Chaos and Complex Systems

Summary


Topics we Covered

  1. Discrete Dynamical Systems. (Logistic equation). Deterministic chaos, quasiperiodicity, 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.

  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. Grassberger ansatz.

  4. Infinite Sets. Cardinality. Countable vs. uncountable infinitites. Construction of the Cantor Set (infinity - infinity = infinity). Metric spaces.

  5. Random Graphs. Random network models (Erdos-Renyi model). Clustering coefficient, degree distribution, average degree, average path length. Small world networks.

  6. Cellular Automata.

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

  8. Agent based models. Compare and contrast with differential equations. Alife simulation of Balinese rice temples. Politics of simulation.

  9. Genetic Algorithms. Royal Road genetic algorithm. Evolving Cellular Automata. Neutral evolution (genetic drift). Fitness landscapes.

  10. Game Theory. Prisoner's dilemma. Payoff matrices. Nash equlibrium. Evolutionarily Stable Strategy.

  11. Tipping Model. Model of neighborhood segretation. 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. Examples: Ben's river network model; tipping models of segregation; ecological iterated prisoner's dilemma.

  4. Universality. Some features of dynamical systems are universal. These features as 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; parity-conserving and directed percolation in CA transitions; percolation in CA epidemic model.

  5. Politics of Simulation. Political assumptions are made when forming models, and, arguably, certain modeling tools have a political stance deeply embedded within them. Examples: Helmreich's critique of Lansing and Kremer's work; unease about game theory.

  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 ]

Web page maintained by dave@hornacek.coa.edu.