The Science of Complexity

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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

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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 rules Linear Nonlinear
Regular Chaotic
Number of variables Many Few
Complex Random
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Typical Experimental Data
x
Time
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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 Map
xn1 Axn(1 - xn)
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The Hénon Attractor
xn1 1 - 1.4xn2 0.3xn-1
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General 2-D Quadratic Map

• xn1 a1 a2xn a3xn2 a4xnyn a5yn a6yn2
• yn1 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
xn1 xn2 - yn2 a yn1 2xnyn b
a
b
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Mandelbrot Images
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Fractals

• Geometrical objects generally with non-integer
  dimension
• Self-similarity (contains infinite copies of
  itself)
• Structure on all scales (detail persists when
  zoomed arbitrarily)

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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
  dont 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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Langtons 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
  (MT Books, 1993)
• Chaos Demonstrations software
• Chaos Data Analyzer software
• sprott_at_juno.physics.wisc.edu