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 3dimensional 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 yaxis is upward and
the
object rotates in the xz 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 timereversal 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  2x

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,
R647R650
(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.
Back to Sprott's
Fractal Gallery