Understanding Complex Systems

Three Tutorials on Mathematical Tools and Their Application

Alfred W. Hubler,
Center for Complex Systems Research
Department of Physics
University of Illinois at Urbana-Champaign

Please use the login: guest and the password:guest
  1. Tutorial I: Chaos and Harmony
    1. Experiment: Chaotic Water Mill
    2. Map Dynamics
      1. Discrete Time Series
      2. Map Dynamics
      3. Regular and Strange Attractors
      4. Density Dynamics: Fixed Points and Noise
      5. Lyapunov Exponent
      6. Chaos
      7. Open Loop Control of Chaos
      8. Modelling of Chaotic Time Series
      9. Self-adjusting dynamics: Harmony and Chaos
      10. Adaptation to the Edge of Chaos
    3. Symbolic Dynamics
      1. Symbol Sequences
      2. Inverting Non-invertible Dynamics
      3. x-Values from a Symbol Sequence
      4. Construction of Epsilon Machines
      5. Prediction with Epsilon Machines
    • Tutorial II: Tree-like Graphs and Fractals
      1. Experiment: Fractal Agglomeration of Ball Bearings in Castor Oil
      2. Fractals
        1. Fractal Dimension
        2. Correlation Dimension
        3. Diffusion Limited Agglomeration
        4. River Networks
        5. Turtle Graphs
    • Tutorial III: Artificial Life - Cellular Automata, Neural Nets, Genetic Algorithms
      1. Experiment: Horizontal Parallel Plate Capacitor partially Filled with Oil
      2. Cellular Automata
        1. Wolfram's CAs
        2. CA Simulation
        3. Game of Life
      3. Neural Nets
        1. A Simple Perceptron
        2. Perceptron Simulation
        3. Back Propagation with Hidden Layer
        4. Hopfield Networks
      4. Genetic Algorithms
        1. Selection, Mutation, and Crossover
Definition:

A Complex System is a System with:

  1. Large Throughput of Energy, Information, Force, .... Through a Well Designed Boundary
  2. Many Parts That Form Emergent Structures (Fractals, Chaos, NNs,GAs, CAs)

Some Complex Systems Paradigms:
  1. Harmony: Self-adjusting Systems Avoid Chaos
  2. Adaptation to the Edge of Chaos
  3. Minimum Resistance: State of Least Resistance is Preferred
  4. Leadership: Agent with the Largest Moment of Surprise Wins Competition
  5. Control of Chaos: Equal Attention to All Relevant Variables => Success
  6. Nonlinear Resonances: Nonlinear Dynamical Systems React Most Sensitiv to their Own Dynamics
  7. The Whole is More Important than the Sum of the Parts

Copyright 2001, Alfred W. Hubler, for this page and all linked pages. All rights reserved.