The Science of Complexity

1. 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

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

3. 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.

4. 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

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

6. Typical Experimental Data x Time

7. 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

8. A Planet Orbiting a Star Elliptical Orbit Chaotic Orbit

9. The Logistic Map xn+1 = Axn(1 - xn)

10. The Hénon Attractor xn+1 = 1 - 1.4xn2 + 0.3xn-1

11. General 2-D Quadratic Map xn+1 = a1 + a2xn + a3xn2 + a4xnyn + a5yn + a6yn2 yn+1 = a7 + a8xn + a9xn2 + a10xnyn + a11yn + a12yn2

12. 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

13. Stretching and Folding

14. Artificial Neural Networks

15. % Chaotic in Neural Networks

16. Mandelbrot Set xn+1 = xn2 - yn2 + a yn+1 = 2xnyn + b a b

17. Mandelbrot Images

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

19. Diffusion-Limited Aggregation

20. Natural Fractals

21. Spatio-Temporal Chaos

22. Diffusion (Random Walk)

23. The Chaos Game

24. 1-D Cellular Automata

25. 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.

26. 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

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

28. Summary • Nature is complicated • Simple models may suffice but

29. 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 References