Bifurcations in systems with Z2 spatio-temporal and O(2) spatial symmetry

This work analyzes the O(2) symmetry breaking bifurcations in systems with an Z2×O(2) symmetry group—where Z2 and O(2) are, respectively, spatio-temporal and spatial symmetries—that are responsible for the transitions from two-dimensional to three-dimensional hydrodynamic states. This symmetry group describes, for example, two-dimensional time-periodic flows past bodies which have reflection symmetry across a wake center plane, such as symmetrical airfoils, circular and square cylinders. Normal form analysis of these systems is based on a joint representation of the monodromy matrix for the half-period-flip map (a composition of a half-period temporal evolution with a spatial reflection) and the spatial O(2) symmetry. There are exactly two kinds of codimension-one synchronous bifurcations in these systems; one preserves the Z2 spatio-temporal symmetry, while the other breaks it. When the Floquet multipliers occur in complex-conjugate pairs (non-resonant with the periodic basic state), there is a single codimension-one bifurcation, and at the bifurcation point two different kind of solutions appear simultaneously: a pair of modulated traveling waves, and a circle of modulated standing waves. At most one of these two types has stable solutions. The symmetries of the system also admit period-doubling bifurcations, but these are codimension-two and the normal form analysis permits specific conclusions regarding these. There are also a number of other codimension-two bifurcations leading to mixed modes and the strong 1:1 and 1:2 resonances. All the codimension-one bifurcations are illustrated with reference to a concrete physical example.