DOCSLIB.ORG

Chaos Theory !

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Chaos Theory ! CHAOSCHAOS THEORYTHEORY !! What is The Butterfly Effect? In 1800 the philosopher Johann Gottlieb Fichte wrote "you could not remove a single grain of sand from its place without thereby... changing something throughout all parts of the immeasurable whole". In 1890 the mathematician Henri Poincaré noticed that very small changes in initial conditions can cause major changes in the output of the model for The Three-Body Problem. In 1952 Ray Bradbury wrote a science fiction story in which the death of one butterfly had a far-reaching ripple effect on later historical events. It was, however, the meteorologist Edward Lorenz who first made popular the term Butterfly Effect, after the publication of his paper in 1972, with the eye-catching title: Predictability: Does the Flap of a Butterfly's Wings in Brazil set off a Tornado in Texas? The flapping wings represent very small changes in the initial conditions of a model, which cause large changes in the outcomes of the model. How did Lorenz discover this Butterfly Effect? In 1961 he was re-running a model to predict the weather for the next two months, the imput data was on punched tape fed into an early valve-powered computer, with no monitor of course, only a printer, those were the hard days (!). For the 2nd run he entered an approximate value of 0.506 from the first run, in- stead of the full precise value 0.506127.. He was astonished to see the new output was now a completely different weather scenario! At first the new values repeated the old ones, but soon the differences steadily doubled every four days, until the original predictions disappeared somewhere in the 2nd month! The initial round-off errors steadily led to very different weather predictions! Here is an actual plot used by Lorenz with predicted values of the weather along the y-axis and time along the x-axis. The red line represents old predictions and the blue line new predictions with new initial conditions which were truncated (or cut off) versions of the old ones. Use estimated temperatures for the days 1-17 to answer these questions: (1) On what day did the lines first start to separate? (2) On what day did they first re-join? (3) What day did they next cross over? (4) Which line had the lower temperature on Day 8? (5) On what day was the temperature difference a maximum? (6) The two lines were quite close until which Day? What was Lorenz's Model? Lorenz used three differential equations to model or simulate the weather conditions in the atmosphere, which was all the time being cooled from above and heated from below. It became clear to him in 1969 that no matter how accurate the measurements were, the weather could NOT be predicted with certainty because of the Butterfly Effect. Weather patterns such as lightning and tornados are chaotic in nature and some are also fractal in appearance! When Lorenz used slightly different initial conditions the values they produced behaved chaotically. Each run of the model never met or crossed another run, and together they formed a beautiful fractal which Lorenz called a strange attractor. Notice also that this image shows self-similarity. Biological Population Dynamics of the micro-organism Yeast The graph opposite shows how the population density (on the y axis) of a micro -organism changes in time (on the x axis). Iteration models can analyse what happens here, and in other animal and bacterial populations, so that we can help populations to survive and not become extinct. First write answers to these questions: (7) Write out in full decimal notation 106 and 104 Can you see why we prefer the shorter power notation? (8) Explain what bistable means. Explain why we might say from the graph that the population is actually tristable. (9) Explain what has happened to the population for the red lines at the bottom of the graph. (10) What do you think will happen after 8, 10, 100 days according to this graph? Can we really predict so far? (11) Explain the difference between population size and population density . In the last Fractal & Chaos Theme using our iterative model you discovered for 0 < b < 2.9 the population of kangaroos on Kangaroo Island converged to a STABLE state of one attractor, either zero or a positive number (like the Yeast in the above graph). So now we need to use your same model to find out what happens for b > 2.9, Do we find stable or unstable states (or both) for the above Yeast population? Back to Kangaroo Island... and the Logistic Equation You may not have realised after all your work on Kangaroo Island that you have actually met the Butterfly Effect already! You investigated how small changes in b in the formula below could lead to period doubling, then four attractors, then 8,16,...128.... and finally to chaos itself! Just like Lorenz, you found that our model of population growth could become unstable and chaotic when you made small changes in the initial conditions (the value of b). This is the Butterfly Effect! The iterative model we used before was x b*x*(1000 - x)/1000 Check the right hand side is the same as b*x* ( 1 - x ) 1000 Notice that x is the fraction or percentage of kangaroos on the island. 