DOCSLIB.ORG

Lecture 3: Dynamical Systems 2

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Lecture 3: Dynamical Systems 2 15-382 COLLECTIVE INTELLIGENCE – S19 LECTURE 3: DYNAMICAL SYSTEMS 2 TEACHER: GIANNI A. DI CARO GENERAL DEFINITION OF DYNAMICAL SYSTEMS A dynamical system is a 3-tuple ", !, Φ : § ! is a set of all possible states of the dynamical system (the state space) § " is the set of values the time (evolution) parameter can take § Φ is the evolution function of the dynamical system, that associates to each $ ∈ ! a unique image in ! depending on the time parameter &, (not all pairs (&, $) are feasible, that requires introducing the subset *) Φ: * ⊆ "×! → ! Ø Φ 0, $ = $1 (the initial condition) Ø Φ &2, 3 &4, $ = Φ(&2 + &4, $), (property of states) for &4, &4+&2 ∈ 6($), &2 ∈ 6(Φ(&4$)), 6 $ = {& ∈ " ∶ (&, $) ∈ *} Ø The evolution function Φ provides the system state (the value) at time & for any initial state $1 Ø :; = {Φ &, $ ∶ & ∈ 6 $ } orbit (flow lines) of the system through $, starting in $ , the set of visited states as a function of time: $(&) 2 TYPES OF DYNAMICAL SYSTEMS § Informally: A dynamical system defines a deterministic rule that allows to know the current state as a function of past states § Given an initial condition !" = !(0) ∈ (, a deterministic trajectory ! ) , ) ∈ + !" , is produced by ,, (, Φ § States can be “anything” mathematically well-behaved that represent situations of interest § The nature of the set , and of the function Φ give raise to different classes of dynamical systems (and resulting properties and trajectories) 3 TYPES OF DYNAMICAL SYSTEMS § Continuous time dynamical systems (Flows): ! open interval of ℝ, Φ continous and differentiable function à Differential equations § Φ represents a flow, defining a smooth (differentiable) continuous curve § The notion of flow builds on and formalizes the idea of the motion of particles in a fluid: it can be viewed as the abstract representation of (continuous) motion of points over time. § Discrete-time dynamical systems (Maps): ! interval of ℤ, Φ a function § Φ, represents an iterated map, which is not a flow (a differentiable curve) anymore, since the trajectory is a discrete set of points § Trajectory is represented through linear interpolation and it can easily present large slope changes at the points (e.g., cuspids) 4 FLOWS VS. ITERATED MAPS Laminar (streamline) flow: No cross-currents or swirls, individual trajectories flow on parallel lines, do not intersect Turbulent flow: Individual trajectories can intersect 5 CONTINUOUS-TIME DYNAMICAL SYSTEMS § Continuous-time dynamical systems (Flows): ! open interval of ℝ, Φ continous and differentiable function § If the flow Φ is generated by a vector field $ on % ⊆ ℝ', then the orbits ((*) of the flow are the images of the integral curves of the vector field § Vector field on a ,-dim space %: assignment of a ,-dim vector to each point of the space, the vector defines a direction and a velocity in the point (that the field would exert on a point-like particle in the point) Vector field on ℝ- Flow orbits $ = (20, −34) 6 CONTINUOUS-TIME DYNAMICAL SYSTEMS § Ordinary Differential Equations (system of ODEs) § Delay models, past state is determining present state "̇ = $(" & − ( ) § Integro-Differential Equations, accounting for history , "̇ = $ " & + + $ " ( .