5 years ago

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

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

Dynamics near Steady

Dynamics near Steady State Bifurcations in Problems with Spherical SymmeL,'y. Reiner Lautelbach Abstract We give a complete description of the dynamics near a bifurcation point where spontaneous symmetry breaking from an 0(3) invariant state occurs. The main hypotheses is that the kernel of the linearized equation is the (natural) irreducible seven dimensional representation of 0(3), D Inlroduction In this note we investigate the complete dynamics near a symmetry breaking bifurcation for problems with spherical symmetry. We assume that the kernel of the Iinearization is irreducible under a natural action of 0(3) and moreover it is a low dimen- sional representation, By this we mean, that the dimension is less or equal to seven. This problem (even up to dimension Q) was also studied by Fiedter and Mischaikov [1989], Their method is based on Conley's approach to connection problems. They use a non- equivafiant version of Conley index and they need some algebraic topological infer- nations on coset spaces of O(3)/G where G is an isotropy subgroup with respect to the given representation. Our approach just uses some informations on fixed point spaces which can be obtained in a rather straightforward manner. Our results look similar to theirs, however due to different methods the results are not identical. A profound compa- rison of the differences is contained in their paper, so I just would like to stress the main points: the heteroclinic solutions obtained in the paper by Fiedler and Mischaikov are stable under symmetry breaking perturbations of the problem, ours are not. In our approach the spatial symmetry of (some of) the heteroclinic orbits is known, their approach does not give informations on the symmetry along the connecting orbit. I would like to thank Bernold Fiedler for several discussions on the subject and for keeping me informed on the progress of his joint work with K. Mischaikov. II) Bifurcation with Symmetry In this section we describe the setup of our problem, More detailed information can be found in Henry [1984] and Golubitsky, Stewart and Schaeffer [lg88]. Consider the differential equation (2.1) u t - G(u.X), where (i) X, Y are Banach spaces, where X c Y is densely embedded, (ii) G:X x R .... Y has the form G(u.X) - A(X)u + f(u).)

257 0ii) A(),)-A 0 + B(X) is a hotomorphic family of type (A). Kato [1976]. where A o is sectorial, B(.) is a family of bounded linear operators X °~- X, ~ ~[0,1) and f.-X~-.-X is a map, such that for all x c ~ the above equation generates a semiflow on an intermediate space X ~' (iv) r(O.x)-0 V), ~ ~. Moreover we assume that we have linear actions of 0(3) on X and Y, such that for all ), c a G(..X) is equivariant with respect to these actions. A typical example of such a situation is a reaction diffusion equation on a spherical domain in R 3, where all the coefficients just depend on the radius, but not on the angle. Other applications we have in mind are buckling of spherical shells (see Knightly and Sather [1980]) or the spherical Benard problem for a fluid between two concentric spheres. We will not specify the assumptions which are necessary to get a similar functional analytic set up, see Chossat [1979]. A typical PDE satisfying (iii) would be a semilinear parabolic equa- tion, where the linear part is strongly elliptic, see Henry [1984] for more details. The fact that A(k) is holomorphic family implies, that the eigenvalues depend smoothly on ),. Due to assumption (iv) u-O is an equilibrium of (21) for all X~R. We assume that the stability of this equilibrium changes as the parameter ), is varied through ),-0. In order to study the branching of solutions near (x,u)-(O,0)~RxX and the local dynamics near the bifurcation point we reduce the system via the center manifold to a finite dimensional problem. The precise requirements for this reduction may be found in Henry [1081] chapters 5 and 6. Henry gives also a method to construct the center ma- nifold. In order to be able to use the center manifold theory we make the following assumptions concerning A_.B(O) (v) the spectrumUo (Ao+ B(O)) f'l ilR- {0}, (vi) 0 is an isolated point in o(A0,-B(O))~ (vii) A0.B(0) is Fredholm of index O. Since A o is sectorial and B(0) is relatively bounded with respect to A o. conditions (v) and (vi) imply that the other parts of the of o(Ao,-B(O) ) are bounded away from the imaginary axis. It is clear that ker (A o + B(O))is invariant under the action of 0(3) and therefore it may be decomposed in sum of irreducible representations. Concerning the action of 0(3) we assume (viii) ker(A0~-B(O))is an absolutely irreducible representation of 0(3). Assuming (i) - (viii) it is possible to reduce the flow to an invariant manifold of dimensi- on d-dim(ker(Ao,-B(O))) which is tangent to ker(Ao,-B(O) ). It is possible to push the flow on this manifold forward to a flow on ker (Ao+B(O)) see Chow and Lauterbach [1988] using the eigenprojector. Using equivariant projectors it is even possible to con- struct an equivariant flow on ker (Ao+B(O)). We will assume that we have clone it that way. Observe that (viii) implies that d is odd, i.e. d-2[ ,-1 for some [ ~N. Therefore we end up assuming to look at a flow on a d dimensional real vector space V. On V we have an absolutely irreducible action of 0(3). The flow on V is given by an ordinary differential equation (2.2) vt- C().)v ,- g(v,),), where v t stands for the derivative of v(t) with respect to time and (I) C(x) is a family of linear mappings V-V. commuting with the action of 0(3), (II) g(.,~.) is 0(3) equivariant and g(0,),)- dvg(0,x)- 0. Due to (I) and the fact that the action of 0(3) is absolutely irreducible we know

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