1000 If we now use x NOT as the number of kangaroos but the decimal fraction of the kangaroo population (i.e. the population density) so that 0< x <1 then the model we used simplifies to: x bx(1-x) For example x=0.6 means 60% of the population and so on. We do not NEED to know the actual population when we know the population density, notice this appeared above in the graph about the micro-organism Yeast. This simpler form of the iteration is called the Logistic Equation which is a model widely used to study the growth of animal and bacterial populations. You will now create spreadsheets and graphs from the Logistic Equation to investigate what happens to the Yeast population: to survive or not to survive? That is the question! Below is some practice just to get used to the NEW values for x in the Logistic Equation formula x bx(1-x) (12) Write the following percentages as decimal fractions, e.g. 67% = 0.67. 32%, 56.1% (be careful here!), 296%, 0.0003%. (13) Write the following decimal fractions as percentages, e.g. 0.04 = 4% 0.99, 0.78, 0.7789, 0.0005, 0.834 (14) Use a calculator to to find the reaults when you substitute x and b in the logistic equation bx(1-x): b = 3,1, x = 0.657, b = 3.111, x = 0.88, b = 0.767, x = 0.967 (15) Explain precisely what is wrong with each of these values of b and x: b = 6.1, b = -4, x = 23.562, x = -0.786 We will now try values of b from 2-4 to see what happens to the Yeast population x after many iterations. If you have problems with the spreadsheet below study the Introduction to Excel help sheets at the end of this theme. Create a new Spreadsheet 7 exactly like the one below. Put the starting population x = 0.6 in cell I1 and copy it into all the cells A5 to L5. Copy the values for b (from 2 - 4) you see entered in cells B3-L3 In cell B6 insert the formula = B$3*B5(1-B5) The $ sign forces the 3 to stay rather than changing as the formula moves down the column. So in this column the b value of 2 remains the same throughout column B. Now fill across from B6 to L6. Then fill down columns B6-L6. You will need to go down a long way to see the full behaviour of ssome of the populations! Save and Store this as Spreadsheet 7 Here is the spreadsheet with the first few rows filled in: What did you discover happens to the Yeast population for various values of b? Here is what should have happened in some cases: Copy and fill in the table below to include all your results from your Spreadsheet 7.: Value of b 2.0 2.2 2.4 2.5 2.6 2,8 3.0 3.2 3.4 3.5 3.6 3.8 4.0 Attractors 1 1 2 4 0 Did you find 8 or more attractors? For some values of b you must go down 1000s of rows to see the pattern! (16) List values of b that produce (i) stable and (ii) unstable Yeast populations. Graphing your results Patterns can often be seen more clearly on a graph rather than looking at reams of numbers. If you can, create your Spreadsheet 8 by copying the spreadsheet below which includes a graph of the results for b = 2.8. More graphs of results showing what is happening to the Yeast population. Now we see that this is a slow moving graph which will eventually end in two attractors. This settles down more quickly to give four attractors. Finally we see eight attractors! Below is one of the most important graphs to help us understand the problem of stable or unstable states for the Yeast population data. We plot b values along the x-axis from 2 to 4, and along the y-axis the number of attractor(s) in the population for that value of b.
Load more

Recommended publications
  • Chaos Theory
    By Nadha CHAOS THEORY What is Chaos Theory? . It is a field of study within applied mathematics . It studies the behavior of dynamical systems that are highly sensitive to initial conditions . It deals with nonlinear systems . It is commonly referred to as the Butterfly Effect What is a nonlinear system? . In mathematics, a nonlinear system is a system which is not linear . It is a system which does not satisfy the superposition principle, or whose output is not directly proportional to its input. Less technically, a nonlinear system is any problem where the variable(s) to be solved for cannot be written as a linear combination of independent components. the SUPERPOSITION PRINCIPLE states that, for all linear systems, the net response at a given place and time caused by two or more stimuli is the sum of the responses which would have been caused by each stimulus individually. So that if input A produces response X and input B produces response Y then input (A + B) produces response (X + Y). So a nonlinear system does not satisfy this principal. There are 3 criteria that a chaotic system satisfies 1) sensitive dependence on initial conditions 2) topological mixing 3) periodic orbits are dense 1) Sensitive dependence on initial conditions . This is popularly known as the "butterfly effect” . It got its name because of the title of a paper given by Edward Lorenz titled Predictability: Does the Flap of a Butterfly’s Wings in Brazil set off a Tornado in Texas? . The flapping wing represents a small change in the initial condition of the system, which causes a chain of events leading to large-scale phenomena.
