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
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Dynamical systems
Chaos and unpredictability
Strange attractors
Artificial neural networks
Mandelbrot set
Fractals
Iterated function systems
Cellular automata
Dynamical Systems
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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
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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 rules
Few
Nonlinear
Regular
Chaotic
Many
Number of variables
Linear
Complex
Random
Typical Experimental Data
x
Time
Characteristics of Chaos
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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 Map
xn+1 = Axn(1 - xn)
The Hénon Attractor
xn+1 = 1 - 1.4xn2 + 0.3xn-1
General 2-D Quadratic Map
xn+1 = a1 + a2xn + a3xn2 +
a4xnyn + a5yn + a6yn2
2
a9xn
yn+1 = a7 + a8xn +
+
a10xnyn + a11yn + a12yn2
Strange Attractors
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Limit set as t  
Set of measure zero
Basin of attraction
Fractal structure
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Chaotic dynamics
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non-integer dimension
self-similarity
infinite detail
sensitivity to initial conditions
topological transitivity
dense periodic orbits
Aesthetic appeal
Stretching and Folding
Artificial Neural Networks
% Chaotic in Neural Networks
Mandelbrot Set
xn+1 = xn2 - yn2 + a
yn+1 = 2xnyn + b
a
b
Mandelbrot Images
Fractals
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Geometrical objects
generally with non-integer
dimension
Self-similarity (contains
infinite copies of itself)
Structure on all scales
(detail persists when
zoomed arbitrarily)
Diffusion-Limited Aggregation
Natural Fractals
Spatio-Temporal Chaos
Diffusion (Random Walk)
The Chaos Game
1-D Cellular Automata
The Game of Life
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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
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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
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Emergent behavior
Self-organization
Evolution
Adaptation
Autonomous agents
Computation
Learning
Artificial intelligence
Extinction
Summary
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Nature is
complicated
but
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Simple models
may suffice
References
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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