An Example of Three-Dimensional Chaos

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This Demonstration illustrates the periodic or chaotic behavior and the sensitivity to initial conditions of three-dimensional maps of the form:

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The functions , , and here are nonlinear differential equations with three parameters , , and . For a particular selection of these, you can observe periodic behavior, period doubling, or chaotic behavior. Both the Rössler (map 4) and Lorentz (map 5) systems can be considered as special cases. Try your own set of equations by changing the functions in the program.

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Contributed by: Erik Mahieu (December 2011)
Open content licensed under CC BY-NC-SA


Snapshots


Details

The systems considered here are as follows:

map 1:

map 2:

map 3 (Rössler):

map 4:

map 5 (Lorentz):

Reference

[1] J. L. Casti, "Order in Chaos," chap. 4 in Reality Rules: I, New York: John Wiley & Sons, 1992 p. 355. books.google.com/books/about/Reality_rules.html?id=nPQIuVYBjBQC.



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