Steady-State bifurcation with 0(3)-Symmetry

Chossat, Pascal; Lauterbach, Reiner; Melbourne, Ian

doi:10.1007/bf00374697

Cited by 94 publications

(153 citation statements)

References 25 publications

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“…First let us remark that some possible quadratic polynomials of the variables x o , x e vanish in the equations. These well-known degeneracies are a consequence of the O(3) symmetry (Chossat et al, 1990). The self-adjoint degeneracy is a supplementary degeneracy which leads to c ee =0 in Eq.…”

Section: Bifurcation Analysis Methodsmentioning

confidence: 99%

GEOFLOW: simulation of convection in a spherical shell under central force field

Beltrame

Travnikov

Gellert

et al. 2006

Nonlin. Processes Geophys.

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Abstract. Time-dependent dynamical simulations related to convective motion in a spherical gap under a central force field due to the dielectrophoretic effect are discussed. This work is part of the preparation of the GEOFLOW-experiment which is planned to run in a microgravity environment. The goal of this experiment is the simulation of large-scale convective motion in a geophysical or astrophysical framework. This problem is new because of, on the one hand, the nature of the force field (dielectrophoretic effect) and, on another hand, the high degree of symmetries of the system, e.g. the top-bottom reflection. Thus, the validation of this simulation with well-known results is not possible. The questions concerning the influence of the dielectrophoretic force and the possibility to reproduce the theoretically expected motions in the astrophysical framework, are open. In the first part, we study the system in terrestrial conditions: the unidirectional Earth's force is superimposed on the central dielectrophoretic force field to compare with the laboratory experiments during the development of the equipment. In the second part, the GEOFLOW-experiment simulations in weightless conditions are compared with theoretical studies in the astrophysical framework's, in the first instance a fluid under a selfgravitating force field. We present complex time-dependent dynamics, where the dielectrophoretic force field causes significant differences in the flow compared to the case that does not involve this force field.

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Section: Bifurcation Analysis Methodsmentioning

confidence: 99%

GEOFLOW: simulation of convection in a spherical shell under central force field

Beltrame

Travnikov

Gellert

et al. 2006

Nonlin. Processes Geophys.

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“…Some additional information about the numbers n(L, H) can be found in [24,53,72]). The case of the equivariant degree without free parameters was discussed by many authors (see, for instance, [74,72,64,68,60,29]).…”

Section: Bibliographical Remarksmentioning

confidence: 97%

A short treatise on the equivariant degree theory and its applications

Balanov

Krawcewicz

Rybicki

et al. 2010

J. Fixed Point Theory Appl.

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The aim of this survey is to give a profound introduction to equivariant degree theory, avoiding as far as possible technical details and highly theoretical background. We describe the equivariant degree and its relation to the Brouwer degree for several classes of symmetry groups, including also the equivariant gradient degree, and particularly emphasizing the algebraic, analytical, and topological tools for its effective calculation, the latter being illustrated by six concrete examples. The paper concludes with a brief sketch of the construction and interpretation of the equivariant degree.Mathematics Subject Classification (2010). Primary 47H11; Secondary 34C60, 37G40, 70F10, 92C15.

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“…Il découle de ces relations que lorsque est pair, S agit trivialement et il existe un terme équivariant quadratique générique dans les équations de bifurcation, tandis que lorsque est impair, les premiers termes nonlinéaires sont cubiques. Dans le cas pair, les branches bifurquées sont transcritiques et de plus les solutions correspondantes sont toujours instables près du seuil (voir [3]). Cependant le problème de convection possède une autre propriété fondamentale qui est liée à la nature de la nonlinéarité dans les équations du problème : le coefficient du terme quadratique s'annule ou est proche de 0, voir par exemple [6].…”

Section: Position Du Problèmeunclassified

“…In general, the plane Fix(H ) contains three axes of symmetry H 0 , H 0 , H 0 ⊂ SO(3), which 'saturates' the quadratic part of (5), (6), see [3] or [2]. Solutions with isotropy type [H ] then result from secondary bifurcations on these axes.…”

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Steady-State bifurcation with 0(3)-Symmetry

Cited by 94 publications

References 25 publications

GEOFLOW: simulation of convection in a spherical shell under central force field

GEOFLOW: simulation of convection in a spherical shell under central force field

A short treatise on the equivariant degree theory and its applications

Une remarque sur les bifurcations avec une singularité quadratique pour les systèmes O(3) invariants

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