( ,- § Partial Differential Equations, accounting for space and time 1 21 21 3 ", & = 3 ", & 01 2&1 2"1 7 DISCRETE-TIME DYNAMICAL SYSTEMS § Discrete-time dynamical systems (Maps): ! interval of ℤ, Φ a function § The iterated map Φ is generated by a set of recurrence equations $ on % ⊆ ℝ( (also referred to as difference equations) § The orbits )(+) are sets of discrete points resulting from the closed-form solution (not always achievable) of the recurrence equations § Example with one single recurrence equation: -( = /(-(01, -(03, … , -(05) § Order-6 Markov states: relevant state information includes all past 6 states § Note that integro-differential equations are in principle order-∞ Markov, since infinite states from the past affect current state § Another, well-known example: Fibonacci recurrence equation § -( = -(01 + -(03 § Initial condition (that uniquely determines the orbit): -9 = :, -1 = ; 8 FROM LOCAL RULES TO GLOBAL BEHAVIORS? '( = )((, ") Flows Maps -. = )(-./0, -./1, … , -./3) '" ∆" = 1, when ∆" → 0 à '" à Differential eq. § For an infinitesimal time, only the instantaneous variation, the velocity, makes sense à The next state is expressed implicitly, and all the instantaneous variations, local in time, must be integrated in order to obtain the global behavior ((") § Also in maps, the time-local iteration rule is a local description that can give rise to extremely complex global behaviors § à How do we integrate the local description into global behaviors? § à How do we predict global behaviors from the local descriptions? 9 CONTINUOUS-TIME DS: VECTOR FIELDS AND ORBITS Uncoupled system § . is a vector field in ℝ0: a function "̇ = 2" = %& (", )) associating a vector to 1-dim point 2 56 786 )̇ = −3) = %- (", )) § Solution: "34 , )34 . = (2", −3)) Orbits / Possible trajectories Direction and speed of solution Φ(:, " : ) for any Flow: 3 (", )) Vector field Phase portrait Rate of change, velocity § Autonomous system à no explicit dependence from time in ., all information about the solution is represented § A fundamental theorem guarantees (under differentiability and continuity assumptions) that two orbits corresponding to two different initial solutions never intersect with each other (laminar flow) 10 VECTOR FIELDS, ORBITS, FIXED POINTS (1,1) "̇ = $ = % (", $) & / = ($, −" − $+) + $̇ = −" − $ = %, (", $) E.g., /(1,1) = (1, −2) (1, −2) (Rescaled) vector field Closed (periodic) orbit Equilibrium point Direction of increasing time § -∗ is an equilibrium (fixed) point of the ODE if / -∗ = 0 § ↔ Once in "∗, the system remains there: -∗ = - 2; -∗ , 2 ≥ 0 11 EXAMPLE: LINEAR MODEL FOR POPULATION GROWTH !̇ = $! § Linear model of population growth (Malthus model, 1798) § Works well for small populations !(0) = !* § ! = size of population, $ = growth rate § This linear equation can easily be integrated by separation of variables: , +, +, +, - = $!, = $./, 0 = $ ∫ ./ +- , , -1 ,1 ! ! 7- ln ! − ln !* = $/ ln = $/ = 6 !* !* 7- ! = !*6 12 LINEAR MODEL FOR POPULATION GROWTH Phase portrait , = *# () Solution orbits / Flow (in !): #(!) = #&' scalar (linear) vector field (a) * > 0: Exponential growth (b) * < 0: Exponential decrease 13 LOGISTIC MODEL FOR POPULATION GROWTH !" § General form for population growth: = %(') !# § What is a good model that captures essential aspects? ü Every living organism must have at least one parent of like kind ü In a finite space, due to the limiting effect of the environment, there is an upper limit to the number of organisms that can occupy that space: resources competition constraint § à Logistic model (1838), non-linear: ) = intrinsic rate of increase [1/t] !" " = )' 1 − - = maX carrying capacity [# individuals] !# , '. = '(0) § à Non-dimensional equation with no parameters: !0 ' τ = )8 (dimensionless time) = 2 1 − 2 2 = . ' !1 . 2 = ∈ [0,1] - - (dimensionless population) 14 LOGISTIC MODEL FOR POPULATION GROWTH § The logistic equation, even if not lineAr, cAn be Also integrAted by separAtion of vAriAbles: !" !" !" = % 1 − % , = +τ, ∫ = ∫ +τ !