Cobweb plot: Difference between revisions

Revision as of 03:39, 20 February 2014

Revision as of 13:29, 19 August 2014

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[[Image:CobwebConstruction.gif|thumb|500px|right|Construction of a cobweb plot of the logistic map, showing an attracting fixed point.]]

[[Image:LogisticCobwebChaos.gif|thumb|500px|right|An animated cobweb diagram of the [[logistic map]], showing [[chaos theory|chaotic]] behaviour for most values of r > 3.57.]]

A '''cobweb plot''', or '''Verhulst diagram''' is a visual tool used in the [[dynamical system]]s field of [[mathematics]] to investigate the qualitative behaviour of one-dimensional [[iterated function]]s, 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.<ref>{{cite book |last1=Stoop |first1= Ruedi |last2=Steeb |first2= Willi-Hans |date=2006 |title=Berechenbares Chaos in dynamischen Systemen |trans_title= Computable Chaos in dynamic systems |url=http://link.springer.com/book/10.1007%2F3-7643-7551-5 |language=german |publisher=Birkhäuser Basel| page=8 |isbn=978-3-7643-7551-5 |accessdate=August 2014 }}</ref>

==Method==

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==Interpretation==

On the cobweb plot, a stable [[fixed point (mathematics)|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 (dynamics)|orbit]] is represented by a rectangle, while greater period cycles produce further, more complex closed loops. A [[chaos theory|chaotic]] orbit would show a 'filled out' area, indicating an infinite number of non-repeating values. <ref>{{cite book |last1=Stoop |first1= Ruedi |last2=Steeb |first2= Willi-Hans |date=2006 |title=Berechenbares Chaos in dynamischen Systemen |trans_title= Computable Chaos in dynamic systems |url=http://link.springer.com/book/10.1007%2F3-7643-7551-5 |language=german |publisher=Birkhäuser Basel| page=8 |isbn=978-3-7643-7551-5 |accessdate=August 2014 }}</ref>

==See also==

* [[Jones diagram]] – similar plotting technique