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

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

36 [B 2] Bruce, J.W.,

36 [B 2] Bruce, J.W., The duals of generic hypersurfaces, Math. Scand. 49 (1981), 36-60. [B 3 ] Bruce, J.W., On singularities, envelopes and elementary differential geometry, Math. Proc. Cambridge Philos. Soc. 89 (1981), 43-48. [B 4] Bruce, J.W., Generic reflections and projections of surfaces, Math. Stand. 54 (1984), 262-278. [B 5 ] Bruce, J.W., Seeing - the mathematical viewpoint, Math. Intelligencer 6 (1984), 18-25. [B 6] Bruce, J.W., Geometry of singular sets, Math. Proc. Cambridge Philos. Soc. 106 (1989), 495-509. [BG I] Bruce, J.W. and Giblin, P.j., On real simple singularities, Math. Proc. Cambridge Philos. Soc. 88 (1980), 273-279. [BG 2] Bruce, J.W. and Giblin, P.3., Curves and singularities, Cambridge University Press 1984. [C I] Chillingworth, D.R.3., Bifurcation from an orbit of symmetry, in Singularities and Dynamical Systems, S.N. Pnevmatikos (ed.), pp. 285-294, North-Holland, Amsterdam 1985. [C 2] Chillingworth, D.R.J., The ubiquitous astroid, in Proc. Int. Symp. on The Physics of Structure Formation: Theory and Simulation, Institut f~r Informationsverarbeltung, Universit~t T~bingen, 1986, W. G~ttinger and G. Dangelmayr (eds.), pp. 372-386, Springer- Verlag, Berlin 1987. [CMW] Chillingworth, D.R.J., Marsden, J.E. and Wan, Y.-H., Symmetry and bifurcation in three-dimensional elasticity, Part I: Arch. Rat. Mech. Anal. 80 (1982), 295-331, Part II: 83 (1983), 363-395. [CH] Chow, S.-N. and Hale, 3.K., Methods of bifurcation theory, Springer-Verlag, New York 1982. [D I] Dancer, E.N., On the existence of bifurcating solutions in the presence of symmetries, Proc. Roy. Soc. Edinburgh 85A (1980), 321-336. [D 2] Dancer, E.N., The G-invariant implicit function theorem in infinite dimensions, Proc. Roy. Soc. Edinburgh 92A (1982), 13-30. [D 3 ] Dancer, E.N., Perturbation of zeros in the presence of symmetries, J. Austral. Math. Soc. (Set. A) 36 (1984), 106-125.

37 [GG] Golubitsky, M. and Guillemin, V., Stable mappings and their singularities, Springer-Verlag, New York 1973. [GS] Golubsky, M. and Schaeffer, D., Imperfect bifurcation in the presence of symmetry, Common. Math. Phys. 67 (1979), 205-232. [GSS] Golubitsky, M., Stewart, I. and Schaeffer, D.G., Singularities and Groups in Bifurcation Theory, Vols. I, II., Springer-Verlag, New York, 1985, 1988. [HT] Hale, J.K. and Taboas, P.Z., Interaction of damping and forcing in a second order equation, Nonlin. Anal. Th. Meth. Appl. 2 (1978), 77-84. [L] Looijenga, E.J.N., Structural stability of smooth families of C m functions, Thesis, Univ. Amsterdam, 1974. [M] Marsden, J.E., Qualitative methods in bifurcation theory, Bull. Amer. Math. Soc. 84 (1978), 1125-1148. [Ma] Mather, J.N., Generic projections, Ann. Math. (2), (1973), 226-245. [Mg] Magnus, R.J., On perturbations of a translationally invariant differential equation, Proc. Roy. Soc. Edinburgh IIOA (1988), 1-25. [R] Reeken, M., Stability of critical points under small perturbations, Part I: Topological theory, Manuscripta Math. 7 (1972), 387-411. [V I] Vanderbauwhede, A., Symmetry and bifurcations near families of solutions, J. Diff. Eqns. 36 (1980), 173-187. IV 2] Vanderbauwhede, A., Local Bifurcation and Symmetry, Research Notes in Math. 75, Pitman, London 1982. [Wa] Wall, C.T.C., Geometric properties of generic differentiable manifolds, in: Geometry and Topology III, Lecture Notes in Math. 597, Springer-Verlag, Berlin 1976. [We] Weinstein, A., Perturbation of periodic manifolds of Hamiltonian systems, Bull. Amer. Math. Soc. 77 (1971), 814-818. [WM] Wan, Y.-H. and Marsden, J.E., Symmetry and bifurcation in three-dimensional elasticity, Part Ill, Arch. Rat. Mech. Anal. 84 (1983), 203-233.

