Chaotic Dynamical Systems

Course Summary

Fall 1999

  1. Finite Difference Equations
    1. Numerical Iteration.
    2. Graphical Iteration ("Cobweb Diagrams").
    3. Chaos and Sensitive Dependence on Initial Conditions.
    4. Stability of periodic orbits.
    5. Bifurcation Diagrams.
    6. Universality of Feigenbaum Constant.

  2. Non-numerical Method to Understand Logistic Map
    1. Mixing: Another way to think about what a chaotic map does.
    2. Topological Conjugacy. Show that f and g are conjugate. Then f and g share properties such as number of fixed points and sdic.
    3. Geometric Views. How does cinnamon move around when dough is kneaded?

  3. Boolean Networks
    1. Strings and loops: Dynamics not that interesting.
    2. Networks: Cycles, fixed points, transient behavior.
    3. Random Boolean Networks: N nodes, each with K connections. The connections and the boolean functions are randomly chosen. If K=N there will be around N/3 cycles in the network. If K=1 there will be almost no cycles. But, if K = 2, there will be around Sqrt(N) cycles. Kaufmann has suggested this may be of relevance to gene regulation.

  4. Cellular Automata
    1. CAs are boolean networks laid out on a grid.
    2. Very simple systems that make neat patterns.
    3. Used to study a wide range of systems.

  5. Fractals
    1. One definition of dimension D: "Bulk"= cl^D, where l is the length of the object.
    2. Box counting dimension.
    3. Fractals can be produced by regular processes. E.g., snowflake exercise.
    4. Fractals and also be produced by stochastic processes. E.g., "the chaos game" produced a Sierpinsky triangle.
    5. The boundaries between different basins of attractions are sometimes fractal. E.g., magnetic pendulum.

  6. Agent-Based Simulation
    1. One traditional approach to modeling: Differential Equations. Eg., Lotka-Volterra systems.
    2. A newer approach to modeling: program a bunch of "agents" -- e.g., grass, sheep wolves -- each obeying some set or rules.

  7. Genetic Algorithms
    1. Natural Evolution: a highly stochastic dynamical system.
    2. Genetic Algorithms: a computer program containing a population of rules that evolve. The rules are selected according to some fitness criteria, rules are then recombined and/or mutated. E.g., evolving cellular automata.
    3. The success of the GA depends on, among other things, the "fitness landscape", how many locally optimal solutions the GA must "travel across" to find the globally optimal solution. However, fitness landscapes are generally not a useful way to think of natural evolution.

  8. Quantifying Uncertainty
    1. Liapunov exponent: average amount of stretching or folding. Positive liapunov exponent indicates chaos.
    2. Information Theory: a broadly applicable mathematical approach to quantifying the uncertainty associated with a probability distribution.
    3. Entropy Rate: For a sequence of symbols, the entropy rate measures the average uncertainty of the next symbol given that an infinite number of symbols have already been observed.
    4. The entropy rate of the logistic equation f(x)=4x(1-x) is 1. The logistic equation at r=4 is as random as a coin toss.

[Dave] [Chaotic Dynamics] [COA]

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