Dynamics and Chaos. Melanie Mitchell. Santa Fe Institute and Portland State University
|
|
- Noreen Nash
- 5 years ago
- Views:
Transcription
1 Dynamics and Chaos Melanie Mitchell Santa Fe Institute and Portland State University Dynamical Systems Theory: The general study of how systems change over time Calculus Differential equations Discrete maps Algebraic topology Vocabulary of change The dynamics of a system: the manner in which the system changes Isaac Newton Dynamical systems theory gives us a vocabulary and set of tools for describing dynamics Chaos: One particular type of dynamics of a system Defined as sensitive dependence on initial conditions Poincaré: Many-body problem in the solar system Henri Poincaré
2 Dr. Ian Malcolm You've never heard of Chaos theory? Non-linear equations? Strange attractors? 2
3 Chaos in Nature Dripping faucets Electrical circuits Solar system orbits Weather and climate (the butterfly effect ) Heart activity (EKG) Computer networks Population growth and dynamics Financial data Brain activity (EEG) What is the difference between chaos and randomness? Notion of deterministic chaos 3
4 A simple example of deterministic chaos: Exponential versus logistic models for population growth n t +1 = 2n t Exponential model: Each year each pair of parents mates, creates four offspring, and then parents die. Linear Behavior n t +1 = 2n t 4
5 Linear Behavior: The whole is the sum of the parts Linear: No interaction among the offspring, except pair-wise mating. More realistic: Introduce limits to population growth. Logistic model Notions of: birth rate death rate (probability an individual will die due to overcrowding) maximum carrying capacity k (upper limit of the population that the habitat will support) n t +1 = (birthrate deathrate)[kn t n t 2 ]/k interac(ons between offspring make this model nonlinear 5
6 Nonlinear Behavior n t +1 = (birthrate deathrate)[kn t n t 2 ]/k Nonlinear behavior of logistic model Nonlinear: The whole is different than the sum of the parts birth rate 2, death rate 0.4, k=32 (keep the same on the two islands) 6
7 Logistic map x t +1 = Raaa x t (1 x t ) Lord Robert May b n t +1 = (birthrate deathrate)[kn t n t 2 ]/k Mitchell Feigenbaum b Let x t = n t /k Let R = birthrate deathrate 1. R = 2 LogisticMap.nlogo 2. R = R = 2.8 Notion of period doubling Notion of attractors 4. R = R = R = R = 4, look at sensitive dependence on initial conditions 7
8 Period Doubling and Universals in Chaos (Mitchell Feigenbaum) R 1 3.0: period 2 R period 4 R period 8 R period 16 R period 32 R period (chaos) Bifurcation Diagram 8
9 Period Doubling and Universals in Chaos (Mitchell Feigenbaum) R 1 3.0: period 2 R period 4 R period 8 R period 16 R period 32 A similar period doubling route to chaos is seen in any one-humped (unimodal) map. R period (chaos) Period Doubling and Universals in Chaos (Mitchell Feigenbaum) R 1 3.0: period 2 R period 4 R period 8 R period 16 R period 32 R period (chaos) Rate at which distance between bifurcations is shrinking: R 2 R = R 3 R = R 3 R = R 4 R = R 4 R = R 5 R = R lim n +1 R n n R n +2 R n +1 9
10 Period Doubling and Universals in Chaos (Mitchell Feigenbaum) In other words, each Rate new at which bifurcation distance appears between about R 1 3.0: period times 2 faster bifurcations than the is shrinking: previous one. R period 4 R 2 R R period 8 = R 3 R = R period 16 R This same period rate 32 of R 3 R = occurs in any unimodal map. R 4 R = R period (chaos) R 4 R = R 5 R = lim R n +1 R n R n +2 R n +1 Significance of dynamics and chaos for complex systems Apparent random behavior from deterministic rules Complexity from simple rules Vocabulary of complex behavior Limits to detailed prediction Universality 10
11 Spatial dynamics: cellular automata Game of Life applications: tumor dynamics modeling in cancer biological pattern formation modeling social systems Martin Nowak s spatial PD model Give an example of modeling with CAs: non-spatial vs. spatial PD Always cooperate, always defect different strategies each individual plays PD with neighbors each individual plays PD with 8 randomly chosen individuals Cellular automata: Spatial dynamical systems Each cell is connected only to neighboring cells Typically periodic boundary conditions, or assume infinite lattice Often can get complex behavior from simple rules 11
12 Example: Game of Life (John Conway, 1970s) Neighborhood: 2 dimensional 3x3 neighborhood: Rules: A dead cell with exactly three live neighbors becomes a live cell (birth). A live cell with two or three live neighbors stays alive (survival). In all other cases, a cell dies or remains dead (overcrowding or loneliness). Demo: Netlogo models library Computer Science Cellular Automata Life What are cellular automaton actually used for? CAs are models of physical (or biological or social) systems fluid dynamics galaxy formation earthquakes biological pattern formation tumor dynamics in cancer social systems etc. CAs are alternative methods for approximating differential equations CAs are devices that can simulate standard computers CAs are parallel computers that can perform image processing, random number generation, cryptography, etc. 12
