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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
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Outline
Dynamical systems Chaos and unpredictability Strange attractors Artificial neural networks Mandelbrot set Fractals Iterated function systems Cellular automata
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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.
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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
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Chaos and Complexity
Complexity of rulesLinear Nonlinear
Nu
mb
er o
f va
riab
les
Man
y
F
ew Regular Chaotic
Complex Random
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Typical Experimental Data
Time
x
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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
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A Planet Orbiting a Star
Elliptical Orbit Chaotic Orbit
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The Logistic Mapxn+1 = Axn(1 - xn)
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The Hénon Attractorxn+1 = 1 - 1.4xn
2 + 0.3xn-1
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General 2-D Quadratic Map
xn+1 = a1 + a2xn + a3xn2 +
a4xnyn + a5yn + a6yn2
yn+1 = a7 + a8xn + a9xn2 +
a10xnyn + a11yn + a12yn2
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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
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Stretching and Folding
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Artificial Neural Networks
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% Chaotic in Neural Networks
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Mandelbrot Set
a
b
xn+1 = xn2 - yn
2 + a
yn+1 = 2xnyn + b
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Mandelbrot Images
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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)
Fractals
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Diffusion-Limited Aggregation
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Natural Fractals
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Spatio-Temporal Chaos
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Diffusion (Random Walk)
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The Chaos Game
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1-D Cellular Automata
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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.
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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
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Dynamics of Complex Systems Emergent behavior Self-organization Evolution Adaptation Autonomous agents Computation Learning Artificial intelligence Extinction
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Summary Nature is complicated
Simple models may suffice
but
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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 [email protected]