# " )*" " )*" +% +% . + . = . +τ ln % − ln 1 − % = τ + 2 % 1 − % 1 − % 1 1 ln = −τ − 2 − 1 = 3*#*4 = 1 + 53*# % % % 1 The integrAtion constAnt 5 depends on %(τ) = 1 + 53*# the initiAl condition %8 ; ; − 98 9(:) = < = 1 + <3*=> 98 15 LOGISTIC MODEL FOR POPULATION GROWTH 1 Equilibrium points: !(τ) = 1 + ()*+ , ! = ! 1 − ! = 0 à ! =1, ! = 0 ! Phase ! = 1 ! portrait ! = 0 ! =1 ! = 0 Asymptotic divergence Flow, different ( values (=1 Flow function /(0, !2) is not defined for all values of 0 16 (BASIC) LOGISTIC MODEL: DOES IT WORK? § Population of the US in 1800: 5.3 millions à Predict population in 1900 and 1950 § Population of the US in 1850: 23.1 millions Answer: 76 (1900), 150.7 (1950) § Let’s look first at what the linear (i.e., exponential growth) model would predict: () !(#) = !&' à We need first to derive an estimate for growth parameter *: !(1850) = !(1800)'() à 23.1 = 5.3'/&( à * = 0.29 ! 1900 = ! 1800 '&.3456&& = 100.7 ! 1950 = ! 1800 '&.3456/& = 438.8 § The non-linear (i.e., logistic growth) model in the dimensional form has two parameters à We need more information: let’s assume we know the 1900 answer: @ !(1850) = = 23.1 6A @=/.B CDEFG//.B : : − !& J = 0.031 ! # = =>) < = 1 + <' ! @ & !(1900) = = 76 : = 189.4 6A @=/.B CDIFFG//.B 189.4 ! # = (the baby boom is 1 + 34.74'=&.&B6) à ! 1950 = 144.7 not accounted!) 17 LOGISTIC MODEL VS. EXPONENTIAL GROWTH ● real population values in the US ▬ Logistic model predictions Exponential explosion Logistic asymptote ● real population values in the US ▬ Logistic model predictions Little difference for small populations Both linear and logistic model work well 18.
Load more

Recommended publications
  • A Gentle Introduction to Dynamical Systems Theory for Researchers in Speech, Language, and Music
    A Gentle Introduction to Dynamical Systems Theory for Researchers in Speech, Language, and Music. Talk given at PoRT workshop, Glasgow, July 2012 Fred Cummins, University College Dublin [1] Dynamical Systems Theory (DST) is the lingua franca of Physics (both Newtonian and modern), Biology, Chemistry, and many other sciences and non-sciences, such as Economics. To employ the tools of DST is to take an explanatory stance with respect to observed phenomena. DST is thus not just another tool in the box. Its use is a different way of doing science. DST is increasingly used in non-computational, non-representational, non-cognitivist approaches to understanding behavior (and perhaps brains). (Embodied, embedded, ecological, enactive theories within cognitive science.) [2] DST originates in the science of mechanics, developed by the (co-)inventor of the calculus: Isaac Newton. This revolutionary science gave us the seductive concept of the mechanism. Mechanics seeks to provide a deterministic account of the relation between the motions of massive bodies and the forces that act upon them. A dynamical system comprises • A state description that indexes the components at time t, and • A dynamic, which is a rule governing state change over time The choice of variables defines the state space. The dynamic associates an instantaneous rate of change with each point in the state space. Any specific instance of a dynamical system will trace out a single trajectory in state space. (This is often, misleadingly, called a solution to the underlying equations.) Description of a specific system therefore also requires specification of the initial conditions. In the domain of mechanics, where we seek to account for the motion of massive bodies, we know which variables to choose (position and velocity).