    [Show full text]
  • Synchronization in Nonlinear Systems and Networks
    Synchronization in Nonlinear Systems and Networks Yuri Maistrenko E-mail: [email protected] you can find me in the room EW 632, Wednesday 13:00-14:00 Lecture 4 - 23.11.2011 1 Chaos actually … is everywhere Web Book CHAOS = BUTTERFLY EFFECT Henri Poincaré (1880) “ It so happens that small differences in the initial state of the system can lead to very large differences in its final state. A small error in the former could then produce an enormous one in the latter. Prediction becomes impossible, and the system appears to behave randomly.” Ray Bradbury “A Sound of Thunder “ (1952) THE ESSENCE OF CHAOS • processes deterministic fully determined by initial state • long-term behavior unpredictable butterfly effect PHYSICAL “DEFINITION “ OF CHAOS “To say that a certain system exhibits chaos means that the system obeys deterministic law of evolution but that the outcome is highly sensitive to small uncertainties in the specification of the initial state. In chaotic system any open ball of initial conditions, no matter how small, will in finite time spread over the extent of the entire asymptotically admissible phase space” Predrag Cvitanovich . Appl.Chaos 1992 EXAMPLES OF CHAOTIC SYSTEMS • the solar system (Poincare) • the weather (Lorenz) • turbulence in fluids • population growth • lots and lots of other systems… “HOT” APPLICATIONS • neuronal networks of the brain • genetic networks UNPREDICTIBILITY OF THE WEATHER Edward Lorenz (1963) Difficulties in predicting the weather are not related to the complexity of the Earths’ climate but to CHAOS in the climate equations! Dynamical systems Dynamical system: a system of one or more variables which evolve in time according to a given rule Two types of dynamical systems: • Differential equations: time is continuous (called flow) dx N f (x), t R dt • Difference equations (iterated maps): time is discrete (called cascade) xn1 f (xn ), n 0, 1, 2,..
    [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]
  • Tom W B Kibble Frank H Ber
    Classical Mechanics 5th Edition Classical Mechanics 5th Edition Tom W.B. Kibble Frank H. Berkshire Imperial College London Imperial College Press ICP Published by Imperial College Press 57 Shelton Street Covent Garden London WC2H 9HE Distributed by World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore 596224 USA office: Suite 202, 1060 Main Street, River Edge, NJ 07661 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE Library of Congress Cataloging-in-Publication Data Kibble, T. W. B. Classical mechanics / Tom W. B. Kibble, Frank H. Berkshire, -- 5th ed. p. cm. Includes bibliographical references and index. ISBN 1860944248 -- ISBN 1860944353 (pbk). 1. Mechanics, Analytic. I. Berkshire, F. H. (Frank H.). II. Title QA805 .K5 2004 531'.01'515--dc 22 2004044010 British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library. Copyright © 2004 by Imperial College Press All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher. For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher. Printed in Singapore. To Anne and Rosie vi Preface This book, based on courses given to physics and applied mathematics stu- dents at Imperial College, deals with the mechanics of particles and rigid bodies.
    [Show full text]
  • 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.
