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Cobweb plot

Visual representation of an iterated function

Wikipedia contributorsAdapted for Fathomly · View authors
1 min read
Source updated September 12, 2026
Image credit is listed at the end of this article.

In mathematics, specifically dynamical systems, a cobweb plot, known also as Lémeray diagram or Verhulst diagram, is a visual tool used to investigate the qualitative behaviour of one-dimensional iterated functions, such as the logistic map. The technique was introduced in 1822 by Adrien-Marie Legendre. Using a cobweb plot, it is possible to infer the long-term status of an initial condition under repeated application of a map.

01Method

For a given iterated function f:\mathbb {R} \rightarrow \mathbb {R}, the plot consists of a diagonal (x=y) line and a curve representing y=f(x). To plot the behaviour of a value x_{0}, apply the following steps.

  1. Find the point on the function curve with an x-coordinate of x_{0}. This has the coordinates (x_{0},f(x_{0})).
  2. Plot horizontally across from this point to the diagonal line. This has the coordinates (f(x_{0}),f(x_{0})).
  3. Plot vertically from the point on the diagonal to the function curve. This has the coordinates (f(x_{0}),f(f(x_{0}))).
  4. Repeat from step 2 as required.
An animated cobweb diagram of the logistic map , showing chaotic behaviour for most values of 3.57" style="vertical-align: -0.338ex; width:8.281ex; height:2.176ex;">.
An animated cobweb diagram of the logistic map , showing chaotic behaviour for most values of 3.57" style="vertical-align: -0.338ex; width:8.281ex; height:2.176ex;">.

02Interpretation

On a cobweb plot, a stable 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 whereas spirals 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.

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Sources and credits

This article is adapted from the Wikipedia article “Cobweb plot”, written by its contributors and licensed under CC BY-SA 4.0. Fathomly has changed the layout, removed citation markers, navigation and maintenance notices, and adjusted punctuation. This adapted version is shared under the same license. For references, see the original article.

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