13 What are cellular automaton actually used for? CAs are a framework for implementing molecular scale computation CAs are a framework for exploring how collective computation might take place in natural systems (and that might be imitated in novel human-made computational systems) Brief History, continued Current renewal of interest due to: Renewed interest in how biological systems compute (and how that can inspire new computer architectures) Reconfigurable computing (FPGAs) Molecular and quantum-scale computation (e.g., quantum dot cellular automata) A New Kind of Science? 13
14 Example of CA-based modeling: Evolution of Cooperation in Spatial Systems (Nowak et al.) Prisoner s dilemma: Player 1 cooperate Player 2 defect cooperate defect 3, 3 0, 5 5, 0 1, 1 Why is it a dilemma? Used extensively in game theory and social-science modeling Example of CA-based modeling: Evolution of Cooperation in Spatial Systems (Nowak et al.) Player 1 cooperate Player 2 defect cooperate defect 3, 3 0, 5 5, 0 1, 1 Group Exercise: Choose a strategy: Always Cooperate or Always Defect Play PD once with each of four others (some people might have to play twice). Keep track of your score. At end, take on strategy of highest scorer among you and the four others you played with. 14
15 Demo: PDCA.nlogo Adapted from: Wilensky, U. (2002). NetLogo PD Basic Evolutionary model. Center for Connected Learning and Computer-Based Modeling, Northwestern University, Evanston, IL. Spatial Prisoner s dilemma model Players arrayed on a two-dimensional lattice, one player per site. Each player either always cooperates or always defects. Players have no memory. Each player plays with eight nearest neighbors. Score is sum of payoffs. Each site is then occupied by highest scoring player in its neighborhood. Updates are done synchronously. 15
16 Interpretation Motivation for this work is primarily biological. We believe that deterministically generated spatial structure within populations may often be crucial for the evolution of cooperation, whether it be among molecules, cells, or organisms. "That territoriality favours cooperation...is likely to remain valid for real-life communities (Karl Sigmund, Nature, 1992, in commentary on Nowak and May paper) Re-implementation by Glance and Huberman (1993) There are important differences between the way a system composed of many interacting elements is simulated by a digital machine and the manner in which it behaves when studied in real experiments." Asked: What role does the assumption of synchronous updating have in the behavior observed by Nowak and May? "In natural social systems... a global clock that causes all the elements of the system to update their state at the same time seldom exists." 16
17 Glance and Huberman s results Synchronous Asynchronous Same ini@al configura@on: a single defector in the center. Blue: cooperator at a site that was a cooperator in previous genera@on. Red: defector following a defector. Yellow: defector following cooperator. Green: cooperator following defector. Glance and Huberman s conclusion "Until it can be demonstrated that global clocks synchronize mutations and chemical reactions among distal elements of biological structures, the patterns and regularities observed in nature will require continuous descriptions and asynchronous simulations." 17
18 Nowak, Bonhoeffer, and May response (PNAS, 1994) Huberman & Glance performed simulation for only one case of the I defect, you cooperate payoff value p. Sequential, rather than synchronous, updating of sites, still leads to persistence of both cooperation and defection for a range of p values. Behavior with other forms of asynchronous updating (e.g., random)? 18
Dynamics and Chaos. Copyright by Melanie Mitchell
Dynamics and Chaos Copyright by Melanie Mitchell Conference on Complex Systems, September, 2015 Dynamics: The general study of how systems change over time Copyright by Melanie Mitchell Conference on Complex
More informationDynamics: The general study of how systems change over time
Dynamics: The general study of how systems change over time Planetary dynamics P http://www.lpi.usra.edu/ Fluid Dynamics http://pmm.nasa.gov/sites/default/files/imagegallery/hurricane_depth.jpg Dynamics
More informationEvolutionary Games and Computer Simulations
