The Science of Complexity J. C. Sprott Department 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 n Dynamical systems n Chaos and unpredictability n Strange attractors n Artificial neural networks n Mandelbrot set n Fractals n Iterated function systems n Cellular automata

Dynamical Systems n The system evolves in time according to a set of rules. n The present conditions determine the future. n The rules are usually nonlinear. n There may be many interacting variables.

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

Chaos and Complexity Complexity of rules Linear Nonlinear Number of variables Many Few Regular Chaotic Complex Random

Typical Experimental Data Time x

Characteristics of Chaos n Never repeats n Depends sensitively on initial conditions (Butterfly effect) n Allows short-term prediction but not long-term prediction n Comes and goes with a small change in some control knob n Usually produces a fractal pattern

A Planet Orbiting a Star Elliptical Orbit Chaotic Orbit

The Logistic Map x n +1 = Ax n (1 - x n )

The Hénon Attractor x n +1 = x n x n -1

General 2-D Quadratic Map x n+1 = a 1 + a 2 x n + a 3 x n 2 + a 4 x n y n + a 5 y n + a 6 y n 2 y n+1 = a 7 + a 8 x n + a 9 x n 2 + a 10 x n y n + a 11 y n + a 12 y n 2

Strange Attractors n Limit set as t   n Set of measure zero n Basin of attraction n Fractal structure u non-integer dimension u self-similarity u infinite detail n Chaotic dynamics u sensitivity to initial conditions u topological transitivity u dense periodic orbits n Aesthetic appeal

Stretching and Folding

Artificial Neural Networks

% Chaotic in Neural Networks

Mandelbrot Set a b x n +1 = x n 2 - y n 2 + a y n +1 = 2x n y n + b

Mandelbrot Images

n Geometrical objects generally with non-integer dimension n Self-similarity (contains infinite copies of itself) n 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 n Individuals live on a 2-D rectangular lattice and don’t move. n Some sites are occupied, others are empty. n If fewer than 2 of your 8 nearest neighbors are alive, you die of isolation. n If 2 or 3 of your neighbors are alive, you survive. n If 3 neighbors are alive, an empty site gives birth. n If more than 3 of your neighbors are alive, you die from overcrowding.

Langton’s Ants n Begin with a large grid of white squares n The ant starts at the center square and moves 1 square to the east n If the square is white, paint it black and turn right n If the square is black, paint it white and turn left n Repeat many times

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

Summary n Nature is complicated n Simple models may suffice but

References n lectures/complex/ lectures/complex/ n Strange Attractors: Creating Patterns in Chaos (M&T Books, 1993) Strange Attractors: Creating Patterns in Chaos n Chaos Demonstrations software Chaos Demonstrations n Chaos Data Analyzer software Chaos Data Analyzer n