A cobweb plot, or Verhulst diagram is a visual tool used in the
dynamical system
In mathematics, a dynamical system is a system in which a function describes the time dependence of a point in an ambient space. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water i ...
s field of
mathematics to investigate the qualitative behaviour of one-dimensional
iterated function
In mathematics, an iterated function is a function (that is, a function from some set to itself) which is obtained by composing another function with itself a certain number of times. The process of repeatedly applying the same function ...
s, such as the
logistic map
The logistic map is a polynomial mapping (equivalently, recurrence relation) of degree 2, often referred to as an archetypal example of how complex, chaotic behaviour can arise from very simple non-linear dynamical equations. The map was popula ...
. Using a cobweb plot, it is possible to infer the long term status of an
initial condition under repeated application of a map.
Method
For a given iterated function ''f'': R → R, the plot consists of a diagonal (x = y) line and a curve representing y = f(x). To plot the behaviour of a value
, apply the following steps.
# Find the point on the function curve with an x-coordinate of
. This has the coordinates (
).
# Plot horizontally across from this point to the diagonal line. This has the coordinates (
).
# Plot vertically from the point on the diagonal to the function curve. This has the coordinates (
).
# Repeat from step 2 as required.
Interpretation
On the cobweb plot, a stable
fixed point corresponds to an inward
spiral
In mathematics, a spiral is a curve which emanates from a point, moving farther away as it revolves around the point.
Helices
Two major definitions of "spiral" in the American Heritage Dictionary are:[orbit
In celestial mechanics, an orbit is the curved trajectory of an object such as the trajectory of a planet around a star, or of a natural satellite around a planet, or of an artificial satellite around an object or position in space such a ...]
is represented by a rectangle, while greater period cycles produce further, more complex closed loops. A
chaotic
Chaotic was originally a Danish trading card game. It expanded to an online game in America which then became a television program based on the game. The program was able to be seen on 4Kids TV (Fox affiliates, nationwide), Jetix, The CW4Kids ...
orbit would show a 'filled out' area, indicating an infinite number of non-repeating values.
See also
*
Jones diagram – similar plotting technique
*
Fixed-point iteration
In numerical analysis, fixed-point iteration is a method of computing fixed points of a function.
More specifically, given a function f defined on the real numbers with real values and given a point x_0 in the domain of f, the fixed-point itera ...
– iterative algorithm to find fixed points (produces a cobweb plot)
References
Plots (graphics)
Dynamical systems
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