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.
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 x0, apply the following steps.
- Find the point on the function curve with an x-coordinate of x0. This has the coordinates (x0,f(x0)).
- Plot horizontally across from this point to the diagonal line. This has the coordinates (f(x0),f(x0)).
- Plot vertically from the point on the diagonal to the function curve. This has the coordinates (f(x0),f(f(x0))).
- 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. 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.
External links
- Emporia State University - A cobweb plotting applet
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