  • Page 1 and 2: Lecture Notes in Mathematics Editor
  • Page 3 and 4: Editors Mark Roberts Ian Stewart Ma
  • Page 5 and 6: Co~e~s P.J. Aston, Scaling laws and
  • Page 7 and 8: Scaling Laws and Bifurcation P.J.As
  • Page 9 and 10: where = It follows immediately from
  • Page 11 and 12: esults for unitary representations
  • Page 13 and 14: and T are orthogonal representation
  • Page 15 and 16: which is also a subgroup of F. Clea
  • Page 17 and 18: Proof 11 The equivalent result that
  • Page 19 and 20: 13 such that the Equlvariant Branch
  • Page 21 and 22: ]5 since h is orthogonal. As b, e,
  • Page 23 and 24: 17 and from Lemma 4.2, hj : X ~k --
  • Page 25 and 26: ]9 in a bifurcating branch of solut
  • Page 27 and 28: 21 Duncan, K. and Eilbeck, 3. C. (1
  • Page 29 and 30: 23 the symmetry arises naturally fr
  • Page 31 and 32: 25 the traction problem in nonlinea
  • Page 33 and 34: 27 First we dispose of the case k =
  • Page 35 and 36: 29 The symbols 0 ..... 4 indicate r
  • Page 37 and 38: 31 In terms of the radially project
  • Page 39 and 40: S I --U 33 A Figure 4 .. • . ".
  • Page 41: 35 k z 4 I For k s 6 the ideas abov
  • Page 45 and 46: 39 small equivariant perturbations
  • Page 47 and 48: 41 It may happen that an obstructio
  • Page 49 and 50: , 43 0(3) {yo} {xO'Yo, Y2+Y-2} D 2
  • Page 51 and 52: 45 stability requires order 5 (Golu
  • Page 53 and 54: 3.1. Phase portrait in FixfD2~.Z ~
  • Page 55 and 56: 49 (11) kl=Z,l+- "-c+ c2~2 2 anothe
  • Page 57 and 58: 51 When c=0, the eigenvalue in the
  • Page 59 and 60: 53 Remark 2. This heteroclinic cycl
  • Page 61 and 62: 55 Armbruster et al. [1988]. This w
  • Page 63 and 64: 57 Notes: 1) each picture shows pro
  • Page 65 and 66: ~< o( 59 i o( o(" ~X1r IX~£~ Figur
  • Page 67 and 68: ¢* 1 61 T ........... "--7 y¢, --
  • Page 69 and 70: Boundary Conditions as Symmetry Con
  • Page 71 and 72: 65 We illustrate this point in the
  • Page 73 and 74: U T(u) 67 m=2 -~ ................ 0
  • Page 75 and 76: 69 be the reduced bifurcation equat
  • Page 77 and 78: 71 3 The Couette-Taylor Experiment
  • Page 79 and 80: 73 00 u(x,y) = v(x,y) = ~-x (x,y) =
  • Page 81 and 82: 75 (dS) F of S at F has an eigenval
  • Page 83 and 84: 77 experimental geometry. 'Upper bo
  • Page 85 and 86: 79 M.G.M. Gomes [1989]. Steady-stat
  • Page 87 and 88: 81 To describe the results, we supp
  • Page 89 and 90: 83 The author's work on these quest
  • Page 91 and 92: 85 certain elements of G may interc
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    87 isolated singularity at 0. We sa

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    in particular, Zp~ = 1 + ~ iii) Vp

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    91 Similarly we let X(G) denote the

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    93 F I Fix(G') x ~ : Fix(G') x $t -

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    Example 4.5: 95 Let G = Z/mE act on

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    97 Thus, we see that X q = 1 + 4V.

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    99 Q(F1) = Cx,~/I(F1) = Cx,~/I(F 0)

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    Corollary 7 101 For i) we know by t

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    103 generator for m A, the module o

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    105 D3 On the number of branches fo

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    On a Codimension-four Bifurcation O

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    109 Setting x = u3 and introducing

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    111 not matter. We choose b > 0. Th

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    113 SLot x SLs Figure 2: Phase port

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    SLs(H} H.(SL s) SNs I SN~ (~6) 115

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    iii 1 SLo/" SNo ,y X 117 i 11 21 .

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    119 extent the dynamics is influenc

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    121 [12] J. Guckenhelmer, SIAM J. M

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    123 produces a continuous function.

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    is generated by 125 V(A~) = {f : f(

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    and the abstract integral equation

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    129 (iii). X® and X+ are finite di

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    131 Definition 6.1 Let E and F be B

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    133 6.3 Contractions on embedded Ba

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    135 Theorem 6.13 (Center Manifold)

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    137 The AIE (6.14) is equivalent to

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    and 139 (q,p) = foh d-"~)p(-r) (7.8

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    [Cha71] [Die87] [Dui76] [DvG84] [Ha

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    143 Much is already known about the

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    145 We refer to ,g = {Su[y E a} as

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    147 ttemark 3.1 It follows by our m

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    149 Definition 4.1 Let 11 be a clos

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    151 Remark 4.3 It follows from Theo

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    (a) f(=) = ~(=)x, au x E GIH. (b) f

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    155 Since ] and ~ are smooth, so ar

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    157 of X at all points z E a. Neces

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    159 In this section we wish to desc

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    161 A straightforward application o

