Fig. 7

Suppose the six-element group \(\varGamma=\mathbb {D}_{3}\) of symmetries of the equilateral triangle acts on \(\mathbb {R}^{2}\), generated by a rotation \(g_{2}\) and a reflection \(g_{1}\) in the x-axis. The group orbit of the point u that is not fixed by any symmetries also has six elements (shown by filled circles), while any group orbit of a point v that is fixed by a symmetry (e.g. \(g_{1}\)) has correspondingly fewer points (shown by open circles) in the group orbit. Bifurcation of equilibria with more symmetry typically leads to several equilibria with less (“broken”) symmetry