Cobweb plot
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]
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Method
For a given iterated function , the plot consists of a diagonal () line and a curve representing . To plot the behaviour of a value , apply the following steps.
- Find the point on the function curve with an x-coordinate of . This has the coordinates ().
- Plot horizontally across from this point to the diagonal line. This has the coordinates ().
- Plot vertically from the point on the diagonal to the function curve. This has the coordinates ().
- 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.[1]
See also
- Jones diagram – similar plotting technique
- Fixed-point iteration – iterative algorithm to find fixed points (produces a cobweb plot)
References
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- Stoop, Ruedi; Steeb, Willi-Hans (2006). Berechenbares Chaos in dynamischen Systemen [Computable Chaos in dynamic systems] (in German). Birkhäuser Basel. p. 8. doi:10.1007/3-7643-7551-5. ISBN 978-3-7643-7551-5.