocallnleractiOn Summer 1993 19 themselves to the edge ofchaos, and so too do may even be legitimate to think of them as In studies ofthe dynamics ofbi ological sys­ Stuart Pinm1's and Jim Drake's ecological behaving and evolvingasa whole, analogous tems, researchers face the dilemma ofde ter­ models. No one can say yet whether individ­ wit11 the superorganism concept that Ed mining from expetimental data whether ob­ ual ecosystems do the same thing. but the Wilson talked about in connection witl1 so­ served vmiations t ~presellf random flucflla­ data from mass extinctions at least suggest cial insect colonies. Coevolving conmmni­ tions or the chaotic state ofa detenninisl1ic system. Ifthey can demonstmte tlrat the sys­ that, globally, they do. 'That's a powerful ties act in concert as a result of the dynamics tem is chaotic rather than roiUlom, tlreyhave message ofa powerful instrinsic dynamic,'' of tl1e system; they do so as a result of a better chance ofdeveloping a strategy to said Chris. 'Systems poised at the edge of individua Is witl1in tl1ecommunity myopical­ understand and controltlris e/7-atic behavior. chaos achieve exquisite control, and I believe ly optimizing U1eir own ends and not as you sec that right the way up to Gaia.' collectiveagrccmenttowardacommon goal; Peterson quotes Leon M. Glass of McGill "If it's true that, for instance, ecological and the communities rc.1llydocome to know University in Montreal: "Complex aperiodic conmmnitics move toward tl1eedgeofchaos , tl1ei r world in a way that was quite unpredict­ rhytltms that arc observed in natural systems where novel properties emerge (such as able before the science of Complexity began might be due to detcrmi nist ic chaos, random foodwebs and tl1e ability of a long-estab­ to illuminate that world. " ' noise.' or some combination of the two lished conmmnity to resist invasion by alien • • • different mechanisms. Thus, the interpreta­ species), then it seems legitimate to talk tion of the dynamical basis of complex Troubling complexities is the title of an about such conmmnities as real systems. It aperiodic rhythms in natural systems is a article by I. Peterson in t11e Sept. i, 1992 issue difficult and hotly debated topic." of Science News (p. 157). In it, Peterson states: .
Load more

Recommended publications
  • Complexity” Makes a Difference: Lessons from Critical Systems Thinking and the Covid-19 Pandemic in the UK
    systems Article How We Understand “Complexity” Makes a Difference: Lessons from Critical Systems Thinking and the Covid-19 Pandemic in the UK Michael C. Jackson Centre for Systems Studies, University of Hull, Hull HU6 7TS, UK; [email protected]; Tel.: +44-7527-196400 Received: 11 November 2020; Accepted: 4 December 2020; Published: 7 December 2020 Abstract: Many authors have sought to summarize what they regard as the key features of “complexity”. Some concentrate on the complexity they see as existing in the world—on “ontological complexity”. Others highlight “cognitive complexity”—the complexity they see arising from the different interpretations of the world held by observers. Others recognize the added difficulties flowing from the interactions between “ontological” and “cognitive” complexity. Using the example of the Covid-19 pandemic in the UK, and the responses to it, the purpose of this paper is to show that the way we understand complexity makes a huge difference to how we respond to crises of this type. Inadequate conceptualizations of complexity lead to poor responses that can make matters worse. Different understandings of complexity are discussed and related to strategies proposed for combatting the pandemic. It is argued that a “critical systems thinking” approach to complexity provides the most appropriate understanding of the phenomenon and, at the same time, suggests which systems methodologies are best employed by decision makers in preparing for, and responding to, such crises. Keywords: complexity; Covid-19; critical systems thinking; systems methodologies 1. Introduction No one doubts that we are, in today’s world, entangled in complexity. At the global level, economic, social, technological, health and ecological factors have become interconnected in unprecedented ways, and the consequences are immense.
