Hopf Bifurcation with O(2) Symmetry

Abstract

The object of this chapter is to study Hopf bifurcation with O(2) symmetry in some depth, including a formal analysis—that is, assuming Birkhoff normal form—of nonlinear degeneracies. The most important case, to which most others reduce, is the standard action of O(2) on ℝ². Since this representation is absolutely irreducible the corresponding Hopf bifurcation occurs on ℝ² ⊕ ℝ². We repeat the calculations of XVI, §7(c), in a more convenient coordinate system and in greater detail. In §1 we find that there are two maximal isotropy subgroups, corresponding to standing and rotating waves, as in the example of a circular hosepipe. We also give a brief discussion of nonstandard actions of O(2), for which the standing and rotating waves acquire extra spatial symmetry. In §2 we derive the generators for the invariants and equivariants of O(2) × S¹ acting on ℝ² ⊕ ℝ². In §3 we apply these results to analyze the branching directions of these solutions in terms of the Taylor expansion of the vector field.