### Abstract

This work analyzes the O(2) symmetry breaking bifurcations in systems with an Z_{2}×O(2) symmetry group - where Z_{2} 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 Z_{2} 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.

Original language | English (US) |
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Pages (from-to) | 247-276 |

Number of pages | 30 |

Journal | Physica D: Nonlinear Phenomena |

Volume | 189 |

Issue number | 3-4 |

DOIs | |

State | Published - Mar 1 2004 |

### Keywords

- Floquet analysis
- Normal forms
- Symmetry breaking

### ASJC Scopus subject areas

- Statistical and Nonlinear Physics
- Mathematical Physics
- Condensed Matter Physics
- Applied Mathematics

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## Cite this

_{2}spatio-temporal and O(2) spatial symmetry.

*Physica D: Nonlinear Phenomena*,

*189*(3-4), 247-276. https://doi.org/10.1016/j.physd.2003.09.041