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Catastrophe Theory
Catastrophe Theory Things that change suddenly, by fits and starts, have long resisted mathematical analysis. A method derived fi-om topology describes these phenomena as examples of seven "elementary catastrophes" by E. C. Zeeman cientists often describe events by con Etudes Scientifique at Bures-sur-Yvette in probable and the dog will most likely flee. structing a mathematical model. In France. He presented his ideas in a book Prediction is also straightforward if neither S deed, when such a model is particular published in 1972, Stabilite Structurelle et stimulus is present; then the dog is likely to ly successful, it is said not only to describe Morphogenese; an English translation by express some neutral kind of behavior unre the events but also to "explain" them; if the David H. Fowler of the University of War lated to either aggression or submission. model can be reduced to a simple equation, wick has recently been published. The the What if the dog is made to feel both rage it may even be called a law of nature. For ory is derived from topology, the branch of and fear simultaneously? The two control 300 years the preeminent method in build mathematics concerned with the properties ling factors are then in direct conflict. Sim ing such models has been the differential of surfaces in many dimensions. Topology ple models that cannot accommodate dis calculus invented by Newton and Leibniz. is involved because the underlying forces in continuity might predict that the two stim Newton himself expressed his laws of mo nature can be described by smooth surfaces uli would cancel each other, leading again tion and gravitation in terms of differential of equilibrium; it is when the equilibrium to neutral behavior. -
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. -
Role of Nonlinear Dynamics and Chaos in Applied Sciences
v.;.;.:.:.:.;.;.^ ROLE OF NONLINEAR DYNAMICS AND CHAOS IN APPLIED SCIENCES by Quissan V. Lawande and Nirupam Maiti Theoretical Physics Oivisipn 2000 Please be aware that all of the Missing Pages in this document were originally blank pages BARC/2OOO/E/OO3 GOVERNMENT OF INDIA ATOMIC ENERGY COMMISSION ROLE OF NONLINEAR DYNAMICS AND CHAOS IN APPLIED SCIENCES by Quissan V. Lawande and Nirupam Maiti Theoretical Physics Division BHABHA ATOMIC RESEARCH CENTRE MUMBAI, INDIA 2000 BARC/2000/E/003 BIBLIOGRAPHIC DESCRIPTION SHEET FOR TECHNICAL REPORT (as per IS : 9400 - 1980) 01 Security classification: Unclassified • 02 Distribution: External 03 Report status: New 04 Series: BARC External • 05 Report type: Technical Report 06 Report No. : BARC/2000/E/003 07 Part No. or Volume No. : 08 Contract No.: 10 Title and subtitle: Role of nonlinear dynamics and chaos in applied sciences 11 Collation: 111 p., figs., ills. 13 Project No. : 20 Personal authors): Quissan V. Lawande; Nirupam Maiti 21 Affiliation ofauthor(s): Theoretical Physics Division, Bhabha Atomic Research Centre, Mumbai 22 Corporate authoifs): Bhabha Atomic Research Centre, Mumbai - 400 085 23 Originating unit : Theoretical Physics Division, BARC, Mumbai 24 Sponsors) Name: Department of Atomic Energy Type: Government Contd...(ii) -l- 30 Date of submission: January 2000 31 Publication/Issue date: February 2000 40 Publisher/Distributor: Head, Library and Information Services Division, Bhabha Atomic Research Centre, Mumbai 42 Form of distribution: Hard copy 50 Language of text: English 51 Language of summary: English 52 No. of references: 40 refs. 53 Gives data on: Abstract: Nonlinear dynamics manifests itself in a number of phenomena in both laboratory and day to day dealings. -
Annotated List of References Tobias Keip, I7801986 Presentation Method: Poster