    [Show full text]
  • A General Representation of Dynamical Systems for Reservoir Computing
    A general representation of dynamical systems for reservoir computing Sidney Pontes-Filho∗,y,x, Anis Yazidi∗, Jianhua Zhang∗, Hugo Hammer∗, Gustavo B. M. Mello∗, Ioanna Sandvigz, Gunnar Tuftey and Stefano Nichele∗ ∗Department of Computer Science, Oslo Metropolitan University, Oslo, Norway yDepartment of Computer Science, Norwegian University of Science and Technology, Trondheim, Norway zDepartment of Neuromedicine and Movement Science, Norwegian University of Science and Technology, Trondheim, Norway Email: [email protected] Abstract—Dynamical systems are capable of performing com- TABLE I putation in a reservoir computing paradigm. This paper presents EXAMPLES OF DYNAMICAL SYSTEMS. a general representation of these systems as an artificial neural network (ANN). Initially, we implement the simplest dynamical Dynamical system State Time Connectivity system, a cellular automaton. The mathematical fundamentals be- Cellular automata Discrete Discrete Regular hind an ANN are maintained, but the weights of the connections Coupled map lattice Continuous Discrete Regular and the activation function are adjusted to work as an update Random Boolean network Discrete Discrete Random rule in the context of cellular automata. The advantages of such Echo state network Continuous Discrete Random implementation are its usage on specialized and optimized deep Liquid state machine Discrete Continuous Random learning libraries, the capabilities to generalize it to other types of networks and the possibility to evolve cellular automata and other dynamical systems in terms of connectivity, update and learning rules. Our implementation of cellular automata constitutes an reservoirs [6], [7]. Other systems can also exhibit the same initial step towards a general framework for dynamical systems. dynamics. The coupled map lattice [8] is very similar to It aims to evolve such systems to optimize their usage in reservoir CA, the only exception is that the coupled map lattice has computing and to model physical computing substrates.
    [Show full text]
  • EMERGENT COMPUTATION in CELLULAR NEURRL NETWORKS Radu Dogaru
    IENTIFIC SERIES ON Series A Vol. 43 EAR SCIENC Editor: Leon 0. Chua HI EMERGENT COMPUTATION IN CELLULAR NEURRL NETWORKS Radu Dogaru World Scientific EMERGENT COMPUTATION IN CELLULAR NEURAL NETWORKS WORLD SCIENTIFIC SERIES ON NONLINEAR SCIENCE Editor: Leon O. Chua University of California, Berkeley Series A. MONOGRAPHS AND TREATISES Volume 27: The Thermomechanics of Nonlinear Irreversible Behaviors — An Introduction G. A. Maugin Volume 28: Applied Nonlinear Dynamics & Chaos of Mechanical Systems with Discontinuities Edited by M. Wiercigroch & B. de Kraker Volume 29: Nonlinear & Parametric Phenomena* V. Damgov Volume 30: Quasi-Conservative Systems: Cycles, Resonances and Chaos A. D. Morozov Volume 31: CNN: A Paradigm for Complexity L O. Chua Volume 32: From Order to Chaos II L. P. Kadanoff Volume 33: Lectures in Synergetics V. I. Sugakov Volume 34: Introduction to Nonlinear Dynamics* L Kocarev & M. P. Kennedy Volume 35: Introduction to Control of Oscillations and Chaos A. L. Fradkov & A. Yu. Pogromsky Volume 36: Chaotic Mechanics in Systems with Impacts & Friction B. Blazejczyk-Okolewska, K. Czolczynski, T. Kapitaniak & J. Wojewoda Volume 37: Invariant Sets for Windows — Resonance Structures, Attractors, Fractals and Patterns A. D. Morozov, T. N. Dragunov, S. A. Boykova & O. V. Malysheva Volume 38: Nonlinear Noninteger Order Circuits & Systems — An Introduction P. Arena, R. Caponetto, L Fortuna & D. Porto Volume 39: The Chaos Avant-Garde: Memories of the Early Days of Chaos Theory Edited by Ralph Abraham & Yoshisuke Ueda Volume 40: Advanced Topics in Nonlinear Control Systems Edited by T. P. Leung & H. S. Qin Volume 41: Synchronization in Coupled Chaotic Circuits and Systems C. W. Wu Volume 42: Chaotic Synchronization: Applications to Living Systems £.