Evolutionary Games and Computer Simulations Bernardo A. Huberman and Natalie S. Glance Dynamics of Computation Group Xerox Palo Alto Research Center Palo Alto, CA 94304 Abstract The prisoner s dilemma
More informationCellular Automata CS 591 Complex Adaptive Systems Spring Professor: Melanie Moses 2/02/09
Cellular Automata CS 591 Complex Adaptive Systems Spring 2009 Professor: Melanie Moses 2/02/09 Introduction to Cellular Automata (CA) Invented by John von Neumann (circa~1950). A cellular automata consists
More informationMitchell Chapter 10. Living systems are open systems that exchange energy, materials & information
Living systems compute Mitchell Chapter 10 Living systems are open systems that exchange energy, materials & information E.g. Erwin Shrodinger (1944) & Lynn Margulis (2000) books: What is Life? discuss
More informationII. Spatial Systems. A. Cellular Automata. Structure. Cellular Automata (CAs) Example: Conway s Game of Life. State Transition Rule
II. Spatial Systems A. Cellular Automata B. Pattern Formation C. Slime Mold D. Excitable Media A. Cellular Automata 1/18/17 1 1/18/17 2 Cellular Automata (CAs) Invented by von Neumann in 1940s to study
More informationIntroduction to Scientific Modeling CS 365, Fall 2011 Cellular Automata
Introduction to Scientific Modeling CS 365, Fall 2011 Cellular Automata Stephanie Forrest ME 214 http://cs.unm.edu/~forrest/cs365/ forrest@cs.unm.edu 505-277-7104 Reading Assignment! Mitchell Ch. 10" Wolfram
More informationCellular Automata. History. 1-Dimensional CA. 1-Dimensional CA. Ozalp Babaoglu
History Cellular Automata Ozalp Babaoglu Developed by John von Neumann as a formal tool to study mechanical self replication Studied extensively by Stephen Wolfram ALMA MATER STUDIORUM UNIVERSITA DI BOLOGNA
More informationCommunities and Populations
ommunities and Populations Two models of population change The logistic map The Lotke-Volterra equations for oscillations in populations Prisoner s dilemma Single play Iterated play ommunity-wide play
More informationInstability in Spatial Evolutionary Games
Instability in Spatial Evolutionary Games Carlos Grilo 1,2 and Luís Correia 2 1 Dep. Eng. Informática, Escola Superior de Tecnologia e Gestão, Instituto Politécnico de Leiria Portugal 2 LabMag, Dep. Informática,
More informationNetworks and sciences: The story of the small-world
Networks and sciences: The story of the small-world Hugues Bersini IRIDIA ULB 2013 Networks and sciences 1 The story begins with Stanley Milgram (1933-1984) In 1960, the famous experience of the submission
More informationII. Spatial Systems A. Cellular Automata 8/24/08 1
II. Spatial Systems A. Cellular Automata 8/24/08 1 Cellular Automata (CAs) Invented by von Neumann in 1940s to study reproduction He succeeded in constructing a self-reproducing CA Have been used as: massively
More informationDynamical Systems: Lecture 1 Naima Hammoud
Dynamical Systems: Lecture 1 Naima Hammoud Feb 21, 2017 What is dynamics? Dynamics is the study of systems that evolve in time What is dynamics? Dynamics is the study of systems that evolve in time a system
More informationDeborah Lacitignola Department of Health and Motory Sciences University of Cassino
DOTTORATO IN Sistemi Tecnologie e Dispositivi per il Movimento e la Salute Cassino, 2011 NONLINEAR DYNAMICAL SYSTEMS AND CHAOS: PHENOMENOLOGICAL AND COMPUTATIONAL ASPECTS Deborah Lacitignola Department
More information15-251: Great Theoretical Ideas in Computer Science Lecture 7. Turing s Legacy Continues
15-251: Great Theoretical Ideas in Computer Science Lecture 7 Turing s Legacy Continues Solvable with Python = Solvable with C = Solvable with Java = Solvable with SML = Decidable Languages (decidable
More information... it may happen that small differences in the initial conditions produce very great ones in the final phenomena. Henri Poincaré
Chapter 2 Dynamical Systems... it may happen that small differences in the initial conditions produce very great ones in the final phenomena. Henri Poincaré One of the exciting new fields to arise out
More informationMotivation. Evolution has rediscovered several times multicellularity as a way to build complex living systems
Cellular Systems 1 Motivation Evolution has rediscovered several times multicellularity as a way to build complex living systems Multicellular systems are composed by many copies of a unique fundamental
More informationCellular automata are idealized models of complex systems Large network of simple components Limited communication among components No central
Cellular automata are idealized models of complex systems Large network of simple components Limited communication among components No central control Complex dynamics from simple rules Capability of information
More informationResource heterogeneity can facilitate cooperation
