Intro to Chaos and Fractals: Summary

Big Ideas

• Chaos: A deterministic system that is bounded and aperiodic and shows sensitive dependence on initial conditions.

• Fractals: Geometric objects that are self-similar across many scales. Usually have non-integer dimension.

• A simple deterministic system can produce unpredictable results. Example: Logistic equation, Lorenz equations.

• A simple deterministic system can produce complicated, ordered behavior, e.g. period 32 or period 17 cycles in logistic equation.

• Mathematical models are sometimes not used for prediction, but instead to gain a qualitative feel for a system.

• Math and physics are living, human endeavors. Math and physics are, in a variety of ways, informed by art, literature, aesthetic concerns.

• Some features of chaotic transitions are universal. Example: Feigenbaum number from the logistic equation is the same as the number measured experimentally in turbulence experiments.

• Fractals can be produced by simple iterated processes. Example: forming fractals by successive removals.

• Fractals can also be produced by random processes. Example: "The Chaos game."

• Strange Attractors: Attractors of dynamical systems that are of a dimension smaller than the phase space, and often fractal. Motion on the attractor, however, is chaotic. This is an example of structure or pattern within chaos or disorder.

• Julia sets: The set of all initial conditions that do not fly off to infinity for a particular iterated function.

• The Mandelbrot set: an encyclopedia of Julia sets. The Mandlebrot set is the set of all C values for which the Julia set of f(x) = x^2 + C is connected. Equivalently, The M-set is the set of all C values for which the orbit of 0 under f(x) = x^2 + C is bounded.

Really Big Idea

• Order? Disorder? Order can give rise to disorder and disorder can give rise to order.

Broader Implications of Chaos + Fractals?

• Definitely started a small trend toward interdisciplinarity. More common now for mathematicians, physicists, biologists, economists, etc., to talk to each other and collaborate.

• Helped return physicists' attention to everyday-scale phenomena, as opposed to the very small, very big, very fast, etc.

• Provided some new methods: phase space embedding, determining dimensions. These methods have proved useful in variety of fields.

• Important lessons about "causes" of order and pattern and disorder. What needs explaining? What sort of explanations should we seek?

• Chaos and fractals provide a language that is appropriate for describing and thinking about a wide range of natural phenomena.

• Chaos is still a relatively minor sub-field of physics. Chaos was not a revolution in the sense that Quantum Mechanics or Evolution or Plate Tectonics. Chaos is not really a theory.

• Nevertheless, chaos and fractals are fun (at least to many people) and are an important and still growing area of study.

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