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

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

192 [9]

192 [9] J-J.Gervals.Bifurcations of subharmonie solutions in reversible systems. J.Diff.Eq.75 (1988),28-42. [1O] M.Golubitsky,M.Krupa,C.C.Lim. Time reversibility and particle sedimentation. Preprint.1989. [11] J.K.Hale. Ordinary Differential Equations.McGraw- Hin.1969. [12] A.Iiummel.Bifureations of periodic points. Thesis.Groningen Univ.1979. [13] K.Kirchg~ssner,J.Scheurle.Global branches of periodic solutions of reversible sys- tems.H.Brezis,H.Berestycki.Eds.Res.Notes.Math.60. Pitmam1981. [14] Y.Kuramoto,T.Yamada. Turbulent states in chemical reaction. Prog.Theo.Phys.56 (1976),679. [15] B.A.Malomed, M.I.Tribel'sldi.Bifurcations in distributed kinetic m/stems ~dth aperi- odic instability.Physica D.14 (1984),67-87. [16] I.Melbourne. The recognition problem for equivariant singularities. Nonlinearity.1 (1988),215-240. [17] D.bfichelson.Steady solutions of the Kuratomo-Shivashinsky equation. PhysicaD.19 (1986),89-111. [18] J.K.Moser.On the theory of quasi-periodic motions. SIAM.Rev.8 (1966),145-172. [19] W.Pluschke.Invariant tori bifurcating from fixed paints of nonanalytic reversible ~9s- terns.Thesis.Stuttgart Univ.1989. [20] M.B.Sevryuk.Reversible Systems.Lec.Notes.Maths.1211. Springer.1986. [21] J.Scheurle. Verzweigung quasiperiodischer LSsungen bei reversiblen dynamisehen Sys- temen.Habilitationschrift.Stuttgart Univ. 1980. [22] A.Vaxtderbauwhede.Local bifurcation and symmetry. tLes.Notes.Math.75.Pitmaa. 1982. [23] A.Vanderbauwhede.Bifurcation of subharmonie solutions in time reversible sys- tems.ZAMP.37 (1986),455-477. [24] A.Vanderbauwhede.Secondary bifurcations of periodic solutions in autonomon.q sys- tems.Can.Matli.Soc.Proc.Conf.8 (1987),693-701. [25] J.H.Wolkowisky.Branches of periodic solutions of the nonlinear Hill's equation. J.Diff.Eq.ll (1972),385-400.

Classification of Symmetric Caustics I: Symplectic Equivalence Staszek Janeczko and Mark Roberts ABSTRACT We generalise the classification theory of Arnold and Zakalyukin for singularities of Lagrange projections to projections that commute with a symplectic action of a compact Lie group. The theory is applied to the classification of infinitesimally stable corank 1 projections with Z 2 symmetry. However examples show that even in very low dimensions there exist genetic projections which are not infinitesimally stable. INTRODUCTION In this paper and its sequel [JR] we describe some general singularity theory machinery which can be used to classify symmetric caustics. Let X be a smooth manifold with a smooth action of the compact Lie group G. This action extends to an action on the cotangent bundle T*X which leaves invariant the natural symplectic form. If L is a G- invariant Lagrange submanifold of T*X then the Lagrange projection rg L : L -. X is G-equivariant and its discriminant, the caustic C L of L, is a G-invatiant subvariety of X. In this paper we consider the classification of the pairs (T*X,L) up to symplectic equivalence, ie symplectomorphisms of T*X which preserve its natural fibration (Definition 2.1). In [JR] we classify just the caustics, up to equivariant diffeomorphisms of X. This caustic equivalence turns out to be a much weaker equivalence relation (see Remarks 4.7). Our approach to symplectic equivalence is a generalization of the non-equivatiant theory of Arnold and Zakalyukin (see [AGV]) in that we use a form of parametrised tight equivalence of Morse families. This should be contrasted with that of [JK1,2] where the emphasis is on classifying generating functions. A major difference between the equivariant and non-equivariant cases is that in the latter the stability of a Morse family

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