Wobbe index: Difference between revisions
changed calorific value in higher calorific value, as Wobbe index is based on gross heating value |
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[[Image:CobwebConstruction.gif|thumb|500px|right|Construction of a cobweb plot of the logistic map, showing an attracting fixed point.]] | |||
[[Image:LogisticCobwebChaos.gif|thumb|500px|right|An animated cobweb diagram of the [[logistic map]], showing [[chaos theory|chaotic]] behaviour for most values of r > 3.57.]] | |||
A '''cobweb plot''', or '''Verhulst diagram''' is a visual tool used in the [[dynamical system]]s field of [[mathematics]] to investigate the qualitative behaviour of one dimensional [[iterated function]]s, 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. | |||
==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 <math>x_0</math>, apply the following steps. | |||
# Find the point on the function curve with an x-coordinate of <math>x_0</math>. This has the coordinates (<math>x_0, f(x_0)</math>). | |||
# Plot horizontally across from this point to the diagonal line. This has the coordinates (<math>f(x_0), f(x_0)</math>). | |||
# Plot vertically from the point on the diagonal to the function curve. This has the coordinates (<math>f(x_0), f(f(x_0))</math>). | |||
# Repeat from step 2 as required. | |||
==Interpretation== | |||
On the cobweb plot, a stable [[fixed point (mathematics)|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 (dynamics)|orbit]] is represented by a rectangle, while greater period cycles produce further, more complex closed loops. A [[chaos theory|chaotic]] orbit would show a 'filled out' area, indicating an infinite number of non-repeating values. | |||
==See also== | |||
* [[Jones diagram]] – similar plotting technique | |||
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[[Category:Plots (graphics)]] | |||
[[Category:Dynamical systems]] |
Revision as of 15:05, 23 January 2014
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.
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, 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.
See also
- Jones diagram – similar plotting technique