    [Show full text]
  • Dissipative Structures, Complexity and Strange Attractors: Keynotes for a New Eco-Aesthetics
    Dissipative structures, complexity and strange attractors: keynotes for a new eco-aesthetics 1 2 3 3 R. M. Pulselli , G. C. Magnoli , N. Marchettini & E. Tiezzi 1Department of Urban Studies and Planning, M.I.T, Cambridge, U.S.A. 2Faculty of Engineering, University of Bergamo, Italy and Research Affiliate, M.I.T, Cambridge, U.S.A. 3Department of Chemical and Biosystems Sciences, University of Siena, Italy Abstract There is a new branch of science strikingly at variance with the idea of knowledge just ended and deterministic. The complexity we observe in nature makes us aware of the limits of traditional reductive investigative tools and requires new comprehensive approaches to reality. Looking at the non-equilibrium thermodynamics reported by Ilya Prigogine as the key point to understanding the behaviour of living systems, the research on design practices takes into account the lot of dynamics occurring in nature and seeks to imagine living shapes suiting living contexts. When Edgar Morin speaks about the necessity of a method of complexity, considering the evolutive features of living systems, he probably means that a comprehensive method should be based on deep observations and flexible ordering laws. Actually designers and planners are engaged in a research field concerning fascinating theories coming from science and whose playground is made of cities, landscapes and human settlements. So, the concept of a dissipative structure and the theory of space organized by networks provide a new point of view to observe the dynamic behaviours of systems, finally bringing their flowing patterns into the practices of design and architecture. However, while science discovers the fashion of living systems, the question asked is how to develop methods to configure open shapes according to the theory of evolutionary physics.
    [Show full text]
  • Fractal Curves and Complexity
    Perception & Psychophysics 1987, 42 (4), 365-370 Fractal curves and complexity JAMES E. CUTI'ING and JEFFREY J. GARVIN Cornell University, Ithaca, New York Fractal curves were generated on square initiators and rated in terms of complexity by eight viewers. The stimuli differed in fractional dimension, recursion, and number of segments in their generators. Across six stimulus sets, recursion accounted for most of the variance in complexity judgments, but among stimuli with the most recursive depth, fractal dimension was a respect­ able predictor. Six variables from previous psychophysical literature known to effect complexity judgments were compared with these fractal variables: symmetry, moments of spatial distribu­ tion, angular variance, number of sides, P2/A, and Leeuwenberg codes. The latter three provided reliable predictive value and were highly correlated with recursive depth, fractal dimension, and number of segments in the generator, respectively. Thus, the measures from the previous litera­ ture and those of fractal parameters provide equal predictive value in judgments of these stimuli. Fractals are mathematicalobjectsthat have recently cap­ determine the fractional dimension by dividing the loga­ tured the imaginations of artists, computer graphics en­ rithm of the number of unit lengths in the generator by gineers, and psychologists. Synthesized and popularized the logarithm of the number of unit lengths across the ini­ by Mandelbrot (1977, 1983), with ever-widening appeal tiator. Since there are five segments in this generator and (e.g., Peitgen & Richter, 1986), fractals have many curi­ three unit lengths across the initiator, the fractionaldimen­ ous and fascinating properties. Consider four. sion is log(5)/log(3), or about 1.47.
    [Show full text]
  • 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
    [Show full text]
  • Measures of Complexity a Non--Exhaustive List
    Measures of Complexity a non--exhaustive list Seth Lloyd d'Arbeloff Laboratory for Information Systems and Technology Department of Mechanical Engineering Massachusetts Institute of Technology [email protected] The world has grown more complex recently, and the number of ways of measuring complexity has grown even faster. This multiplication of measures has been taken by some to indicate confusion in the field of complex systems. In fact, the many measures of complexity represent variations on a few underlying themes. Here is an (incomplete) list of measures of complexity grouped into the corresponding themes. An historical analog to the problem of measuring complexity is the problem of describing electromagnetism before Maxwell's equations. In the case of electromagnetism, quantities such as electric and magnetic forces that arose in different experimental contexts were originally regarded as fundamentally different. Eventually it became clear that electricity and magnetism were in fact closely related aspects of the same fundamental quantity, the electromagnetic field. Similarly, contemporary researchers in architecture, biology, computer science, dynamical systems, engineering, finance, game theory, etc., have defined different measures of complexity for each field. Because these researchers were asking the same questions about the complexity of their different subjects of research, however, the answers that they came up with for how to measure complexity bear a considerable similarity to eachother. Three questions that researchers frequently ask to quantify the complexity of the thing (house, bacterium, problem, process, investment scheme) under study are 1. How hard is it to describe? 2. How hard is it to create? 3. What is its degree of organization? Here is a list of some measures of complexity grouped according to the question that they try to answer.
