August 12, 1997

(Revised July 9, 2004)

On the sci.fractals newsgroup,
Arne
Dehli Halvorsen proposed a 3-D system of chaotic flows that are
symmetric
with respect to cyclic interchanges of *x*, *y*, and *z*.
A
large number of such systems were examined, and their symmetric
attractors
are available as 3-D anaglyphs.

One example of such a system is

d*x*/d*t* = - *ax* - 4*y* - 4*z* - *y*^{2}

d*y*/d*t* = - *ay* - 4*z* - 4*x* - *z*^{2}

d*z*/d*t* = - *az* - 4*x* - 4*y* - *x*^{2}

A bifurcation diagram of the above system, in which successive
maximum
values of *x* are plotted versus *a*, was calculated
using
the
BASIC code SYMMETRY.BAS and compiled
for
DOS
as SYMMETRY.EXE. The result is
shown
below:

The diagram shows a period-doubling route to chaos as *a*
is
decreased,
with a fully chaotic region at about *a* = 1.27. Eight
different
initial conditions chosen randomly over a cube with a sides of
length
10
centered on the origin were used for each value of *a*.
There
appear
to be multiple basins of attraction for some values of *a*.
Below
is an anaglyphic view of the attractor for *a* = 1.27 (use
red/blue
glasses) :

Recently, Rene Thomas in *Int. J. Bifurcation and Chaos* 9,
1889-1905
(1999) proposed an even simpler class of system that possesses the
same
symmetry:

d*x*/d*t* = - *ax* + *f*(*y*)

d*y*/d*t* = - *ay* + *f*(*z*)

d*z*/d*t* = - *az* + *f*(*x*)

where *f* is one of a variety of nonlinear functions.
The
conservative limit of this sytstem with *a* = 0 also has
chaotic
solutions.
A particularly interesting and elegant case is the one with *f*(*x*)
= sin(*x*), which has an infinite number of fixed points
arranged
on a 3-D lattice. The chaotic trajectory percolates within
this
lattice
of unstable steady states. He calls this "Labyrinth Chaos".

Both of these systems, and many more, are described in J.
C. Sprott*, Chaos and Time-Series Analysis*, Oxford
University
Press
(2003) and in J.
C. Sprott, *Elegant Chaos*, World Scientific (2010).

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