Hiroki Sayama sayama@binghamton.edu NECSI Summer School 2008 Week 3: Methods for the Study of Complex Systems Introduction / Iterative Maps Hiroki Sayama sayama@binghamton.edu

Introduction

Course objective To provide an introduction to a variety of mathematical concepts and tools for analysis of complex systems Textbook: Bar-Yam, Y. “Dynamics of Complex Systems” (Perseus Books/Westview Press, 1997)

Topics to be covered Iterative maps Stochastic systems Information theory Computation theory Dynamical systems and phase space Analytical tools for dynamical systems PDEs and reaction-diffusion systems Cellular automata Thermodynamics and statistical mechanics Stochastic fields and mean-field approximation Monte Carlo simulations Scaling, fractals and renormalization

Online resource Course slides for the first two days are available at: http://coco.binghamton.edu/NECSI/ Login name: necsi Password: com3sysB

Course structure Monday ~ Thursday Friday 9:00am~5:00pm: Lectures, discussions 6:00pm~8:00pm: Group projects Friday 9:00am~12:00pm: Presentations (+ optional final exam)

Group projects Presentation (~15 min.) + 5-page paper Option 1: Original research Conduct mathematical analysis of a model of complex systems (either existing or original) and report your findings Option 2: Teaching analytical methods Select some analytical method that is not covered in the classes, prepare teaching materials (including illustrative examples and questions) and deliver a lecture Option 3: Problem sets (for individuals) Go through several problems selected from different sections in textbook, write out your work and hand it in (presentation waived in this case)

Examples of complex systems Chemical networks Gene networks Organisms Physiologies Brains Ecosystems Economies Societies Internet

Several characteristics of complex systems Networks of many components Nonlinear interactions Self-organization Structure/behavior that is neither regular nor random Emergent behavior

Four approaches to complexity Nonlinear Dynamics Complexity = No closed-form solution, Chaos Information Complexity = Length of description, Entropy Computation Complexity = Computational time/space, Algorithmic complexity Collective Behavior Complexity = Multi-scale patterns, Emergence

Dynamical Systems

Dynamical systems theory Considers how systems autonomously change along time Ranges from Newtonian mechanics to modern nonlinear dynamics theories Thinks about underlying dynamical mechanisms, not just static properties of observations Forms the theoretical basis for most of complex systems studies

What is a dynamical system? A system whose state is uniquely specified by a finite set of variables and whose behavior is uniquely determined by predetermined rules Simple population growth Simple pendulum swinging Motion of celestial bodies Behavior of two “rational” agents in a negotiation game

Mathematical formulations of dynamical systems Discrete-time model: xt = F(xt-1, t) Continuous-time model: (differential equations) dx/dt = F(x, t) xt: State variable of the system at time t May take “scalar” or “vector” value F: Some function that determines the rule that the system’s behavior will obey (difference/recurrence equations; iterative maps)

Review of Difference Equations

Difference equation and time series xt = F(xt-1, t) produces series of values of variable x starting with initial condition x0: { x0, x1, x2, x3, … } “time series” A prediction made by the above model (to be compared to experimental data)

Linear vs. nonlinear Linear: xt = a xt-1 + b xt-2 + c xt-3 … Right hand side is just a first-order polynomial of variables xt = a xt-1 + b xt-2 + c xt-3 … Nonlinear: Anything else xt = a xt-1 + b xt-22 + c xt-1 xt-3 …

Single-variable vs. multi-variable Single-variable (univariate): Just one equation given for a series {xt} xt = a xt-1 + b xt-22 + c / xt-3 … Multi-variable (multivariate): Multiple equations given to simultaneously describe multiple series {xt}, {yt}, … xt = a xt-1 + b yt-1 yt = c xt-1 + d yt-1

1st-order vs. higher-order Right hand side refers only to the immediate past xt = a xt-1 ( 1 – xt-1 ) Higher-order: Anything else xt = a xt-1 + b xt-2 + c xt-3 … (Note: this is different from the order of terms in polynomials)