Personal Inquiry – Annotated list of references Tobias Keip, i7801986 Presentation Method: Poster Poster Section 1: What is Chaos? In this section I am introducing the topic. I am describing different types of chaos and how individual perception affects our sense for chaos or chaotic systems. I am also going to define the terminology. I support my ideas with a lot of examples, like chaos in our daily life, then I am going to do a transition to simple mathematical chaotic systems. Larry Bradley. (2010). Chaos and Fractals. Available: www.stsci.edu/~lbradley/seminar/. Last accessed 13 May 2010. This website delivered me with a very good introduction into the topic as there are a lot of books and interesting web-pages in the “References”-Sektion. Gleick, James. Chaos: Making a New Science. Penguin Books, 1987. The book gave me a very general introduction into the topic. Harald Lesch. (2003-2007). alpha-Centauri . Available: www.br-online.de/br- alpha/alpha-centauri/alpha-centauri-harald-lesch-videothek-ID1207836664586.xml. Last accessed 13. May 2010. A web-page with German video-documentations delivered a lot of vivid examples about chaos for my poster. Poster Section 2: Laplace's Demon and the Butterfly Effect In this part I describe the idea of the so called Laplace's Demon and the theory of cause-and-effect chains. I work with a lot of examples, especially the famous weather forecast example. Also too I introduce the mathematical concept of a dynamic system. Jeremy S. Heyl (August 11, 2008). The Double Pendulum Fractal. British Columbia, Canada. -
The Seduction of Curves by Allan Mcrobie
Journal of Humanistic Mathematics Volume 9 | Issue 2 July 2019 Book Review: The Seduction of Curves by Allan McRobie Hans J. Rindisbacher Pomona College Follow this and additional works at: https://scholarship.claremont.edu/jhm Part of the Arts and Humanities Commons, and the Mathematics Commons Recommended Citation Rindisbacher, H. J. "Book Review: The Seduction of Curves by Allan McRobie," Journal of Humanistic Mathematics, Volume 9 Issue 2 (July 2019), pages 323-328. DOI: 10.5642/jhummath.201902.22 . Available at: https://scholarship.claremont.edu/jhm/vol9/iss2/22 ©2019 by the authors. This work is licensed under a Creative Commons License. JHM is an open access bi-annual journal sponsored by the Claremont Center for the Mathematical Sciences and published by the Claremont Colleges Library | ISSN 2159-8118 | http://scholarship.claremont.edu/jhm/ The editorial staff of JHM works hard to make sure the scholarship disseminated in JHM is accurate and upholds professional ethical guidelines. However the views and opinions expressed in each published manuscript belong exclusively to the individual contributor(s). The publisher and the editors do not endorse or accept responsibility for them. See https://scholarship.claremont.edu/jhm/policies.html for more information. Book Review: The Seduction of Curves by Allan McRobie Hans J. Rindisbacher Department of German and Russian, Pomona College, California, USA [email protected] Synopsis This review emphasizes, as does the compelling and beautiful book, The Se- duction of Curves by Allan McRobie, the \lines of beauty" that link art and mathematics. McRobie and his collaborator on the indispensable visuals of the volume, Helena Weightman, succeed admirably in connecting theoretically and visually the mathematical field of singularity or catastrophe theory and its graph- ical representations on the one hand and the seemingly intersecting lines around the volumes of the human body in the artistic representation of the nude. -
Chaos Theory and Its Application to Education: Mehmet Akif Ersoy University Case*
Educational Sciences: Theory & Practice • 14(2) • 510-518 ©2014 Educational Consultancy and Research Center www.edam.com.tr/estp DOI: 10.12738/estp.2014.2.1928 Chaos Theory and its Application to Education: Mehmet Akif Ersoy University Case* Vesile AKMANSOYa Sadık KARTALb Burdur Provincial Education Department Mehmet Akif Ersoy University Abstract Discussions have arisen regarding the application of the new paradigms of chaos theory to social sciences as compared to physical sciences. This study examines what role chaos theory has within the education process and what effect it has by describing the views of university faculty regarding chaos and education. The partici- pants in this study consisted of 30 faculty members with teaching experience in the Faculty of Education, the Faculty of Science and Literature, and the School of Veterinary Sciences at Mehmet Akif Ersoy University in Burdur, Turkey. The sample for this study included voluntary participants. As part of the study, the acquired qualitative data has been tested using both the descriptive analysis method and content analysis. Themes have been organized