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Lecture Notes in Mathematics Editor

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Editors Mark Roberts Ian Stewart Ma

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Co~e~s P.J. Aston, Scaling laws and

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Scaling Laws and Bifurcation P.J.As

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where = It follows immediately from

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esults for unitary representations

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and T are orthogonal representation

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which is also a subgroup of F. Clea

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Proof 11 The equivalent result that

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13 such that the Equlvariant Branch

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]5 since h is orthogonal. As b, e,

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17 and from Lemma 4.2, hj : X ~k --

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]9 in a bifurcating branch of solut

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21 Duncan, K. and Eilbeck, 3. C. (1

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23 the symmetry arises naturally fr

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25 the traction problem in nonlinea

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27 First we dispose of the case k =

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29 The symbols 0 ..... 4 indicate r

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31 In terms of the radially project

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S I --U 33 A Figure 4 .. • . ".

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35 k z 4 I For k s 6 the ideas abov

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37 [GG] Golubitsky, M. and Guillemi

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39 small equivariant perturbations

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41 It may happen that an obstructio

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, 43 0(3) {yo} {xO'Yo, Y2+Y-2} D 2

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45 stability requires order 5 (Golu

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3.1. Phase portrait in FixfD2~.Z ~

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49 (11) kl=Z,l+- "-c+ c2~2 2 anothe

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51 When c=0, the eigenvalue in the

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53 Remark 2. This heteroclinic cycl

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55 Armbruster et al. [1988]. This w

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57 Notes: 1) each picture shows pro

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~< 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
- Page 93 and 94: 87 isolated singularity at 0. We sa
- Page 95 and 96: in particular, Zp~ = 1 + ~ iii) Vp
- Page 97 and 98: 91 Similarly we let X(G) denote the
- Page 99 and 100: 93 F I Fix(G') x ~ : Fix(G') x $t -
- Page 101 and 102: Example 4.5: 95 Let G = Z/mE act on
- Page 103 and 104: 97 Thus, we see that X q = 1 + 4V.
- Page 105 and 106: 99 Q(F1) = Cx,~/I(F1) = Cx,~/I(F 0)
- Page 107 and 108: Corollary 7 101 For i) we know by t
- Page 109 and 110: 103 generator for m A, the module o
- Page 111 and 112: 105 D3 On the number of branches fo
- Page 113 and 114: On a Codimension-four Bifurcation O
- Page 115 and 116: 109 Setting x = u3 and introducing
- Page 117: 111 not matter. We choose b > 0. Th
- Page 121 and 122: SLs(H} H.(SL s) SNs I SN~ (~6) 115
- Page 123 and 124: iii 1 SLo/" SNo ,y X 117 i 11 21 .
- Page 125 and 126: 119 extent the dynamics is influenc
- Page 127 and 128: 121 [12] J. Guckenhelmer, SIAM J. M
- Page 129 and 130: 123 produces a continuous function.
- Page 131 and 132: is generated by 125 V(A~) = {f : f(
- Page 133 and 134: and the abstract integral equation
- Page 135 and 136: 129 (iii). X® and X+ are finite di
- Page 137 and 138: 131 Definition 6.1 Let E and F be B
- Page 139 and 140: 133 6.3 Contractions on embedded Ba
- Page 141 and 142: 135 Theorem 6.13 (Center Manifold)
- Page 143 and 144: 137 The AIE (6.14) is equivalent to
- Page 145 and 146: and 139 (q,p) = foh d-"~)p(-r) (7.8
- Page 147 and 148: [Cha71] [Die87] [Dui76] [DvG84] [Ha
- Page 149 and 150: 143 Much is already known about the
- Page 151 and 152: 145 We refer to ,g = {Su[y E a} as
- Page 153 and 154: 147 ttemark 3.1 It follows by our m
- Page 155 and 156: 149 Definition 4.1 Let 11 be a clos
- Page 157 and 158: 151 Remark 4.3 It follows from Theo
- Page 159 and 160: (a) f(=) = ~(=)x, au x E GIH. (b) f
- Page 161 and 162: 155 Since ] and ~ are smooth, so ar
- Page 163 and 164: 157 of X at all points z E a. Neces
- Page 165 and 166: 159 In this section we wish to desc
- Page 167 and 168: 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