TY - JOUR

T1 - Symmetry-breaking Hopf bifurcation in anisotropic systems

AU - Silber, Mary

AU - Riecke, Hermann

AU - Kramer, Lorenz

N1 - Funding Information:
We would like to thank I. Rehberg and M. de la Torte Ju/trez for discussing their experiments with us. M.S. acknowledges helpful discussions with D. Estep, M. Krupa and I. Melbourne. This work was supported by NATO collaborative research grant #900276. M.S. acknowledges support from the Institute for Mathematics and its Applications and ONR grant N00014-9 l-J-1257. H.R. acknowledges support from the NSF/AFOSR under grant number DMS-9020289. M.S. and H.R. would like to thank the University of Bayreuth, where much of this work was done, for their hospitality.

PY - 1992/12/30

Y1 - 1992/12/30

N2 - Symmetry-breaking Hopf bifurcation from a spatially uniform steady state of a spatially extended anisotropic system is considered. This work is motivated by the experimental observation of a Hopf bifurcation to oblique traveling rolls in electrohydrodynamic convection in planarly aligned nematic liquid crystals. Symmetry forces four traveling rolls to lose stability simultaneously. Four coupled complex ordinary differential equations describing the nonlinear interaction of the traveling rolls are analyzed using methods of equivariant bifurcation theory. Six branches of periodic solutions always bifurcate from the trivial state at the Hopf bifurcation. These correspond to traveling and standing wave patterns. In an open region of coefficient space there is a primary bifurcation to a quasiperiodic standing wave solution. The Hopf bifurcation can also lead directly to an aperiodic attractor in the form of an asymptotically stable, structurally stable heteroclinic cycle. The theory is applied to a model for the transition from normal to oblique traveling rolls.

AB - Symmetry-breaking Hopf bifurcation from a spatially uniform steady state of a spatially extended anisotropic system is considered. This work is motivated by the experimental observation of a Hopf bifurcation to oblique traveling rolls in electrohydrodynamic convection in planarly aligned nematic liquid crystals. Symmetry forces four traveling rolls to lose stability simultaneously. Four coupled complex ordinary differential equations describing the nonlinear interaction of the traveling rolls are analyzed using methods of equivariant bifurcation theory. Six branches of periodic solutions always bifurcate from the trivial state at the Hopf bifurcation. These correspond to traveling and standing wave patterns. In an open region of coefficient space there is a primary bifurcation to a quasiperiodic standing wave solution. The Hopf bifurcation can also lead directly to an aperiodic attractor in the form of an asymptotically stable, structurally stable heteroclinic cycle. The theory is applied to a model for the transition from normal to oblique traveling rolls.

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U2 - 10.1016/0167-2789(92)90170-R

DO - 10.1016/0167-2789(92)90170-R

M3 - Article

AN - SCOPUS:0001416943

SN - 0167-2789

VL - 61

SP - 260

EP - 278

JO - Physica D: Nonlinear Phenomena

JF - Physica D: Nonlinear Phenomena

IS - 1-4

ER -