Chaos and Complex Systems

Fall 2004

Summary


Topics we Covered

  1. Discrete Dynamical Systems. (Logistic equation). Deterministic chaos, attractors and repellors, sources and sinks, sensitive dependence on initial conditions (butterfly effect), bifurcations, orbits, fixed point, periodic points, conditions for stability and instability, strange attractors, Lyapunov exponent.

  2. Universality of the period-doubling route to chaos, Feigenbaum's constant.

  3. Fractals. Power-laws, scale-free behavior, self-similarity. Box-counting dimension. Difficulty of estimating dimension. Averages that "don't exist." St. Petersberg coin-flipping game.

  4. Infinite Sets. Cardinality. Countable vs. uncountable infinities. Construction of the Cantor Set (infinity - infinity = infinity). Borges' "The Aleph."

  5. Complex Graphs. Random network models (Erdos-Renyi model). Clustering coefficient, degree distribution, average degree, average path length. Small world networks. Scale-free networks. Empirical studies of networks.

  6. Cellular Automata.

  7. Random Boolean Networks. (NK-Model, Kauffman Network). Slow growth of number of attractors. Alleged connections with gene regulation networks.

  8. Agent based models. Compare and contrast with differential equations. Alife simulation of Balinese rice temples. Politics of simulation. Simulation of "student's dilemma."

  9. Minority Game. Simple model of strategy, emergence of spontaneous "cooperation."
  10. Tipping Model. Model of neighborhood segregation. Strong history dependence.


Themes and Big Ideas

  1. Vocabulary. These various ideas are used, to varying degrees, in many other fields.

  2. Different styles of math/physics/computer science.

  3. History Matters. The world is more like an egg carton than a salad bowl. Tipping model.

  4. Universality. Some features of dynamical systems are universal. These features ar quantitatively the same for lots of different dynamical systems and can (and have) been measured experimentally. Examples: period-doubling in the logistic equation and Mandlebrot set.

  5. Politics of Simulation. Political assumptions are made when forming models, and, arguably, certain modeling tools have a political stance deeply embedded within them. One must be mindful of the power relations among those being modeled, doing the modeling, and using the results of the model. Example: Helmreich's critique of Lansing and Kremer's work.

  6. Interplay between order and disorder.

    1. Simple, deterministic systems can produce random behavior. Example: logistic equation; three-body problem.

    2. Complicated, large systems can have relatively simple dynamics. Example: Random Boolean networks with K=2 have only sqrt(N) attractors.

    3. Fractals can be produced by deterministic and stochastic systems. Examples: making snowflakes via a deterministic rule; flipping coins to build a Sierpinski triangle.

  7. Models versus Reality. Modeling is fun; how, and to what extent, do we connect models with reality? Do we need to?


[ Dave ] [ Chaos and Complex Systems ] [ COA ]

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