Autonomous vs. non-autonomous Right hand side includes only state variables (x) and not t itself xt = a xt-1 xt-2 + b xt-32 Non-autonomous: Right hand side includes terms that explicitly depend on the value of t xt = a xt-1 xt-2 + b xt-32 + sin(t)

Homogeneous vs. non-homogeneous Every term in the right hand side has the same order xt = a xt-1 + b xt-2 + c xt-3 Non-homogeneous: Anything else (typically has constants) xt = a xt-1 + b xt-2 + c xt-3 + d

Things that you should know (1) Non-autonomous, higher-order equations can always be converted into autonomous, 1st-order equations xt-2 → yt-1, yt = xt-1 t → yt, yt = yt-1 + 1, y0 = 0 Autonomous 1st-order equations (iterative maps) can cover dynamics of any non-autonomous higher-order equations too!

Things that you should know (2) Linear equations are analytically solvable show either equilibrium, exponential growth/decay, periodic oscillation (with >1 variables), or their combination Nonlinear equations may show more complex behaviors do not have analytical solutions in general

Iterative Maps

Iterative map Autonomous, 1st-order difference equation: xt = F(xt-1) Equilibrium points (a.k.a. fixed points, steady states) can be obtained by solving xe = F(xe)

Exercise Obtain equilibrium points of the following discrete-time logistic growth model: Nt = Nt-1 + r Nt-1 ( 1 – Nt-1/K )

Cobweb plot A visual tool to study the behavior of 1-D iterative maps Take xt-1 and xt for two axes Draw the map of interest (xt=F(xt-1)) and the “xt=xt-1” reference line They will intersect at equilibrium points Trace how time series develop from an initial value by jumping between these two curves

Exercise Draw a cobweb plot for each of the following models: xt = xt-1 + 0.1, x0 = 0.1 xt = 1.1 xt-1 , x0 = 0.1

Exercise Draw a cobweb plot of the logistic growth model with r=1, K=1, N0=0.1: Nt = Nt-1 + r Nt-1 ( 1 – Nt-1/K )

Stability of equilibrium points The slope of function F (F/x) at an equilibrium point determines whether the system can converge to or diverge from that point F x -1 1 Unstable Stable Unstable With oscillation No oscillation

Exercise Analyze the stability of the non-zero equilibrium point of the logistic growth model with r=1, K=1: Nt = Nt-1 + r Nt-1 ( 1 – Nt-1/K )

Logistic Map and Chaos

Logistic map *The* most famous single-variable nonlinear difference equation xt = a xt-1 ( 1 – xt-1 ) Similar to (but not quite the same as) the discrete-time logistic growth model (missing first xt on the right hand side) Shows quite complex dynamics as control parameter a is varied

Exercise: Equivalence between logistic growth and logistic map Nt = Nt-1 + r Nt-1 ( 1 – Nt-1/K ) This becomes equivalent to the logistic map if we assume: Nt = xt K (1 + r) / r Show that this is correct Determine the relationship between growth rate r in logistic growth models and coefficient a in logistic maps

Exercise Draw cobweb plots of the logistic map for a = 0.5, 1.5 and 2.5 Try a couple of different initial conditions for each case

Period-doubling bifurcations In discrete-time models, “period-doubling” bifurcations may occur when F/x(x=xe) = -1 The equilibrium point xe is about to destabilize in an oscillatory manner Also possible in multi-dimensional continuous-time models (which will not be covered in class)

Exercise xt = F(xt-1) = a xt-1 (1 – xt-1) Obtain the equilibrium points of the logistic map as a function of a: xt = F(xt-1) = a xt-1 (1 – xt-1) Examine their stability and its dependence on the value of a

Summary of stability analysis Range Equil. of a points 0 < a < 1 1 < a < 3 3 < a x = 0 Stable Unstable x = 1-1/a (Doesn’t exist within positive range) Impossible if this was a continuous-time model

What is going on for a > 3? Example: a = 3.2 Period-doubling bifurcation Both equilibrium points lose stability System starts to oscillate with a doubled period

Exercise: Stability analysis of F2(x) That the system flips back and forth between two points means that they should be equilibrium points of a composite function F2(x) (= F(F(x)) ) Obtain the equilibrium points of F2(x) and examine their stability and its dependence on the value of a What is happening at a=3?

