Exploring Chaos by Brian Davies

The book: Exploring Chaos: Theory and Experiment provides both a fine introduction to Chaos and in addition, a very useful suite of Java applets which let the reader actually experiment and explore.  These utilities not only include implementations of many of the interesting dynamic functions exhibiting chaos, they also include tools for measuring and detecting chaos.  The latter include Bifurcation maps and Lyapunov exponent graphs.

Iteration Map

The Logistics map (  r*x*(1-x)  ) is likely the most studied simple system exhibiting chaotic behavior.  Here is one of the utilities used to iterate/map the function, showing successive values of the input variable.

Cobweb Diagram

An unusually graphic way to show the same data is a "cobweb diagram", which repeatedly reflects the input value of x about the symetric x=y line to find the next value in the iteration.  When run slowly, this very graphically shows the convergence or lack of convergence to stable limit cycles and the like.  The following is the same data as above in the cobweb format.

Bifurcation Diagrams

As the parameter "r" changes, the iteration displays cycles of stability.  Initially, the iterations converge to a single point.  Then they begin converging to an alternating pair of points.  These "split" further into 4, 8, and so on.  These bifurcations turn out to be a common component of chaotic systems.  Indeed, the separation between bifercations have an interesting ratio, the Feigenbaum Constant.  The ratio is not only a common characteristic of these systems, it actually converges to a constant!

Lyapunov Exponents

Another measure of the "chaoticness" of a system is its Lyapunov exponent.  This measures how divergent a pair of points are over time.  Thus if two initial values to an iteration are very close initially, but diverge quickly to be basically independent, then the Lyapunov exponent would be positive.

Lorenz Equations

The systems above were iterations, or discrete.  Continuous systems are in the form of parametric differential equations.  Indeed, the initial study of chaotic systems was by Edward Lorenz, while studying weather systems.  These showed the extreme sensitivity to input variations which became the hallmark of chaotic systems.  Indeed, it took almost two decades for mathematitions to fully appreciate the fundamental importance of his findings, and to realize the lack of convergence was not simple experimental error!