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| [[Image:CobwebConstruction.gif|thumb|500px|right|Construction of a cobweb plot of the logistic map, showing an attracting fixed point.]]
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| [[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.]]
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| 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.
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| ==Method==
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| 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.
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| # 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>).
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| # Plot horizontally across from this point to the diagonal line. This has the coordinates (<math>f(x_0), f(x_0)</math>).
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| # 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>).
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| # Repeat from step 2 as required.
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| ==Interpretation==
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| 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.
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| ==See also== | |
| * [[Jones diagram]] – similar plotting technique
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| [[Category:Plots (graphics)]] | |
| [[Category:Dynamical systems]]
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