    [Show full text]
  • A Cell Dynamical System Model for Simulation of Continuum Dynamics of Turbulent Fluid Flows A
    A Cell Dynamical System Model for Simulation of Continuum Dynamics of Turbulent Fluid Flows A. M. Selvam and S. Fadnavis Email: [email protected] Website: http://www.geocities.com/amselvam Trends in Continuum Physics, TRECOP ’98; Proceedings of the International Symposium on Trends in Continuum Physics, Poznan, Poland, August 17-20, 1998. Edited by Bogdan T. Maruszewski, Wolfgang Muschik, and Andrzej Radowicz. Singapore, World Scientific, 1999, 334(12). 1. INTRODUCTION Atmospheric flows exhibit long-range spatiotemporal correlations manifested as the fractal geometry to the global cloud cover pattern concomitant with inverse power-law form for power spectra of temporal fluctuations of all scales ranging from turbulence (millimeters-seconds) to climate (thousands of kilometers-years) (Tessier et. al., 1996) Long-range spatiotemporal correlations are ubiquitous to dynamical systems in nature and are identified as signatures of self-organized criticality (Bak et. al., 1988) Standard models for turbulent fluid flows in meteorological theory cannot explain satisfactorily the observed multifractal (space-time) structures in atmospheric flows. Numerical models for simulation and prediction of atmospheric flows are subject to deterministic chaos and give unrealistic solutions. Deterministic chaos is a direct consequence of round-off error growth in iterative computations. Round-off error of finite precision computations doubles on an average at each step of iterative computations (Mary Selvam, 1993). Round- off error will propagate to the mainstream computation and give unrealistic solutions in numerical weather prediction (NWP) and climate models which incorporate thousands of iterative computations in long-term numerical integration schemes. A recently developed non-deterministic cell dynamical system model for atmospheric flows (Mary Selvam, 1990; Mary Selvam et.
    [Show full text]
  • Chapter 5. Flow Maps and Dynamical Systems
    Chapter 5 Flow Maps and Dynamical Systems Main concepts: In this chapter we introduce the concepts of continuous and discrete dynamical systems on phase space. Keywords are: classical mechanics, phase space, vector field, linear systems, flow maps, dynamical systems Figure 5.1: A solar system is well modelled by classical mechanics. (source: Wikimedia Commons) In Chapter 1 we encountered a variety of ODEs in one or several variables. Only in a few cases could we write out the solution in a simple closed form. In most cases, the solutions will only be obtainable in some approximate sense, either from an asymptotic expansion or a numerical computation. Yet, as discussed in Chapter 3, many smooth systems of differential equations are known to have solutions, even globally defined ones, and so in principle we can suppose the existence of a trajectory through each point of phase space (or through “almost all” initial points). For such systems we will use the existence of such a solution to define a map called the flow map that takes points forward t units in time from their initial points. The concept of the flow map is very helpful in exploring the qualitative behavior of numerical methods, since it is often possible to think of numerical methods as approximations of the flow map and to evaluate methods by comparing the properties of the map associated to the numerical scheme to those of the flow map. 25 26 CHAPTER 5. FLOW MAPS AND DYNAMICAL SYSTEMS In this chapter we will address autonomous ODEs (recall 1.13) only: 0 d y = f(y), y, f ∈ R (5.1) 5.1 Classical mechanics In Chapter 1 we introduced models from population dynamics.
