TY - JOUR

T1 - Bifurcations in systems with Z2 spatio-temporal and O(2) spatial symmetry

AU - Marques, F.

AU - Lopez, Juan

AU - Blackburn, H. M.

N1 - Funding Information:
This work was partially supported by MCYT grant BFM2001-2350 (Spain), NSF Grant CTS-9908599 (USA), the Australian Partnership for Advanced Computing’s Merit Allocation Scheme, and the Australian Academy of Science’s International Scientific Collaborations Program.
Copyright:
Copyright 2019 Elsevier B.V., All rights reserved.

PY - 2004/3/1

Y1 - 2004/3/1

N2 - 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.

AB - 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.

KW - Floquet analysis

KW - Normal forms

KW - Symmetry breaking

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U2 - 10.1016/j.physd.2003.09.041

DO - 10.1016/j.physd.2003.09.041

M3 - Article

AN - SCOPUS:1042265751

VL - 189

SP - 247

EP - 276

JO - Physica D: Nonlinear Phenomena

JF - Physica D: Nonlinear Phenomena

SN - 0167-2789

IS - 3-4

ER -