Supplementary information for Resource heterogeneity can facilitate cooperation Ádám Kun 1,2,3,4* & Ulf Dieckmann 1 1 Evolution and Ecology Program, International Institute for Advanced System Analysis,
More informationII. Cellular Automata 8/27/03 1
II. Cellular Automata 8/27/03 1 Cellular Automata (CAs) Invented by von Neumann in 1940s to study reproduction He succeeded in constructing a self-reproducing CA Have been used as: massively parallel computer
More informationEvolutionary Games on Networks. Wen-Xu Wang and G Ron Chen Center for Chaos and Complex Networks
Evolutionary Games on Networks Wen-Xu Wang and G Ron Chen Center for Chaos and Complex Networks Email: wenxuw@gmail.com; wxwang@cityu.edu.hk Cooperative behavior among selfish individuals Evolutionary
More informationComplex networks and evolutionary games
Volume 2 Complex networks and evolutionary games Michael Kirley Department of Computer Science and Software Engineering The University of Melbourne, Victoria, Australia Email: mkirley@cs.mu.oz.au Abstract
More informationEvolution of cooperation. Martin Nowak, Harvard University
Evolution of cooperation Martin Nowak, Harvard University As the Fukushima power plant was melting down, a worker in his 20s was among those who volunteered to reenter. In an interview he said: There are
More informationCostly Signals and Cooperation
Costly Signals and Cooperation Károly Takács and András Németh MTA TK Lendület Research Center for Educational and Network Studies (RECENS) and Corvinus University of Budapest New Developments in Signaling
More informationXX Eesti Arvutiteaduse Talvekool
XX Eesti Arvutiteaduse Talvekool Cellular automata, tilings and (un)computability Jarkko Kari Department of Mathematics and Statistics University of Turku Lecture 1: Tutorial on Cellular automata Introduction
More informationEvolutionary prisoner s dilemma game on hierarchical lattices
PHYSICAL REVIEW E 71, 036133 2005 Evolutionary prisoner s dilemma game on hierarchical lattices Jeromos Vukov Department of Biological Physics, Eötvös University, H-1117 Budapest, Pázmány Péter sétány
More informationCellular Automata and Tilings
Cellular Automata and Tilings Jarkko Kari Department of Mathematics, University of Turku, Finland TUCS(Turku Centre for Computer Science), Turku, Finland Outline of the talk (1) Cellular automata (CA)
More informationUnderstanding and Solving Societal Problems with Modeling and Simulation
Understanding and Solving Societal Problems with Modeling and Simulation Lecture 8: The Breakdown of Cooperation ETH Zurich April 15, 2013 Dr. Thomas Chadefaux Why Cooperation is Hard The Tragedy of the
More informationComplexity in social dynamics : from the. micro to the macro. Lecture 4. Franco Bagnoli. Lecture 4. Namur 7-18/4/2008
Complexity in Namur 7-18/4/2008 Outline 1 Evolutionary models. 2 Fitness landscapes. 3 Game theory. 4 Iterated games. Prisoner dilemma. 5 Finite populations. Evolutionary dynamics The word evolution is
More informationBranislav K. Nikolić
Interdisciplinary Topics in Complex Systems: Cellular Automata, Self-Organized Criticality, Neural Networks and Spin Glasses Branislav K. Nikolić Department of Physics and Astronomy, University of Delaware,
More informationCellular Automata: Tutorial
Cellular Automata: Tutorial Jarkko Kari Department of Mathematics, University of Turku, Finland TUCS(Turku Centre for Computer Science), Turku, Finland Cellular Automata: examples A Cellular Automaton
More informationChaos in Dynamical Systems. LIACS Natural Computing Group Leiden University
Chaos in Dynamical Systems Overview Introduction: Modeling Nature! Example: Logistic Growth Fixed Points Bifurcation Diagrams Application Examples 2 INTRODUCTION 3 Linear and Non-linear dynamic systems
More informationToward a Better Understanding of Complexity
Toward a Better Understanding of Complexity Definitions of Complexity, Cellular Automata as Models of Complexity, Random Boolean Networks Christian Jacob jacob@cpsc.ucalgary.ca Department of Computer Science
More informationJEREMIAS EPPERLEIN, STEFAN SIEGMUND, AND PETR STEHLÍK
August 18, 2013 EVOLUTIONARY GAMES ON GRAPHS - MATHEMATICAL FOUNDATIONS JEREMIAS EPPERLEIN, STEFAN SIEGMUND, AND PETR STEHLÍK Abstract. Evolutionary games on graphs play an important role in the study
More informationarxiv: v4 [cs.dm] 20 Nov 2017
THE SPREAD OF COOPERATIVE STRATEGIES ON GRIDS WITH RANDOM ASYNCHRONOUS UPDATING CHRISTOPHER DUFFY 1 AND JEANNETTE JANSSEN 2 1 DEPARTMENT OF MATHEMATICS AND STATISTICS UNIVERSITY OF SASKATCHEWAN 2 DEPARTMENT
More informationEvolutionary Computation. DEIS-Cesena Alma Mater Studiorum Università di Bologna Cesena (Italia)
Evolutionary Computation DEIS-Cesena Alma Mater Studiorum Università di Bologna Cesena (Italia) andrea.roli@unibo.it Evolutionary Computation Inspiring principle: theory of natural selection Species face
