## 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 - y
^{2}
- 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 - x
^{2}
- Case D (77,876
bytes) : dx/dt = -y, dy/dt = x + z, dz/dt = xz + 3y
^{2}
- Case E (24,240
bytes) : dx/dt = yz, dy/dt = x
^{2} - y, dz/dt = 1 - 4x
- Case F (80,564
bytes) : dx/dt = y + z, dy/dt = -x + 0.5y, dz/dt = x
^{2}
- 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 + z
^{2}, 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 + y
^{2}
- z
- Case J (37,512
bytes) : dx/dt = 2z, dy/dt = -2y + z, dz/dt = -x + y + y
^{2}
- 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 = 0.9x
^{2} - y, dz/dt =
1 - x
- Case M (53,162
bytes) : dx/dt = -z, dy/dt = -x
^{2} - y, dz/dt = 1.7 +
1.7x + y
- Case N (65,440
bytes) : dx/dt = -2y, dy/dt = x + z
^{2}, 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 + y
^{2}, dz/dt = x
+ y
- Case Q (60,647
bytes) : dx/dt = -z, dy/dt = x - y, dz/dt = 3.1x + y
^{2}
+ 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 + z
^{2}, 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.

Back to
Sprott's Fractal Gallery