Construction of a cobweb plot of the logistic map y = 2.8 x (1-x), showing an attracting fixed point.

An animated cobweb diagram of the logistic map y = r x (1-x), showing chaotic behaviour for most values of r > 3.57.

A cobweb plot, or Verhulst diagram is a visual tool used in the dynamical systems field of mathematics to investigate the qualitative behaviour of one-dimensional iterated functions, 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.^[1]

Method

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

1. Find the point on the function curve with an x-coordinate of ${\displaystyle x_{0}}$. This has the coordinates (${\displaystyle x_{0},f(x_{0})}$).
2. Plot horizontally across from this point to the diagonal line. This has the coordinates (${\displaystyle f(x_{0}),f(x_{0})}$).
3. Plot vertically from the point on the diagonal to the function curve. This has the coordinates (${\displaystyle f(x_{0}),f(f(x_{0}))}$).
4. Repeat from step 2 as required.

Interpretation

On the cobweb plot, a stable 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 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.^[1]

See also

• Jones diagram – similar plotting technique
• Fixed-point iteration – iterative algorithm to find fixed points (produces a cobweb plot)

References

Wikimedia Commons has media related to Cobweb plots.