More informationA NetLogo Model for the Study of the Evolution of Cooperation in Social Networks
A NetLogo Model for the Study of the Evolution of Cooperation in Social Networks Gregory Todd Jones Georgia State University College of Law Interuniversity Consortium on Negotiation and Conflict Resolution
More informationAsynchronous updating of threshold-coupled chaotic neurons
PRAMANA c Indian Academy of Sciences Vol. 70, No. 6 journal of June 2008 physics pp. 1127 1134 Asynchronous updating of threshold-coupled chaotic neurons MANISH DEV SHRIMALI 1,2,3,, SUDESHNA SINHA 4 and
More informationA Cellular Automata Approach to Population Modeling
A Cellular Automata Approach to Population Modeling Alexa M. Silverman February 24, 2009 Abstract 1 Introduction 1.1 Cellular automata This project provides an agent-based model of the effects of temperature
More informationEvolutionary Dynamics and Extensive Form Games by Ross Cressman. Reviewed by William H. Sandholm *
Evolutionary Dynamics and Extensive Form Games by Ross Cressman Reviewed by William H. Sandholm * Noncooperative game theory is one of a handful of fundamental frameworks used for economic modeling. It
More informationModelling with cellular automata
Modelling with cellular automata Shan He School for Computational Science University of Birmingham Module 06-23836: Computational Modelling with MATLAB Outline Outline of Topics Concepts about cellular
More informationAlana Schick , ISCI 330 Apr. 12, The Evolution of Cooperation: Putting gtheory to the Test
Alana Schick 43320027, ISCI 330 Apr. 12, 2007 The Evolution of Cooperation: Putting gtheory to the Test Evolution by natural selection implies that individuals with a better chance of surviving and reproducing
More informationScale-invariant behavior in a spatial game of prisoners dilemma
PHYSICAL REVIEW E, VOLUME 65, 026134 Scale-invariant behavior in a spatial game of prisoners dilemma Y. F. Lim and Kan Chen Department of Computational Science, National University of Singapore, Singapore
More information16 Period doubling route to chaos
16 Period doubling route to chaos We now study the routes or scenarios towards chaos. We ask: How does the transition from periodic to strange attractor occur? The question is analogous to the study of
More information6. Evolu)on, Co- evolu)on (and Ar)ficial Life) Part 1
6. Evolu)on, Co- evolu)on (and Ar)ficial Life) Part 1 Modelling Social Interac)on in Informa)on Systems hep://davidhales.com/msiis David Hales, University of Szeged dave@davidhales.com 1 Summary How can
More informationIntroduction to Artificial Life and Cellular Automata. Cellular Automata
Introduction to Artificial Life and Cellular Automata CS405 Cellular Automata A cellular automata is a family of simple, finite-state machines that exhibit interesting, emergent behaviors through their
More informationChaotic motion. Phys 750 Lecture 9
Chaotic motion Phys 750 Lecture 9 Finite-difference equations Finite difference equation approximates a differential equation as an iterative map (x n+1,v n+1 )=M[(x n,v n )] Evolution from time t =0to
More informationA Cellular Automata Approach to Population Modeling
A Cellular Automata Approach to Population Modeling Alexa M. Silverman March 31, 2009 Abstract 1 Introduction 1.1 Cellular automata This project provides an agent-based model of the effects of temperature
More informationOutline 1 Introduction Tiling definitions 2 Conway s Game of Life 3 The Projection Method
A Game of Life on Penrose Tilings Kathryn Lindsey Department of Mathematics Cornell University Olivetti Club, Sept. 1, 2009 Outline 1 Introduction Tiling definitions 2 Conway s Game of Life 3 The Projection
More informationChaos & Recursive. Ehsan Tahami. (Properties, Dynamics, and Applications ) PHD student of biomedical engineering
Chaos & Recursive Equations (Properties, Dynamics, and Applications ) Ehsan Tahami PHD student of biomedical engineering Tahami@mshdiau.a.ir Index What is Chaos theory? History of Chaos Introduction of
More informationFrom Last Time. Gravitational forces are apparent at a wide range of scales. Obeys
From Last Time Gravitational forces are apparent at a wide range of scales. Obeys F gravity (Mass of object 1) (Mass of object 2) square of distance between them F = 6.7 10-11 m 1 m 2 d 2 Gravitational
More informationEvolutionary computation
Evolutionary computation Andrea Roli andrea.roli@unibo.it DEIS Alma Mater Studiorum Università di Bologna Evolutionary computation p. 1 Evolutionary Computation Evolutionary computation p. 2 Evolutionary
More informationNonlinear Dynamics. Moreno Marzolla Dip. di Informatica Scienza e Ingegneria (DISI) Università di Bologna.
