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

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

Proposition. For all

Proposition. For all section s of N(M) M = s (M) , and s ~--90 . 24 ~ A with II~I{ sufficiently small there is a such that all zeros of F close to M lie on tends C r uniformly to the zero section s O as Since the zeros of F are trapped on M , the problem has now become: find the zeros of F = F os : M ---> Y , where in fact F (x) E W(x) Obviously x solves F (.) = 0 if and only if s (x) solves F (') = 0 . There are several approaches to this problem in the literature, mostly under the assumptions that for x 0 ~ M the derivative DFo(X O) is a Fredholm operator of index zero, and W(x O) can be identified with ker DFo(X O) in some smoothly-varying way as x 0 varies on M . The problem is then reduced to finding the zeros of a vector field X on M : see e.g. Dancer [DI,D2] , Vanderbauwhede [V1,V2]. Then, as a first step, the Euler characteristic X M of M dictates the existence of at least one zero of X u if XM ~ 0 , and the homology of M gives more detailed information. If F retains some of the symmetry of F 0 then X inherits some symmetry too, and sharper information can be extracted from the topology [D3]. If the original problem is of variational type so that the zeros of F are the critical points of some real-valued function f then further tools are available. Zeros of F are critical points of a function { on M , and the number of critical points of f is bounded below by the Liusternik-Shnirelman category cat(M) This fact has been fruitfully applied in various contexts by many authors, e.g. Reeken [R], Weinstein [We], Chillingworth, Marsden and Wan [CMW], Ambrosetti, Coti-Zelati and Ekeland [ACZE]. Again, any symmetry pays extra dividends as usual: if f retains a symmetry group F then cat(M/F) (often significantly larger than cat(M)) provides the lower bound. In some cases it is straightforward to compute the leading terms in the Taylor expansion of f as a series in ~ , the coefficients being functions on M . Since fo is constant, the bifurcation behaviour is determined by the coefficient of the first nonvanishing power of ~ , provided this coefficient is a Morse function. If it is not, then a further reduction procedure may apply. An application of this method to

25 the traction problem in nonlinear elasticity can be found in [CMW]; see also [C 1] for a general discussion. Remark. If the nondegeneracy condition (2.1) fails there is then room for degeneracy in a direction 'normal' to M, and this may also retain some symmetry. This is where the Golubitsky-Schaeffer machinery could still apply. See observations in [WM]. 3. A generic viewpoint A complementary approach to this (as to many other) bifurcation problems is to exploit the techniques of singularity theory to describe the expected nature of the bifurcation set in ~-space, and the associated bifurcation behaviour, given the dimensions involved. More precisely, after making generic assumptions on F we aim to match the local bifurcation behaviour at each point to one of a finite list of standard models. For the case q = I , A = R2 this was carried out by Hale and Taboas in [HT] (see also Chapter II of [CH]). Here we look at higher-dimensional versions, which, as we see, lead to some interesting differential geometry. These ideas were discussed rather briefly in [C2]. A fuller account will appear elsewhere. We assume as before that DFo(X O) is Fredholm (xo~M) but do not now insist on index zero. However, to avoid one source of possible global topological complication we suppose that the 'cokernel' manifold W forms a trivial sub-bundle of M x Y . (We may in any case do this locally on M, provided we no longer demand compactness.) Then we can identify W(x O) with a single copy of RP (where p = dim W(Xo)) for all x 0 e M and express the reduced problem as where now : Mq x A --) ~ p ~(.,o) • 0 . Suppose now that A = ~k . The vanishing of F on M when ~ = 0 means that F(x,~) may be written as F(x,p.) - G(x,p.).p. (3.1) - (hCx} + OCp.)).p.

  • 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: 23 the symmetry arises naturally fr
  • 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 and 42: 35 k z 4 I For k s 6 the ideas abov
  • Page 43 and 44: 37 [GG] Golubitsky, M. and Guillemi
  • 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) =
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    75 (dS) F of S at F has an eigenval

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    77 experimental geometry. 'Upper bo

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    79 M.G.M. Gomes [1989]. Steady-stat

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    81 To describe the results, we supp

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    83 The author's work on these quest

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    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

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    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

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    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

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    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

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    Proof q 241 aperture aperture apert

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    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

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    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

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    261 denote the generator of the Lie

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

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    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

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    273 Now we return to generating fam

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    ~LLL I 275 ~::'.'~'.C': • ,',': :

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

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    §1. Introduction 279 In this paper

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

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    283 b 0 to avoid negative suffices.

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

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

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    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

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

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    299 4 Description of the proof of t

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    Versal Deformations of Infinitesima

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    303 form. (2) By dropping the sympl

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

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

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

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

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

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

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

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    319 unfolding H(g) of a Hamiltonian

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    ADDRESSES OF CONTRIBUTORS D.Armbrus

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