Simple Chaotic Flow GIF Animations

[Case A]
The image above is the solution of one of the algebraically simplest examples of a chaotic flow. It and the 18 other cases listed below turned up in a search of 3-dimensional systems of ordinary differential equations with either 5 terms and 2 quadratic nonlinearities or 6 terms and 1 quadratic nonlinearity. All of these cases are algebraically simpler than the classic Lorenz and Rossler examples, each of which has 7 terms. In each case, the y-axis is upward and the object rotates in the x-z plane. To view these animations, you need a browser that supports animated GIF files. The software that was used to generate the images is available. Case A (shown above) is a conservative system; all others are dissipative.  Case D has the unusual combination of time-reversal invariance and dissipation arising from a symmetric attractor/repellor pair.
Ref: J. C. Sprott, Phys. Rev. E 50, R647-R650 (1994) 

Since these cases were discovered, an even simpler example of a chaotic flow was found, and it has been rigorously proved that there can be no simpler example of an ordinary differential equation with a quadratic nonlinearity and chaotic solutions.

Most of the above cases have small basins of attraction and are chaotic only over a small region of their parameter space. The different dynamic regions for the cases with six terms and hence two parameters (cases F through S) are shown here.

Electronic circuits corresponding to the above systems have been constructed and animated by Glen K. from Australia as described on his website.


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