Nonlinear Dynamics Moreno Marzolla Dip. di Informatica Scienza e Ingegneria (DISI) Università di Bologna http://www.moreno.marzolla.name/ 2 Introduction: Dynamics of Simple Maps 3 Dynamical systems A dynamical
More informationJustine Seastres. Cellular Automata and the Game of Life
Justine Seastres Saint Mary s College of California Department of Mathematics May 16, 2016 Cellular Automata and the Game of Life Supervisors: Professor Porter Professor Sauerberg 2 Contents 1 Introduction
More informationIntroduction to Nonlinear Dynamics and Chaos
Introduction to Nonlinear Dynamics and Chaos Sean Carney Department of Mathematics University of Texas at Austin Sean Carney (University of Texas at Austin) Introduction to Nonlinear Dynamics and Chaos
More informationRandom Boolean Networks and Evolutionary Game Theory
Random Boolean Networks and Evolutionary Game Theory J. McKenzie Alexander Date: 10 December 2002 Abstract Recent years have seen increased interest in the question of whether it is possible to provide
More informationEvolution of Cooperation in Evolutionary Games for Heterogeneous Interactions
Commun. Theor. Phys. 57 (2012) 547 552 Vol. 57, No. 4, April 15, 2012 Evolution of Cooperation in Evolutionary Games for Heterogeneous Interactions QIAN Xiao-Lan ( ) 1, and YANG Jun-Zhong ( ) 2 1 School
More informationhttp://www.ibiblio.org/e-notes/mset/logistic.htm On to Fractals Now let s consider Scale It s all about scales and its invariance (not just space though can also time And self-organized similarity
More informationKalle Parvinen. Department of Mathematics FIN University of Turku, Finland
Adaptive dynamics: on the origin of species by sympatric speciation, and species extinction by evolutionary suicide. With an application to the evolution of public goods cooperation. Department of Mathematics
More informationThe Evolutionary Design of Collective Computation in Cellular Automata
The Evolutionary Design of Collective Computation in Cellular Automata James P. Crutchfield Santa Fe Institute 1399 Hyde Park Road Santa Fe, NM 8751 chaos@santafe.edu Melanie Mitchell Santa Fe Institute
More informationSpatial Evolutionary Games
Spatial Evolutionary Games Rick Durrett (Duke) ASU 4/10/14 1 / 27 Almost 20 years ago R. Durrett and Simon Levin. The importance of being discrete (and spatial). Theoret. Pop. Biol. 46 (1994), 363-394
More informationarxiv:cond-mat/ v4 [cond-mat.soft] 23 Sep 2002
arxiv:cond-mat/0207679v4 [cond-mat.soft] 23 Sep 2002 A Two-Player Game of Life Mark Levene and George Roussos School of Computer Science and Information Systems Birkbeck College, University of London London
More informationCoalescing Cellular Automata
Coalescing Cellular Automata Jean-Baptiste Rouquier 1 and Michel Morvan 1,2 1 ENS Lyon, LIP, 46 allée d Italie, 69364 Lyon, France 2 EHESS and Santa Fe Institute {jean-baptiste.rouquier, michel.morvan}@ens-lyon.fr
More informationOscillatory Motion. Simple pendulum: linear Hooke s Law restoring force for small angular deviations. small angle approximation. Oscillatory solution
Oscillatory Motion Simple pendulum: linear Hooke s Law restoring force for small angular deviations d 2 θ dt 2 = g l θ small angle approximation θ l Oscillatory solution θ(t) =θ 0 sin(ωt + φ) F with characteristic
More informationarxiv: v1 [math.ds] 15 Nov 2014
EVOLUTIONARY GAMES ON GRAPHS AND DISCRETE DYNAMICAL SYSTEMS JEREMIAS EPPERLEIN, STEFAN SIEGMUND, AND PETR STEHLÍK arxiv:1411.4145v1 [math.ds] 15 Nov 2014 Abstract. Evolutionary games on graphs play an
More informationOscillatory Motion. Simple pendulum: linear Hooke s Law restoring force for small angular deviations. Oscillatory solution
Oscillatory Motion Simple pendulum: linear Hooke s Law restoring force for small angular deviations d 2 θ dt 2 = g l θ θ l Oscillatory solution θ(t) =θ 0 sin(ωt + φ) F with characteristic angular frequency
More informationarxiv: v1 [physics.soc-ph] 27 May 2016
The Role of Noise in the Spatial Public Goods Game Marco Alberto Javarone 1, and Federico Battiston 2 1 Department of Mathematics and Computer Science, arxiv:1605.08690v1 [physics.soc-ph] 27 May 2016 University
More informationGame interactions and dynamics on networked populations
Game interactions and dynamics on networked populations Chiara Mocenni & Dario Madeo Department of Information Engineering and Mathematics University of Siena (Italy) ({mocenni, madeo}@dii.unisi.it) Siena,
More informationChapter 2 Simplicity in the Universe of Cellular Automata
Chapter 2 Simplicity in the Universe of Cellular Automata Because of their simplicity, rules of cellular automata can easily be understood. In a very simple version, we consider two-state one-dimensional
More informationSpatial three-player prisoners dilemma
Spatial three-player prisoners dilemma Rui Jiang, 1 Hui Deng, 1 Mao-Bin Hu, 1,2 Yong-Hong Wu, 2 and Qing-Song Wu 1 1 School of Engineering Science, University of Science and Technology of China, Hefei
More informationNote that numerically, with white corresponding to 0 and black to 1, the rule can be written:
Cellular automata We discuss cellular automata as a simple application of MATLAB programming and as an accessible scientific topic of recent interest. You can find a lot of information on the internet.
