DOCSLIB.ORG

PY2T20: CHAOS and COMPLEXITY (12 Lectures) Stefan Hutzler

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
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) . 35 10.2 Quantum chaos (motivation) . 36 11 Some examples of complex behavior 38 12 Cellular Automata 39 12.1 Concept . 39 12.2 Example in one dimension . 39 12.3 A two-dimensional CA: The game of life . 40 13 Power-laws and self-organised criticality 41 13.1 Power-laws . 41 13.2 A model for Sandpile dynamics . 42 13.3 Earthquake model . 42 14 Pedestrian Dynamics 44 14.1 Observations . 44 14.2 Model of pedestrian motion . 45 14.3 Results of computer simulations . 46 ii 14.4 Optimisation of pedestrian facilities . 47 14.5 Design of new pedestrian facilities . 47 A References 48 iii Chapter 1 A glossary φιλos [philos]: friendly, loving; σoφια [sophia] knowledge ) philosophy: love of wisdom, later: study of “reality” and human nature Aristoteles (384-322 BC) treatise on movement of natural/material bodies: φνσικα [physica] derived from adjective for “natural”, things early names for physics: “natural science”, “natural philosophy”; nature : natural motion TCD School of Physics: Erasmus Smith’s Chair of Natural and Experimental Phi- losophy (1724), currently vacant χαos [chaos]: • disordered formless matter supposed to have existed before the ordered universe (as in Greek and Babylonian mythology, Old Testament “without form, void”) • complete disorder, utter confusion • (Math.) Stochastic behaviour occurring in a deterministic system (Royal Society 1986) stochastic (stochastikos): “skillful in aiming” at a target ’stokhos’, using laws of chance for personal benefit determinism: every event is result of antecedent causes () predictability) 1 in the context of this course: chaos ≡ deterministic chaos: “lawless behaviour governed entirely by law” (Ian Stewart) complexity: Greek [plexus]: braided, [com-]: together ) “braided together” simple: once folded complex system: organised system, at “edge of chaos”, this may lead to pattern formation some comments on (non-)linearity • many equations in physics are linear ) the sum of two solutions of an equation is again a solution (superposition) • this simplifies the mathematics, but might not necessarily describe the actual physics (see treatment of pendulum in chapter 5: sin x ' x gives qualitatively wrong results for large amplitudes) • now: very often non-linearity is at the centre of scientific problems; one gener- ally needs computers for solutions • for non-linear systems small changes in a parameter may lead to sudden dramatic changes in both qualitative and quantitative behaviour of system, ie. change from periodicity to aperiodicity Stanislaw Marcin Ulam (Manhattan project): “Calling the subject non-linear dy- namics is like calling zoology ‘non-elephant studies’ ” 2 Chapter 2 Examples of non-linear and chaotic behaviour 2.1 Population dynamics The model assumptions: population of species (green-flies) whose individuals are born and die in the same year (“breed in summer and leave eggs that hatch in spring”) discrete times ) difference equations simplest model (geometric growth): Pn+1 = aPn (2.1) Pn: population in generation n, a: growth constant, if a > 1: population explosion, a < 1: extinction more realistic growth model: Pn+1 = Pn(a − bPn) (2.2) where b is a constant. This models the fact that population is bounded by a finite carrying capacity of its environment, i.e. overcrowding, diseases, lack of food etc. let’s introduce the rescaled variable xn: a P = x (2.3) n b n 3 thus xn+1 = axn(1 − xn) ≡ fa(x) (2.4) This is the logistic map. (P.F. Verhulst, 1845; french “logis” means “house, ac- commodation”) Note that it is a fully deterministic equation. comments: • xn > 1 ) xn+1 < 0 (unphysical) ) restrict xn[0; 1] and choose 0 ≤ a ≤ 4 • it is a map: range [0,1] is mapped into itself • it is an iterative function: population in N years is found by iterating/repeating the calculation of eqn. 2.4. • the sequence of x values is called the trajectory or orbit of the map ∗ ∗ • fixed points are defined by xa = fa(xa) ∗ ∗ here: there are two fixed points, xa = 0 (trivial) and xa = 1 − 1=a, the latter fixed point is stable only for 1 < a < 3. Discussion of the logistic map study map either by