Simple Chaotic Flow GIF Animations

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.

Case A (157,095 bytes) : dx/dt = y, dy/dt = -x + yz, dz/dt = 1 - y²
Case B (199,857 bytes) : dx/dt = yz, dy/dt = x - y, dz/dt = 1 - xy
Case C (78,515 bytes) : dx/dt = yz, dy/dt = x - y, dz/dt = 1 - x²
Case D (77,876 bytes) : dx/dt = -y, dy/dt = x + z, dz/dt = xz + 3y²
Case E (24,240 bytes) : dx/dt = yz, dy/dt = x² - y, dz/dt = 1 - 4x
Case F (80,564 bytes) : dx/dt = y + z, dy/dt = -x + 0.5y, dz/dt = x² - z
Case G (57,560 bytes) : dx/dt = 0.4x + z, dy/dt = xz - y, dz/dt = -x + y
Case H (82,055 bytes) : dx/dt = -y + z², dy/dt = x + 0.5y, dz/dt = x - z
Case I (45,318 bytes) : dx/dt = -0.2y, dy/dt = x + z, dz/dt = x + y² - z
Case J (37,512 bytes) : dx/dt = 2z, dy/dt = -2y + z, dz/dt = -x + y + y²
Case K (80,325 bytes) : dx/dt = xy - z, dy/dt = x - y, dz/dt = x + 0.3z
Case L (47,827 bytes) : dx/dt = y + 3.9z, dy/dt = 0.9x² - y, dz/dt = 1 - x
Case M (53,162 bytes) : dx/dt = -z, dy/dt = -x² - y, dz/dt = 1.7 + 1.7x + y
Case N (65,440 bytes) : dx/dt = -2y, dy/dt = x + z², dz/dt = 1 + y - 2z
Case O (62,420 bytes) : dx/dt = y, dy/dt = x - z, dz/dt = x + xz + 2.7y
Case P (89,513 bytes) : dx/dt = 2.7y + z, dy/dt = -x + y², dz/dt = x + y
Case Q (60,647 bytes) : dx/dt = -z, dy/dt = x - y, dz/dt = 3.1x + y² + 0.5z
Case R (102,254 bytes) : dx/dt = 0.9 - y, dy/dt = 0.4 + z, dz/dt = xy - z
Case S (57,949 bytes) : dx/dt = -x - 4y, dy/dt = x + z², dz/dt = 1 + x

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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