M. Golubitsky and I.N. Stewart
Hopf bifurcation with dihedral group symmetry: coupled nonlinear oscillators
In: Multiparameter
Bifurcation Theory (M. Golubitsky and J. Guckenheimer, eds)
Contemporary Mathematics 56, AMS (1986) 131-173.
We apply the theory of Hopf bifurcation with symmetry developed in Golubitsky
and Stewart (1985) to systems of ODEs having the symmetries of a regular
polygon, that is, whose symmetry group is dihedral. We consider the
existence and stability of symmetry-breaking branches of periodic solutions.
In particular we apply these results to a general system of n nonlinear
oscillators, coupled symmetrically in a ring, and describe the generic
oscillation patterns. We find, for example, that the symmetry can force
some oscillators to have twice the frequency of others. The case of four
oscillators has exceptional features.