    [Show full text]
  • Instructional Experiments on Nonlinear Dynamics & Chaos (And
    Bibliography of instructional experiments on nonlinear dynamics and chaos Page 1 of 20 Colorado Virtual Campus of Physics Mechanics & Nonlinear Dynamics Cluster Nonlinear Dynamics & Chaos Lab Instructional Experiments on Nonlinear Dynamics & Chaos (and some related theory papers) overviews of nonlinear & chaotic dynamics prototypical nonlinear equations and their simulation analysis of data from chaotic systems control of chaos fractals solitons chaos in Hamiltonian/nondissipative systems & Lagrangian chaos in fluid flow quantum chaos nonlinear oscillators, vibrations & strings chaotic electronic circuits coupled systems, mode interaction & synchronization bouncing ball, dripping faucet, kicked rotor & other discrete interval dynamics nonlinear dynamics of the pendulum inverted pendulum swinging Atwood's machine pumping a swing parametric instability instabilities, bifurcations & catastrophes chemical and biological oscillators & reaction/diffusions systems other pattern forming systems & self-organized criticality miscellaneous nonlinear & chaotic systems -overviews of nonlinear & chaotic dynamics To top? Briggs, K. (1987), "Simple experiments in chaotic dynamics," Am. J. Phys. 55 (12), 1083-9. Hilborn, R. C. (2004), "Sea gulls, butterflies, and grasshoppers: a brief history of the butterfly effect in nonlinear dynamics," Am. J. Phys. 72 (4), 425-7. Hilborn, R. C. and N. B. Tufillaro (1997), "Resource Letter: ND-1: nonlinear dynamics," Am. J. Phys. 65 (9), 822-34. Laws, P. W. (2004), "A unit on oscillations, determinism and chaos for introductory physics students," Am. J. Phys. 72 (4), 446-52. Sungar, N., J. P. Sharpe, M. J. Moelter, N. Fleishon, K. Morrison, J. McDill, and R. Schoonover (2001), "A laboratory-based nonlinear dynamics course for science and engineering students," Am. J. Phys. 69 (5), 591-7. http://carbon.cudenver.edu/~rtagg/CVCP/Ctr_dynamics/Lab_nonlinear_dyn/Bibex_nonline..
    [Show full text]
  • Summary of Unit 3 Chaos and the Butterfly Effect
    Summary of Unit 3 Chaos and the Butterfly Effect Introduction to Dynamical Systems and Chaos http://www.complexityexplorer.org The Logistic Equation ● A very simple model of a population where there is some limit to growth ● f(x) = rx(1-x) ● r is a growth parameter ● x is measured as a fraction of the “annihilation” parameter. ● f(x) gives the population next year given x, the population this year. David P. Feldman Introduction to Dynamical Systems http://www.complexityexplorer.org and Chaos Iterating the Logistic Equation ● We used an online program to iterate the logistic equation for different r values and make time series plots ● We found attracting periodic behavior of different periods and... David P. Feldman Introduction to Dynamical Systems http://www.complexityexplorer.org and Chaos Aperiodic Orbits ● For r=4 (and other values), the orbit is aperiodic. It never repeats. ● Applying the same function over and over again does not result in periodic behavior. David P. Feldman Introduction to Dynamical Systems http://www.complexityexplorer.org and Chaos Comparing Initial Conditions ● We used a different online program to compare time series for two different initial conditions. ● The bottom plot is the difference between the two time series in the top plot. David P. Feldman Introduction to Dynamical Systems http://www.complexityexplorer.org and Chaos Sensitive Dependence on Initial Conditions ● When r=4.0, two orbits that start very close together eventually end up far apart ● This is known as sensitive dependence on initial conditions, or the butterfly effect. David P. Feldman Introduction to Dynamical Systems http://www.complexityexplorer.org and Chaos Sensitive Dependence on Initial Conditions ● For any initial condition x there is another initial condition very near to it that eventually ends up far away ● To predict the behavior of a system with SDIC requires knowing the initial condition with impossible accuracy.
