Cobweb plot: Difference between revisions

Revision as of 11:26, 3 December 2020

Latest revision as of 07:48, 1 September 2025

(20 intermediate revisions by 18 users not shown)

Line 1:

{{Short description|Visual representation of an iterated function}}

{{refimprove|date=August 2014}}

[[Image:CobwebConstruction.gif|class=skin-invert-image|thumb|300px|rightupright=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|300px|rightupright=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, a field of [[mathematics]] to investigate the qualitative behaviour of one-dimensional [[iterated function]]s, such as the [[logistic map]]. UsingThe atechnique cobwebwas plot,introduced itin is1822 possibleby to infer the long term status of an [[initialAdrien-Marie conditionLegendre]] under 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>

{{Cite journal

| date = 2021

| doi = 10.14708/am.v15i1.7056

| journal = Antiquitates Mathematicae

| last = Rosa

| first = Alessandro

| pages = 3–90

| title = An episodic history of the staircased iteration diagram

| url = https://wydawnictwa.ptm.org.pl/index.php/antiquitates-mathematicae/article/viewArticle/7056

| volume = 15

| url-access= subscription

}}

</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

| date = 2006

| doi = 10.1007/3-7643-7551-5

| isbn = 978-3-7643-7551-5

| language = german

| last1 = Stoop

| first1 = Ruedi

| last2 = Steeb

| first2 = Willi-Hans

| page = 8

| publisher = Birkhäuser Basel

| title = Berechenbares Chaos in dynamischen Systemen

| trans-title = Computable Chaos in dynamic systems

}}

</ref>

==Method==

For a given iterated function ''<math>f'':&nbsp;'''\mathbb{R'''&nbsp;→&nbsp;'''}\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>).

Line 15 ⟶ 44:

==Interpretation==

On the cobwebLémeray plotdiagram, 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 onespiral. It follows from the definition of a fixed point that thesethe spiralsstaircases will[[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)

<!-- Link is broken ~~~~ -->

==References==

{{Commons category|Cobweb plots}}

{{Reflist}}

[[Category:Plots (graphics)]]

[[Category:Dynamical systems]]

== External links ==

{{Sister project links|auto=yes}}

{{math-physics-stub}}