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    8.1 Poincard maps 163 We review the

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    165 Proposition 8.3 Let P. be a rel

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    On the Bifurcations of Subharmonics

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    169 and so we give in a fourth part

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    171 of KerL, the elements of B are

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    173 From the fixed-point subspaces

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    175 and the following curve is the

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    q even q=3 a g 177 s~llx-ag odd Fig

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    q=3 S 179 s=lk-aa q ~ - - q>5 ~ a ~

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    3.2 dimB--1 181 We are now in the s

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    183 Puting them back into (14) and

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    4.2 The recognition problem 185 Thi

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    case I II 1V V VI VII IX case I U 1

  • Page 195 and 196:

    qo r0 =0 =O #o ~o Pa. ~o 1=° top-c

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    qo =0 #o References .p~ =0 191 ~o ~

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    Classification of Symmetric Caustic

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    195 internal variables p and extern

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    197 In the classification of Lagran

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    Remark 1.3 More generally, the orga

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    (W*~V,0) (v,o) (where n2 is the nat

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    G $~G-versal unfolding of F(.,0) in

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    Z (V,U) as an E q module. It follow

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    3 FINITE DETERMINACY 207 Good deter

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    209 Definition 3.3 G Let ~ be any g

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    211 where q%(x,y) = Z Vbc(X'y)xc an

  • Page 219 and 220:

    Theorem 4.5 213 (i) If r >_. s then

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    215 (4) In [JR] we show that the ca

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    217 In terms of the invariants the

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    R~P'ERF~CES [AI [ADI [AGVI [BPW] [D

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    221 instrument in terms of composed

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    ~p'_-sin0' ~0' sin0' 3p -sin0 30 si

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    225 perfect gas with V = volume, S

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    227 according to co and all maximal

  • Page 235 and 236:

    a) a submersion p : X -~ Y, 229 b)

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    231 Then the reduced symplectic spa

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    233 Definition The phase space of a

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    where m ~t-- 1 +,~2 = a9 '2 - 2 9'

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    237 The Billiard Map as an Optical

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    exists a local generating Morse fam

  • Page 247 and 248:

    Proof q 241 aperture aperture apert

  • Page 249 and 250:

    A1 :FlCO = -7 i2" A3 : F3f/) = - ¼

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    245 Now taking an inflection point

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    247 1 5 2 3 q2)+kl(ql~.2+q2)2 H 4 :

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    249 A2(k+l) singularities by specif

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    251 Assume that the surfaces have t

  • Page 259 and 260:

    253 -- tsin @ )+(t-'+~in~>t_~os2q>

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    255 Poston, T. and Stewart, I. [197

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    257 0ii) A(),)-A 0 + B(X) is a hoto

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    259 their lists. The correct lists

  • Page 267 and 268:

    261 denote the generator of the Lie

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    263 4 ÷ D6 d O(3), O(2)-, O- O(2)-

  • Page 271 and 272:

    References 265 Chossat, P. [1Q70]:

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    267 In this paper I consider invari

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    269 be the set of critical points i

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    271 Figure 1: Two trajectories in a

  • Page 279 and 280:

    273 Now we return to generating fam

  • Page 281 and 282:

    ~LLL I 275 ~::'.'~'.C': • ,',': :

  • Page 283 and 284:

    277 Let L = £(J20) and m E M. Work

  • Page 285 and 286:

    §1. Introduction 279 In this paper

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    where 281 Im(ei~R0) - 0 , (l.4)b M

  • Page 289 and 290:

    283 b 0 to avoid negative suffices.

  • Page 291 and 292:

    and where 285 r r r i i i r i Ro ~

  • Page 293 and 294:

    with R 0 as before in (2.8), but no

  • Page 295 and 296:

    is always real. such that is real,

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    291 I. The fixed point (2.9) remain

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    References 293 [i] R. W. Lucky (196

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    295 -- ba+~+~t h For small values o

  • Page 303 and 304:

    h =0 XX ~0 (h~ 1, h~L 2) H = (0, O)

  • Page 305 and 306:

    299 4 Description of the proof of t

  • Page 307 and 308:

    Versal Deformations of Infinitesima

  • Page 309 and 310:

    303 form. (2) By dropping the sympl

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    305 Jij q + L~j=l Ji]~j )' we have

  • Page 313 and 314:

    307 For (0) n, n--even,~= 1, set Ix

  • Page 315 and 316:

    D% I -% ,,~ Fig. 1 309 Each oblique

  • Page 317 and 318:

    311 I -- T/ Fig. 2 7~-form Now, def

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    313 b'st = r (-1)s-t[; s t' case(c)

  • Page 321 and 322:

    H i [X22"FILXl 2] ~ [(2x2x4-x32) -F

  • Page 323 and 324:

    H 2Re[+2AZlZ2+Z2Z2 +#zlT" l ] 2E--R

  • Page 325 and 326:

    319 unfolding H(g) of a Hamiltonian

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    ADDRESSES OF CONTRIBUTORS D.Armbrus

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