Understanding Complex Systems

Three Lectures on Mathematical Tools and their Application

Alfred W. Hubler, a-hubler@uiuc.edu
Center for Complex Systems Research
Department of Physics
University of Illinois at Urbana-Champaign

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  1. Lecture I: Chaos
    1. Experiments: Chaotic Water Mill, 2 Double Pendula, Inlined Air Track with two spring-coupled gliders, bouncing ball, Vibrated Water Droplets
    2. Map Dynamics
      1. Discrete Time Series: Observing Chaos
      2. Map Dynamics: Dynamical Systems Paradigm and Short Term Prediction of Chaos
      3. Regular and Strange Attractors - Stable Unstable Motion
      4. Lyapunov Exponent: Separation of Nearby Trajectories - A Measure of Predictability
      5. Chaos
      6. Density Dynamics: Fixed Points and Noise - Medium and Long Term Prediction of Chaos
      7. Open Loop Control of Chaos - Control of Chaos Paradigm
      8. Modelling of Chaotic Time Series
      9. Resonances of Dynamical Systems - Principle of the Dynamical Key
      10. Self-adjusting dynamics: Harmony and Chaos
      11. Adaptation to the Edge of Chaos

  2. Lecture II: Tree-like Graphs, Fractals, and Networks
    1. Experiment: Fractal Growth of Oil Droplet, Rotated Drum, Thermo-acoustic Resonator
    2. Fractals
      1. Turtle Graphs
      2. Probabelistic Turtle Graphs
      3. Fractal Dimension
      4. Correlation Dimension
      5. Diffusion Limited Agglomeration
      6. Eden Networks
    3. Neural Nets
      1. A Simple Perceptron
      2. Perceptron Simulation
      3. Back Propagation with Hidden Layer
      4. Hopfield Networks

  3. Lecure III: Artificial Life - Cellular Automata, Genetic Algorithma, Symbolic Dynamics
    1. Experiment: Horizontal Parallel Plate Capacitor partially Filled with Oil, hardware implementation of a CA, Thermo-acoustic Resonantor
    2. Cellular Automata
      1. Wolfram's CAs
      2. Statistical Properties of CAs
      3. Reconstructing CAs from Experimental Data: Probabelistic CAs
      4. Specific Properties of Certain Cellular Automata
      5. Specific Two Dimensional Cellular Automata Game of Life Multi-Color Cellular Automata
    3. Genetic Algorithms
      1. Selection, Mutation, and Crossover
    4. Symbolic Dynamics
      1. Symbol Sequences
      2. Inverting Non-invertible Dynamics
      3. x-Values from a Symbol Sequence
      4. Properties of Minimal Markov Models
      5. Prediction with Minimal Marlov Models
  4. Applications: Non-equilibrium Materials, High-capacity Condensator, Fractal Absorbers, Resonance Spectroscopy, Chaotic Power Lines


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. Dynamical Systems Paradigm: When a System is Unstable or Chaotic, Discrete Master Equations privide more Accurate Models than the Correspnding Differential Equations (A. Gerig and A. Hubler, preprint)

  2. Harmony: Self-adjusting Systems Avoid Chaos => Adaptation to the Edge of Chaos
    P. Melby, J. Kaidel, N. Weber, A. Hubler, Adaptation to the Edge of Chaos in the Self-Adjusting Logistic Map, Phys.Rev.Lett 84 5991-5993 (2000)
    P. Melby, N. Weber, A. Hubler, Robustness of Adaptation in Controlled Self-adjusting Chaotic Systems, Phys. Fluctuation and Noise Lett. 2, L285-L292 (2002)

  3. Minimum Resistance: State of Least Resistance is Preferred
    M. Dueweke, U. Dierker, A. Hubler, Self-assembling Electrical Connections Based on the Principle of Minimum Resistance, Phys.Rev.E 54, 496-506 (1996)
    M. Sperl, A Chang, N. Weber, A. Hubler, Hebbian Learning in the Agglomeration of Conducting Particles, Phys.Rev.E. 59, 3165-3168 (1999)
    D. Smyth, A. Hubler, A Conductivity-Dependent Phase Transition from Closed-Loop to Open-Loop Dendritic Networks, Complexity 9, 56-60(2003). Joseph K. Jun and Alfred W. Hubler, Formation and structure of ramified charge transportation networks in an electromechanical system, PNAS 102, 536–540 (2005).

  4. Prediction of Chaos: Long Term Behavior Originates at Singular Points (V. Strelioff and A. Hubler, preprint)

  5. Control of Chaos: Equal Attention to All Relevant Variables => Success
    A. Hubler, Adaptive Control of Chaotic Systems, Helv.Phys,Acta 62,343 (1989)
    B. Plapp, A. Hubler, Nonlinear Resonances and Suppression of Chaos in the rf-Biased Josephson Junction, Phys.Rev.Lett 65, 2303-2306 (1990)

  6. Leadership: Agent with the Largest Moment of Surprise Wins Competition
    A. Hubler, D. Pines, Prediction and Adaptation in an Evolving Chaotic Environment, in Complexity: From Metapher to Reality, G. Cowan, D. Pines, G. Meltzer (Eds.), Adison-Wesley, 1994, 42 pages
    D. Pierre, A. Hubler, A Theory for Adaptation and Competition Applied to Logistic Map Dynamics, Physica D 75, 343-360 (1994)

  7. Nonlinear Resonances: Nonlinear Dynamical Systems React Most Sensitiv to their Own Dynamics
    G. Foster, A. Hubler, Robust and Efficient Interaction with Complex Systems, Proceedings of 2003 IEEE International Conference on Systems, Man & Cybernetics, 2029-2034(2003)
    J.Xu, A. Hubler, Enhanced Diffraction Pattern from a Fibonnaci Chain, Phys.Rev.B 67, 184202(2003)
    A. Hubler, U. Kuhl, R. Wittmann, T. Nagata, Sharp Diffraction Peaks from Chaotic Structures, CHAOS 7, 577-589 (1997)
    C. Wargitsch, A. Hubler, Resonances of Nonlinear Oscillators, Phys.Rev.E 51 1508-1519 (1995)
    Glenn Foster, Alfred W. Hübler, Karin Dahmen, Resonant forcing of multi-dimensional chaotic map dynamics, to appesr in Phys. Rev. E 2007
    G. Foster, A. Hubler, K. Dahmen, Resonance Spectroscopy with Chaotic Forcing Functions, preprint 2007

  8. Clock Paradigm: The fast clock locks the slow clock.

  9. Quatization of Emergent Structures: Dissipative Wave Particle Systems tend to have Attractors which are Separated in Space.

  10. The Whole is More than the Sum of the Parts: When the Top-Down and the Bottom-Up Sequences of Symmetry Breakings Match the Must Stable Structures Emerge.
    Alfred W. Hubler, Predicting Complex Systems with a Holistic Approach, Complexity 10, 11-16 (2005)

Copyright 2007, Alfred W. Hubler, for this page and all linked pages. All rights reserved.