    [Show full text]
  • Thermodynamic Properties of Coupled Map Lattices 1 Introduction
    Thermodynamic properties of coupled map lattices J´erˆome Losson and Michael C. Mackey Abstract This chapter presents an overview of the literature which deals with appli- cations of models framed as coupled map lattices (CML’s), and some recent results on the spectral properties of the transfer operators induced by various deterministic and stochastic CML’s. These operators (one of which is the well- known Perron-Frobenius operator) govern the temporal evolution of ensemble statistics. As such, they lie at the heart of any thermodynamic description of CML’s, and they provide some interesting insight into the origins of nontrivial collective behavior in these models. 1 Introduction This chapter describes the statistical properties of networks of chaotic, interacting el- ements, whose evolution in time is discrete. Such systems can be profitably modeled by networks of coupled iterative maps, usually referred to as coupled map lattices (CML’s for short). The description of CML’s has been the subject of intense scrutiny in the past decade, and most (though by no means all) investigations have been pri- marily numerical rather than analytical. Investigators have often been concerned with the statistical properties of CML’s, because a deterministic description of the motion of all the individual elements of the lattice is either out of reach or uninteresting, un- less the behavior can somehow be described with a few degrees of freedom. However there is still no consistent framework, analogous to equilibrium statistical mechanics, within which one can describe the probabilistic properties of CML’s possessing a large but finite number of elements.
    [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]
  • A Simple Scalar Coupled Map Lattice Model for Excitable Media
    This is a repository copy of A simple scalar coupled map lattice model for excitable media. White Rose Research Online URL for this paper: http://eprints.whiterose.ac.uk/74667/ Monograph: Guo, Y., Zhao, Y., Coca, D. et al. (1 more author) (2010) A simple scalar coupled map lattice model for excitable media. Research Report. ACSE Research Report no. 1016 . Automatic Control and Systems Engineering, University of Sheffield Reuse Unless indicated otherwise, fulltext items are protected by copyright with all rights reserved. The copyright exception in section 29 of the Copyright, Designs and Patents Act 1988 allows the making of a single copy solely for the purpose of non-commercial research or private study within the limits of fair dealing. The publisher or other rights-holder may allow further reproduction and re-use of this version - refer to the White Rose Research Online record for this item. Where records identify the publisher as the copyright holder, users can verify any specific terms of use on the publisher’s website. Takedown If you consider content in White Rose Research Online to be in breach of UK law, please notify us by emailing [email protected] including the URL of the record and the reason for the withdrawal request. [email protected] https://eprints.whiterose.ac.uk/ A Simple Scalar Coupled Map Lattice Model for Excitable Media Yuzhu Guo, Yifan Zhao, Daniel Coca, and S. A. Billings Research Report No. 1016 Department of Automatic Control and Systems Engineering The University of Sheffield Mappin Street, Sheffield, S1 3JD, UK 8 September 2010 A Simple Scalar Coupled Map Lattice Model for Excitable Media Yuzhu Guo, Yifan Zhao, Daniel Coca, and S.A.
    [Show full text]
  • Flows of Continuous-Time Dynamical Systems with No Periodic Orbit As an Equivalence Class Under Topological Conjugacy Relation
    Journal of Mathematics and Statistics 7 (3): 207-215, 2011 ISSN 1549-3644 © 2011 Science Publications Flows of Continuous-Time Dynamical Systems with No Periodic Orbit as an Equivalence Class under Topological Conjugacy Relation Tahir Ahmad and Tan Lit Ken Department of Mathematics, Faculty of Science, Nanotechnology Research Alliance, Theoretical and Computational Modeling for Complex Systems, University Technology Malaysia 81310 Skudai, Johor, Malaysia Abstract: Problem statement: Flows of continuous-time dynamical systems with the same number of equilibrium points and trajectories, and which has no periodic orbit form an equivalence class under the topological conjugacy relation. Approach: Arbitrarily, two trajectories resulting from two distinct flows of this type of dynamical systems were written as a set of points (orbit). A homeomorphism which maps between these two sets is then built. Using the notion of topological conjugacy, they were shown to conjugate topologically. By the arbitrariness in selection of flows and their respective initial states, the results were extended to all the flows of dynamical system of that type. Results: Any two flows of such dynamical systems were shown to share the same dynamics temporally along with other properties such as order isomorphic and homeomorphic. Conclusion: Topological conjugacy serves as an equivalence relation in the set of flows of continuous-time dynamical systems which have same number of equilibrium points and trajectories, and has no periodic orbit. Key words: Dynamical system, equilibrium points, trajectories, periodic orbit, equivalence class, topological conjugacy, order isomorphic, Flat Electroencephalography (Flat EEG), dynamical systems INTRODUCTION models. Nandhakumar et al . (2009) for example, the dynamics of robot arm is studied.
