The Science of Complexity

J. C. SprottDepartment of Physics

University of Wisconsin - Madison

Presented to the

First National Conference on Complexity and Health Care

in Princeton, New Jersey

on December 3, 1997

Outline

Dynamical systems Chaos and unpredictability Strange attractors Artificial neural networks Mandelbrot set Fractals Iterated function systems Cellular automata

Dynamical Systems

The system evolves in time according to a set of rules.

The present conditions determine the future.

The rules are usually nonlinear. There may be many interacting

variables.

Examples of Dynamical Systems The Solar System The atmosphere (the weather) The economy (stock market) The human body (heart, brain, lungs, ...) Ecology (plant and animal populations) Cancer growth Spread of epidemics Chemical reactions The electrical power grid The Internet

Chaos and Complexity

Complexity of rulesLinear Nonlinear

Nu

mb

er o

f va

riab

les

Man

y

F

ew Regular Chaotic

Complex Random

Typical Experimental Data

Time

x

Characteristics of Chaos Never repeats Depends sensitively on initial

conditions (Butterfly effect) Allows short-term prediction

but not long-term prediction Comes and goes with a small

change in some control knob Usually produces a fractal

pattern

A Planet Orbiting a Star

Elliptical Orbit Chaotic Orbit

The Logistic Mapxn+1 = Axn(1 - xn)

The Hénon Attractorxn+1 = 1 - 1.4xn

2 + 0.3xn-1

General 2-D Quadratic Map

xn+1 = a1 + a2xn + a3xn2 +

a4xnyn + a5yn + a6yn2

yn+1 = a7 + a8xn + a9xn2 +

a10xnyn + a11yn + a12yn2

Strange Attractors Limit set as t Set of measure zero Basin of attraction Fractal structure

non-integer dimension self-similarity infinite detail

Chaotic dynamics sensitivity to initial conditions topological transitivity dense periodic orbits

Aesthetic appeal

Stretching and Folding

Artificial Neural Networks

% Chaotic in Neural Networks

Mandelbrot Set

a

b

xn+1 = xn2 - yn

2 + a

yn+1 = 2xnyn + b

Mandelbrot Images

Geometrical objects generally with non-integer dimension

Self-similarity (contains infinite copies of itself)

Structure on all scales (detail persists when zoomed arbitrarily)

Fractals

Diffusion-Limited Aggregation

Natural Fractals

Spatio-Temporal Chaos

Diffusion (Random Walk)

The Chaos Game

1-D Cellular Automata

The Game of Life Individuals live on a 2-D rectangular

lattice and don’t move. Some sites are occupied, others are

empty. If fewer than 2 of your 8 nearest

neighbors are alive, you die of isolation. If 2 or 3 of your neighbors are alive, you

survive. If 3 neighbors are alive, an empty site

gives birth. If more than 3 of your neighbors are

alive, you die from overcrowding.

Langton’s Ants Begin with a large grid of white

squares The ant starts at the center

square and moves 1 square to the east

If the square is white, paint it black and turn right

If the square is black, paint it white and turn left

Repeat many times

Dynamics of Complex Systems Emergent behavior Self-organization Evolution Adaptation Autonomous agents Computation Learning Artificial intelligence Extinction

Summary Nature is complicated

Simple models may suffice

but

References

http://sprott.physics.wisc.edu/ lectures/complex/

Strange Attractors: Creating Patterns in Chaos (M&T Books, 1993)

Chaos Demonstrations software Chaos Data Analyzer software sprott@juno.physics.wisc.edu