Cobweb plot: Difference between revisions

Revision as of 10:12, 25 November 2024

Revision as of 15:02, 4 December 2024

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{{refimprove|date=August 2014}}

[[Image:CobwebConstruction.gif|thumb|upright=1.2|Construction of a cobweb plot of the logistic map <math>y = x^2+3</math>, showing an [[attracting fixed point]].]]

[[Image:LogisticCobwebChaos.gif|thumb|upright=1.2|An animated cobweb diagram of the [[logistic map]] <math>y = r x (1-x)</math>, showing [[chaosChaos theory|chaotic behaviour]] behaviour for most values of <math>r > 3.57</math>.]]

A '''cobweb plot''', known also as '''Lémeray Diagram''' 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]]. The technique was introduced in the 1890s by E.-M. Lémeray.<ref>{{Cite journal |last=Lémeray |first=E.-M. |date=1897 |title=Sur la convergence des substitutions uniformes. |url=http://www.numdam.org/item/NAM_1898_3_17__75_1.pdf |journal=Nouvelles annales de mathématiques, 3e série. |volume=16 |pages=306–319}}</ref> Using a cobweb plot, it is possible to infer the long -term status of an [[initial condition]] under [[Recurrence relation|repeated application]] of a map.<ref name="stoop">{{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 |language=german |publisher=Birkhäuser Basel| page=8 |isbn=978-3-7643-7551-5 |doi= 10.1007/3-7643-7551-5 }}</ref>

==Method==