direct computation/iteration or graphically [figure: Hilborn 1.12 + applet http://brain.cc.kogakuin.ac.jp/ kanamaru/Chaos/e/Logits/]; fixed points are the intersections of the parabola fa(x) with y = x example: a=0.6 figure: evolution of a population here: 0 is the attractor, interval [0:1] is the basin of attraction; effect: population dies out for any size of initial population attractor: set of points to which trajectories tend as number of iterations goes to infinity example a=2: ∗ ∗ ∗ 2 fixed points, x1 = 0; x2 = 1 − 1=a = 0:5, here x2 = 0:5 is an attractor 4 example a = 3.1 [Hilborn, table 1.1] x0 = 0:250 : observation of period doubling (bifurcation = splitting into two parts): population is high in one year and low in the following year two-point attractor, basis of attraction ]0,1[, except x∗ = 1=a = 1:=3:1, which is a fixed point four-point attractor at a = 3.44948 8-point attractor at ... at a ≥ 3:5699: trajectory values do no longer repeat: chaotic behaviour [Table 1.2 (Hilborn)] shows the evolution of similar starting values x0 in the chaotic regime. What’s going on? The above observations may be summarised in a bifurcation diagram where for a given value of the parameter a one plots the attracting points of the map. [figure: Hilborn fig. 1.14, Gould-Tobochnik fig 6.2] (pitch-fork bifurcation applet at http://brain.cc.kogakuin.ac.jp/ kanamaru/Chaos/e/BifArea/) questions: relevance to actual biological system? complicated assessment; see figure of evolution of blowflies [Kendall 2002] relevance to physical systems? 5 2.2 Non-linear electrical circuit [Hilborn fig. 1.1] components: • ac signal generator (voltage V0) • diode: valve • inductor (coil): produces electrical potential difference proportional to the rate of change of current • bias dc generator (voltage Vdc) measure diode voltage as function of time; control parameter: driving voltage V0 of the periodic signal initially diode voltage has same frequency as signal generator; increase V0: obser- vation of period doubling etc. [Hilborn figs 1.2-1.8] important features: • diode: reverse recovery time (micro-seconds); use appropriate frequency of sig- nal generator, i.e. kHz • closing time depends on strength of current • energy stored in capacitance of diode (100pF) • inductor: extra degree of freedom; picks out special frequencies for oscillations in circuit (resonance) 6 2.3 Lorenz model of atmospheric convection Edward Lorenz, meteorologist (model published in 1963) historical importance: this highlights the importance of initial conditions, “but- terfly effect”: a butterfly flapping its wings could change the course of the weather... model atmosphere treated as fluid layer, heated at bottom, cooled at top. Navier-Stokes equa- tions for motion of the fluid + thermal diffusion ... =) 3 dimensionless variables X,Y,Z Lorenz equations, final form: X_ = p(Y − X) Y_ = −XZ + rX − Y Z_ = XY − bZ (2.5) [figures: Hilborn 1.19, 1.22] 2.4 Summary of observations • sudden changes in qualitative behaviour as control parameters are slowly varied • chaos is not random noise: divergence of nearby trajectories • well defined changes from regular motion via period doubling to chaos 7 Chapter 3 Universal properties and self-similarity 3.1 Feigenbaum constants idea of Mitchell Feigenbaum 1978: let’s look again at the period doubling scenario from chapter 2.1: define a1: value of control parameter at change from period 1 to period 2 behaviour a2: value of parameter at change from period 2 to period 4 a3 etc. an−an−1 compute δn = an+1−an Feigenbaum found that δn is approximately the same for all n; it can be shown that this holds for all maps which are parabolic near the maximum, as for example the map xn+1 = bsin(πx)(b < 1) Feigenbaum δ = limn!1 δn = 4:6692016091::: δ belongs to a group of fundamental numbers such as π ' 3:1415927; e ' 2:7182818, Pn 1 Euler-Mascheroni constant γ = limn!1 k=1 − ln(n) ' 0:57721, golden mean p k φ = 1+ 5 ' 1:618034, fine structure constant e2 ' 1 2 4π¯hc0 137:03599976 8 experimental data? comparison is limited due to small n < 7, nevertheless there is reasonable agreement (error < 20%) [Hilborn table 2.3, p.52] comment: use δn to predict period doubling note: other Feigenbaum constants are easier to compute: [figure: Gould/Tobochnic, Fig.