General condition for stability of periodic trajectories Consider a periodic trajectory: { x0, x1, x2, …, xt } = { x0, F(x0), F(F(x0)), … Ft(x0) } (xt = x0) This trajectory is stable if & only if: |Ft/x(x=x0) | < 1, or |F/x(x=x0) F/x(x=x1) … F/x(x=xt-1) | < 1

What is going on for a > 3.6? Example: a = 3.8 Chaos The system loses periodicity after a cascade of period doubling events

Drawing bifurcation diagrams using numerical results For systems with periodic (or chaotic) long-term behavior, it is useful to draw a bifurcation diagram using numerical simulation results instead of analytical results Plot actual system states sampled after a long period of time has passed Can capture period-doubling bifurcation by a “set” of points

Cascade of period-doubling bifurcations leading to chaos Doubling periods diverge to  at a ≈ 3.599692…! Period- doubling bifurcation Chaos Period-doubling bifurcation Period-doubling bifurcation Transcritical bifurcation

Discovery of chaos Discovered in early 1960’s by Edward N. Lorenz (in a 3-D continuous-time model) Popularized in 1976 by Sir Robert M. May as an example of complex dynamics caused by simple rules (he used a 1-D discrete-time logistic map)

Chaos in dynamical systems A long-term behavior of a dynamical system that never falls in any static or periodic trajectories Looks like a random fluctuation, but still occurs in completely deterministic simple systems Exhibits sensitivity to initial conditions Can be found everywhere

Reinterpreting chaos Where a period diverges to infinity Periodic behavior with infinitely long periods means “aperiodic” behavior Where (almost) no periodic trajectories are stable No fixed points or periodic trajectories you can sit in (you have to always deviate from your past course!!)

Exercise Draw a more detailed bifurcation diagram for the chaotic regime and see how those chaotic trajectories are structured there Can you find any “order” in chaos?

Exercise What happens if you try plotting cobweb plots and bifurcation diagrams of the logistic map for negative a? Determine the range of a where chaotic dynamics occurs

Characteristics of Chaos

Common mechanism of chaos: Stretch and folding in phase space In chaotic systems, it is generally seen that a phase space is stretched and folded, like dough kneading 1 1 0.2 0.4 0.6 0.8 1 1 1

Example

Other examples of stretching and folding: tent map, saw map 0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1 These maps also have chaotic dynamics if the initial condition is an irrational number What kind of numerical operation is carried out when one applies either of these maps to a number?

Sensitivity to initial conditions Stretching and folding mechanisms always dig out microscale details hidden in a system’s state and expand them to macroscale visible levels This causes chaotic systems very sensitive to initial conditions Think about where a grain in a dough will eventually move during kneading

Lyapunov exponent (1) A quantitative measure of a system’s sensitivity to small differences in initial condition Characterizes how the distance between initially close two points will grow over time | Ft(x0+e) - Ft(x0) | ~ e etl (for large t)

Lyapunov exponent (2) | Ft(x0+e) - Ft(x0) | ~ e etl (for large t) l = limt log = limt log = limt Si=0~t-1 log |F’(xi)| Ft(x0+e) - Ft(x0) e t 1 e0 dx d Ft(x0)

Exercise Plot how the Lyapunov exponent of the logistic map changes over varying a Compare the result with its bifurcation diagram

Summary Iterative maps xt = F(xt-1) are simple yet general discrete-time dynamical systems Stability of trajectories of period n can be analyzed by studying dFn/dx Even simple iterative maps may produce very complex, unpredictable (chaotic) behavior