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 V1 + V0, which is a faithful representation space of D6 x Z2.
In all the figures, the top view is the representation space V1,
and the bottom is a `side view' of V1 + V0.
The symmetry group of the dynamical system is
D6 x Z2.
The dihedral group D6 acts on the top view in the usual way,
acts as inversion through the origin in both views.
Note that the bottom view is not a representation space of D6 x Z2, but it is a reprsentation space of Z2 x Z2, where the first Z2 is the 180 degree rotation in D6.
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.