under each discourse question after checking and defining the processes. To test the data, fre- quency and percentage, statistical techniques were used. The views of the attendees were stated verbatim in the Turkish version, then translated into English by the researchers. The findings of this study indicate the presence of a “butterfly effect” within educational organizations, whereby a small failure in the education process causes a bigger failure later on. Key Words Chaos, Chaos and Education, Chaos in Social Sciences, Chaos Theory. The term chaos continues to become more and that can be called “united science,” characterized by more prominent within the various fields of its interdisciplinary approach (Yeşilorman, 2006). -
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). -
Strategies and Rubrics for Teaching Chaos and Complex Systems Theories As Elaborating, Self-Organizing, and Fractionating Evolutionary Systems Lynn S
Strategies and Rubrics for Teaching Chaos and Complex Systems Theories as Elaborating, Self-Organizing, and Fractionating Evolutionary Systems Lynn S. Fichter1,2, E.J. Pyle1,3, S.J. Whitmeyer1,4 ABSTRACT To say Earth systems are complex, is not the same as saying they are a complex system. A complex system, in the technical sense, is a group of ―agents‖ (individual interacting units, like birds in a flock, sand grains in a ripple, or individual units of friction along a fault zone), existing far from equilibrium, interacting through positive and negative feedbacks, forming interdependent, dynamic, evolutionary networks, that possess universality properties common to all complex systems (bifurcations, sensitive dependence, fractal organization, and avalanche behaviour that follows power- law distributions.) Chaos/complex systems theory behaviors are explicit, with their own assumptions, approaches, cognitive tools, and models that must be taught as deliberately and systematically as the equilibrium principles normally taught to students. We present a learning progression of concept building from chaos theory, through a variety of complex systems, and ending with how such systems result in increases in complexity, diversity, order, and/or interconnectedness with time—that is, evolve. Quantitative and qualitative course-end assessment data indicate that students who have gone through the rubrics are receptive to the ideas, and willing to continue to learn about, apply, and be influenced by them. The reliability/validity is strongly supported by open, written student comments. INTRODUCTION divided into fractions through the addition of sufficient Two interrelated subjects are poised for rapid energy because of differences in the size, weight, valence, development in the Earth sciences. -
Bifurcations, Global Change, Tipping Points and All That
TiPES IHP, Paris, Kick-off 16 October 2019 Meeeting Bifurcations, Global Change, Tipping Points and All That Michael Ghil Ecole Normale Supérieure, Paris, and University of California, Los Angeles Please visit these sites for more info. on the talk http://www.atmos.ucla.edu/tcd/, http://www.environnement.ens.fr/, and https://www.researchgate.net/profile/Michael_Ghil SC1/NP1.5: Tipping Points in the Geosciences Michael Ghil, Peter Ditlevsen and Henk Dijkstra NP division, EGU-GA 2012 Wednesday, April 25, 2012 Outline • Intrinsic vs. forced variability – short-, intermediate, & long-term prediction – multiple scales of motion – IPCC & the uncertainties • Time-dependent forcing – pullback and random attractors (PBAs & RAs) – tipping points (TPs) • An illustrative example – the Lorenz convection model with stochastic forcing – LORA – its topological analysis (BraMaH) – “grand unification” = deterministic + stochastic • Conclusions and references – what do we & don’t we know? – selected bibliography Motivation • There’s a lot of talk about “tipping points.” • It sounds threatening, like falling off a cliff: that’s why we care! • But what are they, and what do we know about them? • Here’s a disambiguation page (cf. Wikipedia), first. • Sociology: “the moment of critical mass, the threshold, the boiling point” (Gladwell, 2000); a previously rare phenomenon becomes rapidly and dramatically more common. • Physics: the point at which a system changes from a stable equilibrium into a new, qualitatively dissimilar equilibrium (throwing a switch, tilting a plank, boiling water, etc.). • Climatology: “A climate tipping point is a somewhat ill-defined concept […]”— so we’ll try to actually define it better. • Catastrophe theory: branch of bifurcation theory in the study of dynamical systems; here, a tipping point is “a parameter value at which the set of equilibria abruptly change.” è Let’s see! M. -