Load more

Recommended publications
  • Secretaria De Estado Da Educação Do Paraná Programa De Desenvolvimento Educacional - Pde
    SECRETARIA DE ESTADO DA EDUCAÇÃO DO PARANÁ PROGRAMA DE DESENVOLVIMENTO EDUCACIONAL - PDE JOÃO VIEIRA BERTI A GEOMETRIA DOS FRACTAIS PARA O ENSINO FUNDAMENTAL CASCAVEL – PR 2008 JOÃO VIEIRA BERTI A GEOMETRIA DOS FRACTAIS PARA O ENSINO FUNDAMENTAL Artigo apresentado ao Programa de Desenvolvimento Educacional do Paraná – PDE, como requisito para conclusão do programa. Orientadora: Dra. Patrícia Sândalo Pereira CASCAVEL – PR 2008 A GEOMETRIA DOS FRACTAIS PARA O ENSINO FUNDAMENTAL João Vieira Berti1 Patrícia Sândalo Pereira2 Resumo O seguinte trabalho tem a finalidade de apresentar a Geometria Fractal segundo a visão de Benoit Mandelbrot, considerado o pai da Geometria Fractal, bem como a sua relação como a Teoria do Caos. Serão também apresentadas algumas das mais notáveis figuras fractais, tais como: Conjunto ou Poeira de Cantor, Curva e Floco de Neve de Koch, Triângulo de Sierpinski, Conjunto de Mandelbrot e Julia, entre outros, bem como suas propriedades e possíveis aplicações em sala de aula. Este trabalho de pesquisa foi desenvolvido com professores de matemática da rede estadual de Foz do Iguaçu e Região e também com professores de matemática participantes do Programa de Desenvolvimento Educacional do Paraná – PDE da Região Oeste e Sudoeste do Paraná a fim de lhes apresentar uma nova forma de trabalhar a geometria fractal com a utilização de softwares educacionais dinâmicos. Palavras-chave: Geometria, Fractais, Softwares Educacionais. Abstract The pourpose of this paper is to present Fractal Geometry according the vision of Benoit Mandelbrot´s, the father of Fractal Geometry, and it´s relationship with the Theory of Chaos as well. Also some of the most notable fractals figures, such as: Cantor Dust, Koch´s snowflake, the Sierpinski Triangle, Mandelbrot Set and Julia, among others, are going to be will be presented as well as their properties and potential classroom applications.
    [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]
  • 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.
    [Show full text]
  • Alwyn C. Scott
    the frontiers collection the frontiers collection Series Editors: A.C. Elitzur M.P. Silverman J. Tuszynski R. Vaas H.D. Zeh The books in this collection are devoted to challenging and open problems at the forefront of modern science, including related philosophical debates. In contrast to typical research monographs, however, they strive to present their topics in a manner accessible also to scientifically literate non-specialists wishing to gain insight into the deeper implications and fascinating questions involved. Taken as a whole, the series reflects the need for a fundamental and interdisciplinary approach to modern science. Furthermore, it is intended to encourage active scientists in all areas to ponder over important and perhaps controversial issues beyond their own speciality. Extending from quantum physics and relativity to entropy, consciousness and complex systems – the Frontiers Collection will inspire readers to push back the frontiers of their own knowledge. Other Recent Titles The Thermodynamic Machinery of Life By M. Kurzynski The Emerging Physics of Consciousness Edited by J. A. Tuszynski Weak Links Stabilizers of Complex Systems from Proteins to Social Networks By P. Csermely Quantum Mechanics at the Crossroads New Perspectives from History, Philosophy and Physics Edited by J. Evans, A.S. Thorndike Particle Metaphysics A Critical Account of Subatomic Reality By B. Falkenburg The Physical Basis of the Direction of Time By H.D. Zeh Asymmetry: The Foundation of Information By S.J. Muller Mindful Universe Quantum Mechanics and the Participating Observer By H. Stapp Decoherence and the Quantum-to-Classical Transition By M. Schlosshauer For a complete list of titles in The Frontiers Collection, see back of book Alwyn C.