More informationResearch Article Snowdrift Game on Topologically Alterable Complex Networks
Mathematical Problems in Engineering Volume 25, Article ID 3627, 5 pages http://dx.doi.org/.55/25/3627 Research Article Snowdrift Game on Topologically Alterable Complex Networks Zhe Wang, Hong Yao, 2
More informationGenerative urban design with Cellular Automata and Agent Based Modelling
Generative urban design with Cellular Automata and Agent Based Modelling Nikolay Popov Unitec Institute of Technology, Auckland, New Zealand ABSTRACT: This paper reports on initial findings of a bigger
More informationA Very Brief and Shallow Introduction to: Complexity, Chaos, and Fractals. J. Kropp
A Very Brief and Shallow Introduction to: Complexity, Chaos, and Fractals J. Kropp Other Possible Titles: Chaos for Dummies Learn Chaos in 1 hour All you need to know about Chaos Definition of Complexity
More informationCellular Automata. Jason Frank Mathematical Institute
Cellular Automata Jason Frank Mathematical Institute WISM484 Introduction to Complex Systems, Utrecht University, 2015 Cellular Automata Game of Life: Simulator: http://www.bitstorm.org/gameoflife/ Hawking:
More informationLevels of Ecological Organization. Biotic and Abiotic Factors. Studying Ecology. Chapter 4 Population Ecology
Chapter 4 Population Ecology Lesson 4.1 Studying Ecology Levels of Ecological Organization Biotic and Abiotic Factors The study of how organisms interact with each other and with their environments Scientists
More informationChapter 4 Population Ecology
Chapter 4 Population Ecology Lesson 4.1 Studying Ecology Levels of Ecological Organization The study of how organisms interact with each other and with their environments Scientists study ecology at various
More informationEvolution & Learning in Games
1 / 27 Evolution & Learning in Games Econ 243B Jean-Paul Carvalho Lecture 2. Foundations of Evolution & Learning in Games II 2 / 27 Outline In this lecture, we shall: Take a first look at local stability.
More informationONE DIMENSIONAL CELLULAR AUTOMATA(CA). By Bertrand Rurangwa
ONE DIMENSIONAL CELLULAR AUTOMATA(CA). By Bertrand Rurangwa bertrand LUT, 21May2010 Cellula automata(ca) OUTLINE - Introduction. -Short history. -Complex system. -Why to study CA. -One dimensional CA.
More informationChaotic motion. Phys 420/580 Lecture 10
Chaotic motion Phys 420/580 Lecture 10 Finite-difference equations Finite difference equation approximates a differential equation as an iterative map (x n+1,v n+1 )=M[(x n,v n )] Evolution from time t
More informationCollective Evolution of Turn-taking Norm in Dispersion Games
ollective Evolution of Turn-taking Norm in ispersion Games Akira NAMATAME ept. of omputer Science National efense Academy Yokosuka,239-8686,Japan E-mail: nama@nda.ac.jp http//www.nda.ac.jp/~nama/ Outline
More informationSPATIOTEMPORAL CHAOS IN COUPLED MAP LATTICE. Itishree Priyadarshini. Prof. Biplab Ganguli
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
More informationCellular Automata. and beyond. The World of Simple Programs. Christian Jacob
Cellular Automata and beyond The World of Simple Programs Christian Jacob Department of Computer Science Department of Biochemistry & Molecular Biology University of Calgary CPSC / MDSC 605 Fall 2003 Cellular
More informationIntroduction to Dynamical Systems Basic Concepts of Dynamics
Introduction to Dynamical Systems Basic Concepts of Dynamics A dynamical system: Has a notion of state, which contains all the information upon which the dynamical system acts. A simple set of deterministic
More informationNon-local interactions in spatial evolutionary games
Graduate Theses and Dissertations Iowa State University Capstones, Theses and Dissertations 23 Non-local interactions in spatial evolutionary games Ozgur Hakan Aydogmus Iowa State University Follow this
More informationarxiv: v2 [q-bio.pe] 18 Dec 2007
The Effect of a Random Drift on Mixed and Pure Strategies in the Snowdrift Game arxiv:0711.3249v2 [q-bio.pe] 18 Dec 2007 André C. R. Martins and Renato Vicente GRIFE, Escola de Artes, Ciências e Humanidades,
More informationRevisiting the Edge of Chaos: Evolving Cellular Automata to Perform Computations
Revisiting the Edge of Chaos: Evolving Cellular Automata to Perform Computations Melanie Mitchell 1, Peter T. Hraber 1, and James P. Crutchfield 2 Abstract We present results from an experiment similar
More informationHow Do Things Evolve? How do things change, become more complex, through time?