Complex Adaptive Systems
Evidence scan: Complex adaptive systems August 2010 Identify Innovate Demonstrate Encourage Contents Key messages 3 1. Scope 4 2. Concepts 6 3. Sectors outside of healthcare 10 4. Healthcare 13 5. Practical examples 18 6. Usefulness and lessons learnt 24 References 28 Health Foundation evidence scans provide information to help those involved in improving the quality of healthcare understand what research is available on particular topics. Evidence scans provide a rapid collation of empirical research about a topic relevant to the Health Foundation's work. Although all of the evidence is sourced and compiled systematically, they are not systematic reviews. They do not seek to summarise theoretical literature or to explore in any depth the concepts covered by the scan or those arising from it. This evidence scan was prepared by The Evidence Centre on behalf of the Health Foundation. © 2010 The Health Foundation Previously published as Research scan: Complex adaptive systems Key messages Complex adaptive systems thinking is an approach that challenges simple cause and effect assumptions, and instead sees healthcare and other systems as a dynamic process. One where the interactions and relationships of different components simultaneously affect and are shaped by the system. This research scan collates more than 100 articles The scan suggests that a complex adaptive systems about complex adaptive systems thinking in approach has something to offer when thinking healthcare and other sectors. The purpose is to about leadership and organisational development provide a synopsis of evidence to help inform in healthcare, not least of which because it may discussions and to help identify if there is need for challenge taken for granted assumptions and further research or development in this area. -
Deterministic Chaos: Applications in Cardiac Electrophysiology Misha Klassen Western Washington University, [email protected]
Occam's Razor Volume 6 (2016) Article 7 2016 Deterministic Chaos: Applications in Cardiac Electrophysiology Misha Klassen Western Washington University, [email protected] Follow this and additional works at: https://cedar.wwu.edu/orwwu Part of the Medicine and Health Sciences Commons Recommended Citation Klassen, Misha (2016) "Deterministic Chaos: Applications in Cardiac Electrophysiology," Occam's Razor: Vol. 6 , Article 7. Available at: https://cedar.wwu.edu/orwwu/vol6/iss1/7 This Research Paper is brought to you for free and open access by the Western Student Publications at Western CEDAR. It has been accepted for inclusion in Occam's Razor by an authorized editor of Western CEDAR. For more information, please contact [email protected]. Klassen: Deterministic Chaos DETERMINISTIC CHAOS APPLICATIONS IN CARDIAC ELECTROPHYSIOLOGY BY MISHA KLASSEN I. INTRODUCTION Our universe is a complex system. It is made up A system must have at least three dimensions, of many moving parts as a dynamic, multifaceted and nonlinear characteristics, in order to generate machine that works in perfect harmony to create deterministic chaos. When nonlinearity is introduced the natural world that allows us life. e modeling as a term in a deterministic model, chaos becomes of dynamical systems is the key to understanding possible. ese nonlinear dynamical systems are seen the complex workings of our universe. One such in many aspects of nature and human physiology. complexity is chaos: a condition exhibited by an is paper will discuss how the distribution of irregular or aperiodic nonlinear deterministic system. blood throughout the human body, including factors Data that is generated by a chaotic mechanism will a ecting the heart and blood vessels, demonstrate appear scattered and random, yet can be dened by chaotic behavior. -
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.