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{{Short description|Visual representation of an iterated function}}
{{refimprove|date=August 2014}}
[[Image:CobwebConstruction.gif|class=skin-invert-image|thumb|upright=1.2|Construction of a cobweb plot of the logistic map <math>y = 2.8 x (1-x)</math>, showing an [[attracting fixed point]].]]
[[Image:LogisticCobwebChaos.gif|class=skin-invert-image|thumb|upright=1.2|An animated cobweb diagram of the [[logistic map]] <math>y = r x (1-x)</math>, showing [[Chaos theory|chaotic 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 [[dynamical system]]s, a 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 |series=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==

For a given iterated function <math>f:\mathbb{R}\rightarrow\mathbb{R}</math>, the plot consists of a diagonal (<math>x=y</math>) line and a curve representing <math>y = f(x)</math>. To plot the behaviour of a value <math>x_0</math>, apply the following steps.

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

==Interpretation==

On the Lémeray diagram, a stable [[fixed point (mathematics)|fixed point]] corresponds to the segment of the staircase with progressively decreasing stair lengths or to an inward [[spiral]], while an unstable fixed point is the segment of the staircase with growing stairs or an outward spiral. It follows from the definition of a fixed point that the staircases [[Converge (mathematics)|converge]] whereas spirals center at a point where the [[Identity function|diagonal]] <math>y=x</math> 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 name="stoop" />

==See also==
* [[Jones diagram]] – similar plotting technique
* [[Fixed-point iteration]] – iterative algorithm to find fixed points (produces a cobweb plot)



==References==
{{Reflist}}

[[Category:Plots (graphics)]]
[[Category:Dynamical systems]]

== External links ==
{{Sister project links|auto=yes}}{{math-physics-stub}}

Cobweb plot - Revision history

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