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SN~ (~6) lff 2It 3k_~ , 5 ",,x_J {b)

SN~ (~6) lff 2It 3k_~ , 5 ",,x_J {b)

(7a) (7b) (8a) (8b) (8c)

(7a) (7b) (8a) (8b) (8c) 44 Ao(x,y ) = 3/2xoYo 2 - 2xoYlY.1 - xoY2Y_ 2 - Nt3-/2(xlYoY_l+x_lYoYl) + 3~/2-/2(xl y ay_2+x_lYoY2), Al(X,y ) = N/3/2x0Y0Yl - 3"~-/2x0Y2Y_l - 1/2XlYlY_l + 2xay2y. 2 + ~f-6X_l Do(x,y ) = -xo2y 0 - 4XlX_ly o + ~/-6(x_12y2+x12y_2) + "x/-3(xoxlY_l+ Xox_lYl), Dl(x,y) = -~/3xoxly o -3xlxqy 1 + 3x12y.1 + 3~Xox_ly 2, D2(x,y ) = -~x12y 0 - 3~r2X0XlYl + 3xo2y2 . 2.2 The bifurcation of equilibria and their stability The bifurcating equilibria of F fall in general into three classes defined by their isotropy type: Type 1: isotropy type E1=O(2)~7 ~ (axisymmetric solutions). The Z 1 representatives occur on the line {Yo} and are called "pure modes" by Friedrich and Haken because these solutions are just those of the unmixed 5-dimensional bifurcation problem (1=2). Type 2: isotropy type E2=O(2)- (axisymmetric solutions). The Z 2 representatives occur in the plane {x o, Yo} with yo,YO and are called "mixed modes" Friedrich and Haken. These solutions branch off the trivial one and off the type 1 branch (secondary bifurcation). Type 3: isotropy type Z3=D ~ (non-axisymmetric solutions). The Z 3 representatives occur in the space {x 0, Y0, Y2r} with Y2r=Rey2#0. They have not been observed by Friedrich and Haken. In order to allow a "bending back" of the (transcritical) branch of type 1 equilibria, we assume in the following: (H1) Icl

45 stability requires order 5 (Golubitsky, Schaeffer [1982], Golubitsky et al. [1988]). We come back to these solutions at the end of section 4.1. In the next section we state precisely the hypotheses that we need and we give some information on the bifurcation diagrams and the associated dynamics in the low- dimensional spaces Fix(O(2)-), Fix(D2~ 7~) and Fix(D~). In table 1 are listed the branching equations for each type of equilibrium and we give the leading part of the eigenvalues of DxF(X, g) evaluated on each branch. Most of these eigenvalues can be evaluated explicitly thanks to the isotypic decomposition of the representation induced in V by the action of the isotropy subgroup for each solution. One defines the "isotypic decomposition" of a linear group representation as being its decomposition into blocks of equivalent irreducible representations. Since the linearized operator at an equilibrium commutes with the action of its isotropy subgroup, the isotypic decomposition leads to a block decomposition of the associated jacobian matrix (see Golubitsky, Stewart, Schaeffer [1988]). These blocks are low dimensional and allow a direct computation of the eigenvalues, except for type 3 solutions for which one 3×3 matrix remains undecomposable. In the table we list the invariant subspaces resulting from this decomposition and the corresponding eigenvalues (to leading order). We denote the coordinates of the equilibria with a "~" and the eigenvalues for type j branches by c~ ). 1 2 Table 1. Bifurcating equilibria (Y2 is real in the equations for type 3). S EQuations { Eigendirections }: eigenvalues O(2)@Z~. ~L2+cyo+dy02=O {Yl,Y-1}: O, {Xo}: G(ll)=~l+~Yo, {Xl,X_l }: 0/21)=~,1--(1~/2)y0, { Y2,Y-2 } :(s(41)=-3cyo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0(2)- ~.l+13yo+'YXo2=O { xl,Yl } @ { X-l,Y-1 ]: O,

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