    [Show full text]
  • Montana Throne Molly Gupta Laura Brannan Fractals: a Visual Display of Mathematics Linear Algebra
    Montana Throne Molly Gupta Laura Brannan Fractals: A Visual Display of Mathematics Linear Algebra - Math 2270 Introduction: Fractals are infinite patterns that look similar at all levels of magnification and exist between the normal dimensions. With the advent of the computer, we can generate these complex structures to model natural structures around us such as blood vessels, heartbeat rhythms, trees, forests, and mountains, to name a few. I will begin by explaining how different linear transformations have been used to create fractals. Then I will explain how I have created fractals using linear transformations and include the computer-generated results. A Brief History: Fractals seem to be a relatively new concept in mathematics, but that may be because the term was coined only 43 years ago. It is in the century before Benoit Mandelbrot coined the term that the study of concepts now considered fractals really started to gain traction. The invention of the computer provided the computing power needed to generate fractals visually and further their study and interest. Expand on the ideas by century: 17th century ideas • Leibniz 19th century ideas • Karl Weierstrass • George Cantor • Felix Klein • Henri Poincare 20th century ideas • Helge von Koch • Waclaw Sierpinski • Gaston Julia • Pierre Fatou • Felix Hausdorff • Paul Levy • Benoit Mandelbrot • Lewis Fry Richardson • Loren Carpenter How They Work: Infinitely complex objects, revealed upon enlarging. Basics: translations, uniform scaling and non-uniform scaling then translations. Utilize translation vectors. Concepts used in fractals-- Affine Transformation (operate on individual points in the set), Rotation Matrix, Similitude Transformation Affine-- translations, scalings, reflections, rotations Insert Equations here.
    [Show full text]
  • Bachelorarbeit Im Studiengang Audiovisuelle Medien Die
    Bachelorarbeit im Studiengang Audiovisuelle Medien Die Nutzbarkeit von Fraktalen in VFX Produktionen vorgelegt von Denise Hauck an der Hochschule der Medien Stuttgart am 29.03.2019 zur Erlangung des akademischen Grades eines Bachelor of Engineering Erst-Prüferin: Prof. Katja Schmid Zweit-Prüfer: Prof. Jan Adamczyk Eidesstattliche Erklärung Name: Vorname: Hauck Denise Matrikel-Nr.: 30394 Studiengang: Audiovisuelle Medien Hiermit versichere ich, Denise Hauck, ehrenwörtlich, dass ich die vorliegende Bachelorarbeit mit dem Titel: „Die Nutzbarkeit von Fraktalen in VFX Produktionen“ selbstständig und ohne fremde Hilfe verfasst und keine anderen als die angegebenen Hilfsmittel benutzt habe. Die Stellen der Arbeit, die dem Wortlaut oder dem Sinn nach anderen Werken entnommen wurden, sind in jedem Fall unter Angabe der Quelle kenntlich gemacht. Die Arbeit ist noch nicht veröffentlicht oder in anderer Form als Prüfungsleistung vorgelegt worden. Ich habe die Bedeutung der ehrenwörtlichen Versicherung und die prüfungsrechtlichen Folgen (§26 Abs. 2 Bachelor-SPO (6 Semester), § 24 Abs. 2 Bachelor-SPO (7 Semester), § 23 Abs. 2 Master-SPO (3 Semester) bzw. § 19 Abs. 2 Master-SPO (4 Semester und berufsbegleitend) der HdM) einer unrichtigen oder unvollständigen ehrenwörtlichen Versicherung zur Kenntnis genommen. Stuttgart, den 29.03.2019 2 Kurzfassung Das Ziel dieser Bachelorarbeit ist es, ein Verständnis für die Generierung und Verwendung von Fraktalen in VFX Produktionen, zu vermitteln. Dabei bildet der Einblick in die Arten und Entstehung der Fraktale
    [Show full text]
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • Chaos Theory and Its Application in the Atmosphere
    Chaos Theory and its Application in the Atmosphere by XubinZeng Department of Atmospheric Science Colorado State University Fort Collins, Colorado Roger A. Pielke, P.I. NSF Grant #ATM-8915265 CHAOS THEORY AND ITS APPLICATION IN THE ATMOSPHERE Xubin Zeng Department of Atmospheric Science CoJorado State University Fort Collins, Colorado Summer, 1992 Atmospheric Science Paper No. 504 \llIlll~lIl1ll~I""I1~II~'I\1 U16400 7029194 ABSTRACT CHAOS THEORY AND ITS APPLICATION IN THE ATMOSPHERE Chaos theory is thoroughly reviewed, which includes the bifurcation and routes to tur­ bulence, and the characterization of chaos such as dimension, Lyapunov exponent, and Kolmogorov-Sinai entropy. A new method is developed to compute Lyapunov exponents from limited experimental data. Our method is tested on a variety of known model systems, and it is found that our algorithm can be used to obtain a reasonable Lyapunov­ exponent spectrum from only 5000 data points with a precision of 10-1 or 10-2 in 3- or 4-dimensional phase space, or 10,000 data points in 5-dimensional phase space. On 1:he basis of both the objective analyses of different methods for computing the Lyapunov exponents and our own experience, which is subjective, this is recommended as a good practical method for estiIpating the Lyapunov-exponent spectrum from short time series of low precision. The application of chaos is divided into three categories: observational data analysis, llew ideas or physical insights inspired by chaos, and numerical model output analysis. Corresponding with these categories, three subjects are studied. First, the fractal dimen­ sion, Lyapunov-exponent spectrum, Kolmogorov entropy, and predictability are evaluated from the observed time series of daily surface temperature and pressure over several regions of the United States and the North Atlantic Ocean with different climatic signal-to-noise ratios.