    [Show full text]
  • Edge of Spatiotemporal Chaos in Cellular Nonlinear Networks
    1998 Fifth IEEE International Workshop on Cellular Neural Networks and their Applications, London, England, 14-17 April, 1998 EDGE OF SPATIOTEMPORAL CHAOS IN CELLULAR NONLINEAR NETWORKS Alexander S. Dmitriev and Yuri V. Andreyev Institute of Radioengineering and Electronics of the Russian Academy of Sciences Mokhovaya st., 11, Moscow, GSP-3, 103907, Russia Email: [email protected] ABSTRACT: In this report, we investigate phenomena on the edge of spatially uniform chaotic mode and spatial temporal chaos in a lattice of chaotic 1-D maps with only local connections. It is shown that in autonomous lattice with local connections, spatially uni- form chaotic mode cannot exist if the Lyapunov exponent l of the isolated chaotic map is greater than some critical value lcr > 0. We proposed a model of a lattice with a pace- maker and found a spatially uniform mode synchronous with the pacemaker, as well as a spatially uniform mode different from the pacemaker mode. 1. Introduction A number of arguments indicates that along with the three well-studied types of motion of dynamic systems — stationary state, stable periodic and quasi-periodic oscillations, and chaos — there is a special type of the system behavior, i.e., the behavior at the boundary between the regular motion and chaos. As was noticed, pro- cesses similar to evolution or information processing can take place at this boundary, as is often called, on the "edge of chaos". Here are some remarkable examples of the systems demonstrating the behavior on the edge of chaos: cellu- lar automata with corresponding rules [1], self-organized criticality [2], the Earth as a natural system with the main processes taking place on the boundary between the globe and the space.
    [Show full text]
  • Lecture Notes on Descriptional Complexity and Randomness
    Lecture notes on descriptional complexity and randomness Peter Gács Boston University [email protected] A didactical survey of the foundations of Algorithmic Information The- ory. These notes introduce some of the main techniques and concepts of the field. They have evolved over many years and have been used and ref- erenced widely. “Version history” below gives some information on what has changed when. Contents Contents iii 1 Complexity1 1.1 Introduction ........................... 1 1.1.1 Formal results ...................... 3 1.1.2 Applications ....................... 6 1.1.3 History of the problem.................. 8 1.2 Notation ............................. 10 1.3 Kolmogorov complexity ..................... 11 1.3.1 Invariance ........................ 11 1.3.2 Simple quantitative estimates .............. 14 1.4 Simple properties of information ................ 16 1.5 Algorithmic properties of complexity .............. 19 1.6 The coding theorem....................... 24 1.6.1 Self-delimiting complexity................ 24 1.6.2 Universal semimeasure.................. 27 1.6.3 Prefix codes ....................... 28 1.6.4 The coding theorem for ¹Fº . 30 1.6.5 Algorithmic probability.................. 31 1.7 The statistics of description length ............... 32 2 Randomness 37 2.1 Uniform distribution....................... 37 2.2 Computable distributions..................... 40 2.2.1 Two kinds of test..................... 40 2.2.2 Randomness via complexity............... 41 2.2.3 Conservation of randomness............... 43 2.3 Infinite sequences ........................ 46 iii Contents 2.3.1 Null sets ......................... 47 2.3.2 Probability space..................... 52 2.3.3 Computability ...................... 54 2.3.4 Integral.......................... 54 2.3.5 Randomness tests .................... 55 2.3.6 Randomness and complexity............... 56 2.3.7 Universal semimeasure, algorithmic probability . 58 2.3.8 Randomness via algorithmic probability........
    [Show full text]
  • The Edge of Chaos – an Alternative to the Random Walk Hypothesis
    THE EDGE OF CHAOS – AN ALTERNATIVE TO THE RANDOM WALK HYPOTHESIS CIARÁN DORNAN O‟FATHAIGH Junior Sophister The Random Walk Hypothesis claims that stock price movements are random and cannot be predicted from past events. A random system may be unpredictable but an unpredictable system need not be random. The alternative is that it could be described by chaos theory and although seem random, not actually be so. Chaos theory can describe the overall order of a non-linear system; it is not about the absence of order but the search for it. In this essay Ciarán Doran O’Fathaigh explains these ideas in greater detail and also looks at empirical work that has concluded in a rejection of the Random Walk Hypothesis. Introduction This essay intends to put forward an alternative to the Random Walk Hypothesis. This alternative will be that systems that appear to be random are in fact chaotic. Firstly, both the Random Walk Hypothesis and Chaos Theory will be outlined. Following that, some empirical cases will be examined where Chaos Theory is applicable, including real exchange rates with the US Dollar, a chaotic attractor for the S&P 500 and the returns on T-bills. The Random Walk Hypothesis The Random Walk Hypothesis states that stock market prices evolve according to a random walk and that endeavours to predict future movements will be fruitless. There is both a narrow version and a broad version of the Random Walk Hypothesis. Narrow Version: The narrow version of the Random Walk Hypothesis asserts that the movements of a stock or the market as a whole cannot be predicted from past behaviour (Wallich, 1968).
