These figures accompany the article ``Observing the symmetry of attractors,'' by Schenker and Swift. Physica D 114 (1998) pp. 315-337. If you don't have the paper, you may send Jim.Swift@nau.edu e-mail requesting a hard copy reprint.

The phase space of the 6 coupled oscillators is 12-dimensional.
Click here for the equations.
The figure shows two views of a 3-dimensional projection of the attractor
into the space V_{1} + V_{0}, which is a
** faithful** representation space of
D_{6} x Z_{2}.

In all the figures, the top view is the representation space V_{1},
and the bottom is a `side view' of V_{1} + V_{0}.
The symmetry group of the dynamical system is
D_{6} x Z_{2}.
The dihedral group D_{6} acts on the top view in the usual way,
and Z_{2}
acts as inversion through the origin in both views.

Note that the bottom view is not a representation space of
D_{6} x Z_{2}, but it ** is **
a reprsentation space of Z_{2} x Z_{2},
where the first Z_{2} is the 180 degree rotation in D_{6}.

The parameter in the problem is gamma. You can choose a figure by clicking its icon, or by clicking the parameter value. You can also look at two figures at a time, showing the bifurcations.

gam = -2.2 gam = -2.3 gam = -2.4 gam = -2.5 gam = -4.2765 gam = -4.2766 gam = -4.5 gam = -4.6

gamma = -2.2 /-2.3 gamma = -2.4 /-2.5 gamma = -4.2765 /-4.2766 gamma = -4.5 /-4.6

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