    [Show full text]
  • Toward a Theory of Chaos
    December 3, 2003 12:13 00851 Tutorials and Reviews International Journal of Bifurcation and Chaos, Vol. 13, No. 11 (2003) 3147{3233 c World Scientific Publishing Company TOWARD A THEORY OF CHAOS A. SENGUPTA Department of Mechanical Engineering, Indian Institute of Technology Kanpur, Kanpur 208016, India [email protected] Received February 23, 2001; Revised September 19, 2002 This paper formulates a new approach to the study of chaos in discrete dynamical systems based on the notions of inverse ill-posed problems, set-valued mappings, generalized and multi- valued inverses, graphical convergence of a net of functions in an extended multifunction space [Sengupta & Ray, 2000], and the topological theory of convergence. Order, chaos and complexity are described as distinct components of this unified mathematical structure that can be viewed as an application of the theory of convergence in topological spaces to increasingly nonlinear mappings, with the boundary between order and complexity in the topology of graphical con- vergence being the region in (Multi(X)) that is susceptible to chaos. The paper uses results from the discretized spectral approximation in neutron transport theory [Sengupta, 1988, 1995] and concludes that the numerically exact results obtained by this approximation of the Case singular eigenfunction solution is due to the graphical convergence of the Poisson and conjugate Poisson kernels to the Dirac delta and the principal value multifunctions respectively. In (Multi(X)), the continuous spectrum is shown to reduce to a point spectrum, and we introduce a notion of latent chaotic states to interpret superposition over generalized eigenfunctions. Along with these latent states, spectral theory of nonlinear operators is used to conclude that nature supports complexity to attain efficiently a multiplicity of states that otherwise would remain unavailable to it.
    [Show full text]
  • Study of Behavior of Human Capital from a Fractal Perspective
    http://ijba.sciedupress.com International Journal of Business Administration Vol. 8, No. 1; 2017 Study of Behavior of Human Capital from a Fractal Perspective Leydi Z. Guzmán-Aguilar1, Ángel Machorro-Rodríguez2, Tomás Morales-Acoltzi3, Miguel Montaño-Alvarez1, Marcos Salazar-Medina2 & Edna A. Romero-Flores2 1 Maestría en Ingeniería Administrativa, Tecnológico Nacional de México, Instituto Tecnológico de Orizaba, Colonia Emiliano Zapata, México 2 División de Estudios de Posgrado e Investigación, Tecnológico Nacional de México, Instituto Tecnológico de Orizaba, Colonia Emiliano Zapata, México 3 Centro de Ciencias de la Atmósfera, UNAM, Circuito exterior, CU, CDMX, México Correspondence: Leydi Z. Guzmán-Aguilar, Maestría en Ingeniería Administrativa, Tecnológico Nacional de México, Instituto Tecnológico de Orizaba, Ver. Oriente 9 No. 852 Colonia Emiliano Zapata, C.P. 94320, México. Tel: 1-272-725-7518. Received: November 11, 2016 Accepted: November 27, 2016 Online Published: January 4, 2017 doi:10.5430/ijba.v8n1p50 URL: http://dx.doi.org/10.5430/ijba.v8n1p50 Abstract In the last decade, the application of fractal geometry has surged in all disciplines. In the area of human capital management, obtaining a relatively long time series (TS) is difficult, and likewise nonlinear methods require at least 512 observations. In this research, we therefore suggest the application of a fractal interpolation scheme to generate ad hoc TS. When we inhibit the vertical scale factor, the proposed interpolation scheme has the effect of simulating the original TS. Keywords: human capital, wage, fractal interpolation, vertical scale factor 1. Introduction In nature, irregular processes exist that by using Euclidean models as their basis for analysis, do not capture the variety and complexity of the dynamics of their environment.
    [Show full text]