Cobweb diagram

Construction of a cobweb diagram of a logistic map logistic map showing an attractive fixed point

An animated cobweb diagram of the logistic map showing chaotic behavior for most values ​​of r> 3.57

A cobweb diagram , cobweb plot or Verhulst diagram is a visual tool used in the field of dynamic systems of mathematics to consider the qualitative behavior of one-dimensional iterated functions such as logistic mapping . Using the cobweb diagram, it is possible to infer the long-term status of an initial condition by repeatedly applying the map.

method

For a given function to be iterated , the plot consists of the diagonal and the curve . The following steps are used to show the behavior of a start value . ${\displaystyle f\colon \mathbb {R} \to \mathbb {R} }$ ${\displaystyle x=y}$ ${\displaystyle y=f(x)}$ ${\displaystyle x_{0}}$

1. Find the point on the function curve with the x-coordinate . This point has the coordinates ( ). ${\displaystyle x_{0}}$ ${\displaystyle x_{0},f(x_{0})}$
2. Draw a horizontal line through this point to the diagonal. This point has the coordinates ( ). ${\displaystyle f(x_{0}),f(x_{0})}$
3. Draw a vertical line from this point on the diagonal to the function curve. This point has the coordinates ( ). ${\displaystyle f(x_{0}),f(f(x_{0}))}$
4. Repeat these steps from step 2.

interpretation

On the cobweb diagram, a stable fixed point corresponds to an inward spiral, an unstable fixed point to an outward spiral. It follows from the definition of a fixed point that these spirals have a center at which the diagonal line intersects the function graph. An orbit with period 2 is represented by a rectangle, with larger period cycles forming further, more complex, closed loops. A chaotic orbit shows up as a "filled in" area that shows an infinite number of non-repeating numbers. ${\displaystyle y=x}$

See also

• Cobweb theorem

Individual evidence

1. ^ ^{A b} Ruedi Stoop, Willi-Hans Steeb: Computable chaos in dynamic systems . Birkhäuser Basel, 2006, ISBN 3-7643-7551-5 , p. 8 .