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 1643 1727 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 1854 1912 1
Dr. Ian Malcolm You've never heard of Chaos theory? Non-linear equations? Strange attractors? 2
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
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
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
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
Logistic map x t +1 = Raaa x t (1 x t ) Lord Robert May b. 1936 n t +1 = (birthrate deathrate)[kn t n t 2 ]/k Mitchell Feigenbaum b. 1944 Let x t = n t /k Let R = birthrate deathrate 1. R = 2 LogisticMap.nlogo 2. R = 2.5 3. R = 2.8 Notion of period doubling Notion of attractors 4. R = 3.1 5. R = 3.49 6. R = 3.56 7. R = 4, look at sensitive dependence on initial conditions 7
Period Doubling and Universals in Chaos (Mitchell Feigenbaum) R 1 3.0: period 2 R 2 3.44949 period 4 R 3 3.54409 period 8 R 4 3.564407 period 16 R 5 3.568759 period 32 R 3.569946 period (chaos) Bifurcation Diagram 8
Period Doubling and Universals in Chaos (Mitchell Feigenbaum) R 1 3.0: period 2 R 2 3.44949 period 4 R 3 3.54409 period 8 R 4 3.564407 period 16 R 5 3.568759 period 32 A similar period doubling route to chaos is seen in any one-humped (unimodal) map. R 3.569946 period (chaos) Period Doubling and Universals in Chaos (Mitchell Feigenbaum) R 1 3.0: period 2 R 2 3.44949 period 4 R 3 3.54409 period 8 R 4 3.564407 period 16 R 5 3.568759 period 32 R 3.569946 period (chaos) Rate at which distance between bifurcations is shrinking: R 2 R 1 3.44949 3.0 = R 3 R 2 3.54409 3.44949 = 4.75147992 R 3 R 2 3.54409 3.44949 = R 4 R 3 3.564407 3.54409 = 4.65619924 R 4 R 3 3.564407 3.54409 = R 5 R 4 3.568759 3.564407 = 4.66842831 R lim n +1 R n 4.6692016 n R n +2 R n +1 9
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: 4.6692016 period times 2 faster bifurcations than the is shrinking: previous one. R 2 3.44949 period 4 R 2 R 1 3.44949 3.0 R 3 3.54409 period 8 = R 3 R 2 3.54409 3.44949 = 4.75147992 R 4 3.564407 period 16 R 5 3.568759 This same period rate 32 of 4.6692016 R 3 R 2 3.54409 3.44949 = occurs in any unimodal map. R 4 R 3 3.564407 3.54409 = 4.65619924 R 3.569946 period (chaos) R 4 R 3 3.564407 3.54409 = R 5 R 4 3.568759 3.564407 = 4.66842831 lim R n +1 R n 4.6692016 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
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
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
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
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
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
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
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
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