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Bifurcation diagrams
Bifurcation diagrams
A bifurcation diagram can be drawn for chaotic systems (such as the Lorenz and Rössler attractors and the Mandelbrot set). It shows the system changing from periodic behaviour to chaotic behaviour.
A simple map for drawing a bifurcation diagram is the logistic map:
x → a x ( 1 - x )
A zoom into the bifurcation diagram of the logistic map is shown here:
Many properties of the diagram were investigated by Mitchell Feigenbaum and Robert May.
A strange and significant property is the rate at which branches of the tree diagram split. The rate gives a constant called the Feigenbaum constant with the value
δ = 4.669 201 609 102 990 671 853 203 821 578….
The same property appears in many very different chaotic systems and is known as a constant of universality in dynamical systems.
How the diagram is drawn
How the diagram is drawn
An initial value of x between 0 and 1 is chosen. The calculation above is repeated many times, giving a series of values of x. These are plotted on a vertical axis. The process is repeated for a range of values of a control parameter, a (plotted along a horizontal axis).
An example of x values with a = 3.3 are:
0.55, 0.817, 0.494, 0.825, 0.477, 0.823, 0.480, 0.824, 0.479, 0.824, 0.479, ...
With this value of a the x values end up switching between 0.824 and 0.479. These are shown with red and blue marks on a pair of branches in the movie.
A list like this can be made in a spreadsheet as follows:
At different values of a the branching multiplies, even to an infinite number of branches.
The first point at which there are an infinite number of branches is at about
a = 3.5699... .
An x value is never revisited and the behaviour is said to be chaotic.
Example spreadsheet cells
Example spreadsheet cells
A sequence of values xi are shown for a = 4, x0 = 0.1, for the first 100 values and 3,000 values (with a moving average):
Limit cycles
Limit cycles
For different values of the control parameter iterations can be plotted as 'limit cycles'. The system behaves chaotically when a limit cycle does not cycle and never revisits a previously calculated value. (Note that numerically there will a repeat will eventually occur due to a finite resolution in calculations). Four limit cycles are displayed:
How to calculate the Feigenbaum constant, δ
How to calculate the Feigenbaum constant, δ
The Feigenbaum constant is calculated by taking the ratio of distances between successive bifurcations using the values of a. The figure below is used to estimate three bifurcation points at:
a ≈ 3.56875, a ≈ 3.569687
and a ≈ 3.569888.
These three points produce an estimate of δ:
δ ≈ (3.569687 - 3.56875)
÷ (3.569888 - 3.569687)
= 4.662
At a ≈ 3.56995 an infinite number of bifurcations has occurred where the sequence of x values never repeats and is chaotic (that is to within the accuracy of decimal numbers stored by the computer).
Further information is at Wikipedia - Logistic map and Feigenbaum constants.
Slides
Slides
Below are two images from the bifurcation diagram of the logistic map (drawn using this java app.)
Even something as simple as repeatedly adding and multiplying four numbers results in unexpected patterns that have infinite structure to them - an 'edge' of logic. The writer of Ecclesiastes writes:
[God] has made everything beautiful in its time. He has also set eternity in the human heart; yet no one can fathom what God has done from beginning to end.
Beautiful curves in the density of iterated points can be seen in this slide with the following parameter values and x range:
a: 3.5818 to 3.58203
x: 0.797 to 0.856
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