    [Show full text]
  • 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.
    [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]
  • Attractors: Nonstrange to Chaotic
    1 Attractors: Nonstrange to Chaotic Robert L. V. Taylor The College of Wooster [email protected] Advised by Dr. John David The College of Wooster [email protected] Abstract—The theory of chaotic dynamical systems can particular, a strange attractor. It is a common miscon- be a tricky area of study for a non-expert to break into. ception among non-experts that the term strange attractor Because the theory is relatively recent, the new student finds is simply another way of saying chaotic attractor. We himself immersed in a subject with very few clear and intuitive definitions. This paper aims to carve out a small will show why this is not necessarily true. In Section section of the theory of chaotic dynamical systems – that III, we apply these ideas by classifying and identifying of attractors – and outline its fundamental concepts from a properties of several example attractors. computational mathematics perspective. The motivation for It is assumed that the reader has a rudimentary under- this paper is primarily to define what an attractor is and standing of dynamical systems. For example, the reader to clarify what distinguishes its various types (nonstrange, strange nonchaotic, and strange chaotic). Furthermore, by should be familiar with basic concepts concerning initial providing some examples of attractors and explaining how conditions, orbits, and the differences between real and and why they are classified, we hope to provide the reader discrete systems. with a good feel for the fundamental connection between fractal geometry and the existence of chaos. II. Mathematical Concepts A. Fractals and Dimension I. Introduction Simply put, a fractal is a geometric object that is self- HE theory of dynamical systems is an extremely similar on all scales.
    [Show full text]
  • Dynamical Systems Theory
    Dynamical Systems Theory Bjorn¨ Birnir Center for Complex and Nonlinear Dynamics and Department of Mathematics University of California Santa Barbara1 1 c 2008, Bjorn¨ Birnir. All rights reserved. 2 Contents 1 Introduction 9 1.1 The 3 Body Problem . .9 1.2 Nonlinear Dynamical Systems Theory . 11 1.3 The Nonlinear Pendulum . 11 1.4 The Homoclinic Tangle . 18 2 Existence, Uniqueness and Invariance 25 2.1 The Picard Existence Theorem . 25 2.2 Global Solutions . 35 2.3 Lyapunov Stability . 39 2.4 Absorbing Sets, Omega-Limit Sets and Attractors . 42 3 The Geometry of Flows 51 3.1 Vector Fields and Flows . 51 3.2 The Tangent Space . 58 3.3 Flow Equivalance . 60 4 Invariant Manifolds 65 5 Chaotic Dynamics 75 5.1 Maps and Diffeomorphisms . 75 5.2 Classification of Flows and Maps . 81 5.3 Horseshoe Maps and Symbolic Dynamics . 84 5.4 The Smale-Birkhoff Homoclinic Theorem . 95 5.5 The Melnikov Method . 96 5.6 Transient Dynamics . 99 6 Center Manifolds 103 3 4 CONTENTS 7 Bifurcation Theory 109 7.1 Codimension One Bifurcations . 110 7.1.1 The Saddle-Node Bifurcation . 111 7.1.2 A Transcritical Bifurcation . 113 7.1.3 A Pitchfork Bifurcation . 115 7.2 The Poincare´ Map . 118 7.3 The Period Doubling Bifurcation . 119 7.4 The Hopf Bifurcation . 121 8 The Period Doubling Cascade 123 8.1 The Quadradic Map . 123 8.2 Scaling Behaviour . 130 8.2.1 The Singularly Supported Strange Attractor . 138 A The Homoclinic Orbits of the Pendulum 141 List of Figures 1.1 The rotation of the Sun and Jupiter in a plane around a common center of mass and the motion of a comet perpendicular to the plane.
    [Show full text]