    [Show full text]
  • Chaos: the Mathematics Behind the Butterfly Effect
    Chaos: The Mathematics Behind the Butterfly E↵ect James Manning Advisor: Jan Holly Colby College Mathematics Spring, 2017 1 1. Introduction A butterfly flaps its wings, and a hurricane hits somewhere many miles away. Can these two events possibly be related? This is an adage known to many but understood by few. That fact is based on the difficulty of the mathematics behind the adage. Now, it must be stated that, in fact, the flapping of a butterfly’s wings is not actually known to be the reason for any natural disasters, but the idea of it does get at the driving force of Chaos Theory. The common theme among the two is sensitive dependence on initial conditions. This is an idea that will be revisited later in the paper, because we must first cover the concepts necessary to frame chaos. This paper will explore one, two, and three dimensional systems, maps, bifurcations, limit cycles, attractors, and strange attractors before looking into the mechanics of chaos. Once chaos is introduced, we will look in depth at the Lorenz Equations. 2. One Dimensional Systems We begin our study by looking at nonlinear systems in one dimen- sion. One of the important features of these is the nonlinearity. Non- linearity in an equation evokes behavior that is not easily predicted due to the disproportionate nature of inputs and outputs. Also, the term “system”isoftenamisnomerbecauseitoftenevokestheideaof asystemofequations.Thiswillbethecaseaswemoveourfocuso↵ of one dimension, but for now we do not want to think of a system of equations. In this case, the type of system we want to consider is a first-order system of a single equation.
    [Show full text]
  • SPATIOTEMPORAL CHAOS in COUPLED MAP LATTICE Itishree
    SPATIOTEMPORAL CHAOS IN COUPLED MAP LATTICE By Itishree Priyadarshini Under the Guidance of Prof. Biplab Ganguli Department of Physics National Institute of Technology, Rourkela CERTIFICATE This is to certify that the project thesis entitled ” Spatiotemporal chaos in Cou- pled Map Lattice ” being submitted by Itishree Priyadarshini in partial fulfilment to the requirement of the one year project course (PH 592) of MSc Degree in physics of National Institute of Technology, Rourkela has been carried out under my super- vision. The result incorporated in the thesis has been produced by developing her own computer codes. Prof. Biplab Ganguli Dept. of Physics National Institute of Technology Rourkela - 769008 1 ACKNOWLEDGEMENT I would like to acknowledge my guide Prof. Biplab Ganguli for his help and guidance in the completion of my one-year project and also for his enormous moti- vation and encouragement. I am also very much thankful to research scholars whose encouragement and support helped me to complete my project. 2 ABSTRACT The sensitive dependence on initial condition, which is the essential feature of chaos is demonstrated through simple Lorenz model. Period doubling route to chaos is shown by analysis of Logistic map and other different route to chaos is discussed. Coupled map lattices are investigated as a model for spatio-temporal chaos. Diffusively coupled logistic lattice is studied which shows different pattern in accordance with the coupling constant and the non-linear parameter i.e. frozen random pattern, pattern selection with suppression of chaos , Brownian motion of the space defect, intermittent collapse, soliton turbulence and travelling waves. 3 Contents 1 Introduction 3 2 Chaos 3 3 Lorenz System 4 4 Route to Chaos 6 4.1 PeriodDoubling.............................
    [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]
  • THE BUTTERFLY EFFECT Étienne GHYS CNRS-UMPA ENS Lyon [email protected]
    12th International Congress on Mathematical Education Program Name XX-YY-zz (pp. abcde-fghij) 8 July – 15 July, 2012, COEX, Seoul, Korea (This part is for LOC use only. Please do not change this part.) THE BUTTERFLY EFFECT Étienne GHYS CNRS-UMPA ENS Lyon [email protected] It is very unusual for a mathematical idea to disseminate into the society at large. An interesting example is chaos theory, popularized by Lorenz’s butterfly effect: “does the flap of a butterfly’s wings in Brazil set off a tornado in Texas?” A tiny cause can generate big consequences! Can one adequately summarize chaos theory in such a simple minded way? Are mathematicians responsible for the inadequate transmission of their theories outside of their own community? What is the precise message that Lorenz wanted to convey? Some of the main characters of the history of chaos were indeed concerned with the problem of communicating their ideas to other scientists or non-scientists. I’ll try to discuss their successes and failures. The education of future mathematicians should include specific training to teach them how to explain mathematics outside their community. This is more and more necessary due to the increasing complexity of mathematics. A necessity and a challenge! INTRODUCTION In 1972, the meteorologist Edward Lorenz gave a talk at the 139th meeting of the American Association for the Advancement of Science entitled “Does the flap of a butterfly’s wings in Brazil set off a tornado in Texas?”. Forty years later, a google search “butterfly effect” generates ten million answers.
    [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]