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 (mathematics), function describes the time dependence of a Point (geometry), point in an ambient space. Examples include the mathematical models that describe the swinging of a ...
s field of
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern 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 is ...
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 popular ...
. Using a cobweb plot, it is possible to infer the long term status of an
initial condition
In mathematics and particularly in dynamic systems, an initial condition, in some contexts called a seed value, is a value of an evolving variable at some point in time designated as the initial time (typically denoted ''t'' = 0). For ...
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 as a p ...]
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 CW4Kid ...
orbit would show a 'filled out' area, indicating an infinite number of non-repeating values.
See also
*
Jones diagram
A Jones diagram is a type of Cartesian graph developed by Loyd A. Jones in the 1940s, where each axis represents a different variable. In a Jones diagram opposite directions of an axis represent different quantities, unlike in a Cartesian graph wh ...
– 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 iterat ...
– iterative algorithm to find fixed points (produces a cobweb plot)
References
Plots (graphics)
Dynamical systems
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