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

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

62 Aknowledgements.

62 Aknowledgements. Pictures 6,7,8,9,10 were produced on a Sun station during the visit of P. Chossat at the Institute of Mathematics, University of Warwick, in July 1989. References Armbruster D, Chossat P [1989]. Structurally stable heteroclinic cycles in mode interaction with O(3)-symmetry, to appear. Armbruster D, Guckenheimer J, Holmes P [1988]. Heteroclinic cycles and modulated traveling waves in systems with 0(2) symmetry, Physica 29D, 257-282. Chossat P [1982]. de Nice. Le probl~me de B6nard dans une couche sphfrique, thesis, Universit6 Chossat P [1983]. Int6raction entres bifurcations de modes 1=1 et 1=2 dans les probl~mes invariants par sym6trie O(3), Comptes-Rendus de l'Acad6mie des Sciences de Paris, s6rie I, 297, 469-471. Dos Reis G.L [1984]. Structural stability of equivariant vector fields on two manifolds, Trans. Amer. Math. Soc., 283, 633-643. Friedrich R, Haken H [1986]. Static, wavelike and chaotic convection in spherical geometries, Phys. Rev. A, 34, 3, 2100-2120. Golubitsky M, Schaeffer D [1982]. Bifurcation with 0(3) symmetry including applications to the spherical B6nard problem, Comm. Pure Appl. Math., 35-81. Golubitsky M, Stewart I, Schaeffer D [1988]. Singularities and groups in Bifurcation theory vol. II, Appl. Math. Sci. Ser. 69, Springer Verlag, New-York. Guckenheimer J, Holmes P [1988]. Structurally stable heteroclinic cycles, Math. Proc. Camb. Phil. Soc., 103, 183-192. Melbourne I. Chossat P, Golubitsky M [1988]. Heteroclinic cycles involving periodic solutions in mode interaction with 0(2) symmetry, Preprint, University of Houston. Moutrane E [1988]. Int6raction de modes sph6riques dans le probl~me de B6nard entre deux spheres, thesis, Universit6 de Nice.

Boundary Conditions as Symmetry Constraints J.D.Crawford ,M.Golubitsky, M.G.M.Gomes, E.Knobloch, I.N.Stewart ABSTRACT Fujii, Mimura, and Nishiura [1985] and Armbruster and Dangelmayr [1986, 1987] have observed that reaction-diffusion equations on the interval with Neumann boundary conditions can be viewed as restrictions of similar problems with periodic boundary conditions; and that this extension reveals the presence of additional symmetry constraints which affect the generic bifurcation phenomena. We show that, more generally, similar observations hold for multi-dimensional rectangular domains with either Neumann or Dirichlet boundary conditions, and analyse the group-theoretic restrictions that this structure imposes upon bifurcations. We discuss a number of examples of these phenomena that arise in applications, including the Taytor-Couette experiment, Rayleigh-B6nard convection, and the Faraday experiment. 0 Introduction Let P(u) denote a reaction-diffusion equation on the line. Then P(u) is invariant under translations and reflections. It is well known that a solution u(x) to (u) = 0 on the interval [0,rc] with Neumann boundary conditions (NBC) may be extended to a solution of the same PDE on the whole line that satisfies periodic boundary conditions (PBC) on the interval [-n,~]. This extension is accomplished by reflection across the boundaries, that is, by defining u(-x) = u(x) for x ~ [-n,0] and then extending u to be 2n-periodic on R. By using Euclidean invariance, the Neumann boundary conditions, and the second order structure of the PDE, it is not hard to show that this extension procedure preserves regularity of the solutions: for example C OO solutions remain C °°, provided the operator P is itself C °°. Fujii, Mimura, and Nishiura [1985] and Armbruster and Dangelmayr [1986, 1987] observe that this extension property changes, in a subtle way, the generic behaviour of codimension two steady-state mode interactions. In this note we give a straightforward group-theoretic description of why genericity is affected by this extension property: simply put, NBC can be thought of as a symmetry constraint on the PBC problem. Using this general construction we indicate several ways in which this idea may be extended. In particular, we show how the same general construction can be applied to Dirichlet boundary conditions (DBC) and to Euclidean-invariant PDEs in several spatial variables, defined on generalized rectangles. We also remark on several physical systems whose analyses illustrate these ideas. These include the Taylor-

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