How Do Things Evolve? How do things change, become more complex, through time? Earth about 4.0 Ga. Ok, we have created the Earth Modeling an Evolutionary System Bifurcation Diagram And we have observed
More informationarxiv: v1 [physics.soc-ph] 11 Nov 2007
Cooperation enhanced by the difference between interaction and learning neighborhoods for evolutionary spatial prisoner s dilemma games Zhi-Xi Wu and Ying-Hai Wang Institute of Theoretical Physics, Lanzhou
More informationEvolution of Diversity and Cooperation 2 / 3. Jorge M. Pacheco. Departamento de Matemática & Aplicações Universidade do Minho Portugal
Evolution of Diversity and Cooperation 2 / 3 Jorge M. Pacheco Departamento de Matemática & Aplicações Universidade do Minho Portugal Luis Santaló School, 18 th of July, 2013 prisoner s dilemma C D C (
More informationTuring s Legacy Continues
15-251: Great Theoretical Ideas in Computer Science Lecture 6 Turing s Legacy Continues Solvable with Python = Solvable with C = Solvable with Java = Solvable with SML = Decidable Languages (decidable
More informationBINARY MORPHOLOGY AND CELLULAR AUTOMATA
BINARY MORPHOLOGY AND CELLULAR AUTOMATA I can't leave this subject without mentioning cellular automata (CAs). Conway's "Game of Life" is an example of a cellular automaton (CA). In each generation (or
More informationCHAPTER 2 FEIGENBAUM UNIVERSALITY IN 1-DIMENSIONAL NONLINEAR ALGEBRAIC MAPS
CHAPTER 2 FEIGENBAUM UNIVERSALITY IN 1-DIMENSIONAL NONLINEAR ALGEBRAIC MAPS The chief aim of this chapter is to discuss the dynamical behaviour of some 1-dimensional discrete maps. In this chapter, we
More informationCellular Automata. ,C ) (t ) ,..., C i +[ K / 2] Cellular Automata. x > N : C x ! N. = C x. x < 1: C x. = C N+ x.
and beyond Lindenmayer Systems The World of Simple Programs Christian Jacob Department of Computer Science Department of Biochemistry & Molecular Biology University of Calgary CPSC 673 Winter 2004 Random
More informationAn Introduction to Evolutionary Game Theory: Lecture 2
An Introduction to Evolutionary Game Theory: Lecture 2 Mauro Mobilia Lectures delivered at the Graduate School on Nonlinear and Stochastic Systems in Biology held in the Department of Applied Mathematics,
More informationbiologically-inspired computing lecture 12 Informatics luis rocha 2015 INDIANA UNIVERSITY biologically Inspired computing
lecture 12 -inspired Sections I485/H400 course outlook Assignments: 35% Students will complete 4/5 assignments based on algorithms presented in class Lab meets in I1 (West) 109 on Lab Wednesdays Lab 0
More informationAll living organisms are limited by factors in the environment
All living organisms are limited by factors in the environment Monday, October 30 POPULATION ECOLOGY Monday, October 30 POPULATION ECOLOGY Population Definition Root of the word: The word in another language
More informationSimple models for complex systems toys or tools? Katarzyna Sznajd-Weron Institute of Theoretical Physics University of Wrocław
Simple models for complex systems toys or tools? Katarzyna Sznajd-Weron Institute of Theoretical Physics University of Wrocław Agenda: Population dynamics Lessons from simple models Mass Extinction and
More informationPhase transitions in social networks
Phase transitions in social networks Jahan Claes Abstract In both evolution and economics, populations sometimes cooperate in ways that do not benefit the individual, and sometimes fail to cooperate in
More informationProblems on Evolutionary dynamics
Problems on Evolutionary dynamics Doctoral Programme in Physics José A. Cuesta Lausanne, June 10 13, 2014 Replication 1. Consider the Galton-Watson process defined by the offspring distribution p 0 =
More information