    [Show full text]
  • Nonlinear Dynamics and Entropy of Complex Systems with Hidden and Self-Excited Attractors
    entropy Editorial Nonlinear Dynamics and Entropy of Complex Systems with Hidden and Self-Excited Attractors Christos K. Volos 1,* , Sajad Jafari 2 , Jacques Kengne 3 , Jesus M. Munoz-Pacheco 4 and Karthikeyan Rajagopal 5 1 Laboratory of Nonlinear Systems, Circuits & Complexity (LaNSCom), Department of Physics, Aristotle University of Thessaloniki, Thessaloniki 54124, Greece 2 Nonlinear Systems and Applications, Faculty of Electrical and Electronics Engineering, Ton Duc Thang University, Ho Chi Minh City 700000, Vietnam; [email protected] 3 Department of Electrical Engineering, University of Dschang, P.O. Box 134 Dschang, Cameroon; [email protected] 4 Faculty of Electronics Sciences, Autonomous University of Puebla, Puebla 72000, Mexico; [email protected] 5 Center for Nonlinear Dynamics, Institute of Research and Development, Defence University, P.O. Box 1041 Bishoftu, Ethiopia; [email protected] * Correspondence: [email protected] Received: 1 April 2019; Accepted: 3 April 2019; Published: 5 April 2019 Keywords: hidden attractor; complex systems; fractional-order; entropy; chaotic maps; chaos In the last few years, entropy has been a fundamental and essential concept in information theory. It is also often used as a measure of the degree of chaos in systems; e.g., Lyapunov exponents, fractal dimension, and entropy are usually used to describe the complexity of chaotic systems. Thus, it will be important to study entropy in nonlinear systems. Additionally, there has been an increasing interest in a new classification of nonlinear dynamical systems including two kinds of attractors: self-excited attractors and hidden attractors. Self-excited attractors can be localized straightforwardly by applying a standard computational procedure.
    [Show full text]
  • Transformations)
    TRANSFORMACJE (TRANSFORMATIONS) Transformacje (Transformations) is an interdisciplinary refereed, reviewed journal, published since 1992. The journal is devoted to i.a.: civilizational and cultural transformations, information (knowledge) societies, global problematique, sustainable development, political philosophy and values, future studies. The journal's quasi-paradigm is TRANSFORMATION - as a present stage and form of development of technology, society, culture, civilization, values, mindsets etc. Impacts and potentialities of change and transition need new methodological tools, new visions and innovation for theoretical and practical capacity-building. The journal aims to promote inter-, multi- and transdisci- plinary approach, future orientation and strategic and global thinking. Transformacje (Transformations) are internationally available – since 2012 we have a licence agrement with the global database: EBSCO Publishing (Ipswich, MA, USA) We are listed by INDEX COPERNICUS since 2013 I TRANSFORMACJE(TRANSFORMATIONS) 3-4 (78-79) 2013 ISSN 1230-0292 Reviewed journal Published twice a year (double issues) in Polish and English (separate papers) Editorial Staff: Prof. Lech W. ZACHER, Center of Impact Assessment Studies and Forecasting, Kozminski University, Warsaw, Poland ([email protected]) – Editor-in-Chief Prof. Dora MARINOVA, Sustainability Policy Institute, Curtin University, Perth, Australia ([email protected]) – Deputy Editor-in-Chief Prof. Tadeusz MICZKA, Institute of Cultural and Interdisciplinary Studies, University of Silesia, Katowice, Poland ([email protected]) – Deputy Editor-in-Chief Dr Małgorzata SKÓRZEWSKA-AMBERG, School of Law, Kozminski University, Warsaw, Poland ([email protected]) – Coordinator Dr Alina BETLEJ, Institute of Sociology, John Paul II Catholic University of Lublin, Poland Dr Mirosław GEISE, Institute of Political Sciences, Kazimierz Wielki University, Bydgoszcz, Poland (also statistical editor) Prof.
    [Show full text]