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Rhombic Periodicity Lattice in Bifurcational Symmetry Breaking Problems
Author(s) -
Konopleva Irina,
Loginov Boris
Publication year - 2003
Publication title -
pamm
Language(s) - English
Resource type - Journals
ISSN - 1617-7061
DOI - 10.1002/pamm.200310496
Subject(s) - mathematics , fredholm determinant , symmetry group , lattice (music) , invariant (physics) , symmetry breaking , bifurcation , explicit symmetry breaking , euclidean geometry , pure mathematics , spontaneous symmetry breaking , mathematical physics , mathematical analysis , physics , quantum mechanics , geometry , nonlinear system , acoustics
General theory of symmetry breaking problems in branching theory is presented in the works [1–3]. Such problems are invariant relative to Euclidean space motions group and their solution which is invariant relatively this group is the rest state or uniform linear motion. At the stability loss cell structure solutions arise, which are invariant relative to the group of the definite period shifts along the definite directions, passing mutually under discrete subgroup transformations that is defined by the symmetry of elementary cell of the periodicity. Thus the Euclidean space motions group is changed by the symmetry of a certain crystallographic group. In this article for the case of 4–dimensional degeneracy of the linearized Fredholm operator abstract bifurcational symmetry breaking problems both for stationary and Andronov‐Hopf bifurcation with planar rhombic periodicity lattice are considered. Applications to hydrodynamical problems are given.