Cobweb plot

Construction of a cobweb plot of the logistic map, showing an attracting fixed point.
An animated cobweb diagram of the logistic map, showing chaotic behaviour for most values of r > 3.57.

A cobweb plot, or Verhulst diagram is a visual tool used in the dynamical systems field of mathematics to investigate the qualitative behaviour of one-dimensional iterated functions, such as the logistic map. Using a cobweb plot, it is possible to infer the long term status of an initial condition under repeated application of a map.[1]

Method

For a given iterated function f: R → R, the plot consists of a diagonal (x = y) line and a curve representing y = f(x). To plot the behaviour of a value x_0, apply the following steps.

  1. Find the point on the function curve with an x-coordinate of x_0. This has the coordinates (x_0, f(x_0)).
  2. Plot horizontally across from this point to the diagonal line. This has the coordinates (f(x_0), f(x_0)).
  3. Plot vertically from the point on the diagonal to the function curve. This has the coordinates (f(x_0), f(f(x_0))).
  4. Repeat from step 2 as required.

Interpretation

On the cobweb plot, a stable fixed point corresponds to an inward spiral, while an unstable fixed point is an outward one. It follows from the definition of a fixed point that these spirals will center at a point where the diagonal y=x line crosses the function graph. A period 2 orbit is represented by a rectangle, while greater period cycles produce further, more complex closed loops. A chaotic orbit would show a 'filled out' area, indicating an infinite number of non-repeating values.[2]

See also


References

  1. ↑ Stoop, Ruedi; Steeb, Willi-Hans (2006). Berechenbares Chaos in dynamischen Systemen [Computable Chaos in dynamic systems] (in German). Birkhäuser Basel. p. 8. ISBN 978-3-7643-7551-5. Retrieved August 2014.
  2. ↑ Stoop, Ruedi; Steeb, Willi-Hans (2006). Berechenbares Chaos in dynamischen Systemen [Computable Chaos in dynamic systems] (in German). Birkhäuser Basel. p. 8. ISBN 978-3-7643-7551-5. Retrieved August 2014.
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