- 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 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) =
- 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: 83 The author's work on these quest
- 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 and 118: 111 not matter. We choose b > 0. Th
- Page 119 and 120: 113 SLot x SLs Figure 2: Phase port
- 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

- Page 169 and 170:
8.1 Poincard maps 163 We review the

- Page 171 and 172:
165 Proposition 8.3 Let P. be a rel

- Page 173 and 174:
On the Bifurcations of Subharmonics

- Page 175 and 176:
169 and so we give in a fourth part

- Page 177 and 178:
171 of KerL, the elements of B are

- Page 179 and 180:
173 From the fixed-point subspaces

- Page 181 and 182:
175 and the following curve is the

- Page 183 and 184:
q even q=3 a g 177 s~llx-ag odd Fig

- Page 185 and 186:
q=3 S 179 s=lk-aa q ~ - - q>5 ~ a ~

- Page 187 and 188:
3.2 dimB--1 181 We are now in the s

- Page 189 and 190:
183 Puting them back into (14) and

- Page 191 and 192:
4.2 The recognition problem 185 Thi

- Page 193 and 194:
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

- Page 197 and 198:
qo =0 #o References .p~ =0 191 ~o ~

- Page 199 and 200:
Classification of Symmetric Caustic

- Page 201 and 202:
195 internal variables p and extern

- Page 203 and 204:
197 In the classification of Lagran

- Page 205 and 206:
Remark 1.3 More generally, the orga

- Page 207 and 208:
(W*~V,0) (v,o) (where n2 is the nat

- Page 209 and 210:
G $~G-versal unfolding of F(.,0) in

- Page 211 and 212:
Z (V,U) as an E q module. It follow

- Page 213 and 214:
3 FINITE DETERMINACY 207 Good deter

- Page 215 and 216:
209 Definition 3.3 G Let ~ be any g

- Page 217 and 218:
211 where q%(x,y) = Z Vbc(X'y)xc an

- Page 219 and 220:
Theorem 4.5 213 (i) If r >_. s then

- Page 221 and 222:
215 (4) In [JR] we show that the ca

- Page 223 and 224:
217 In terms of the invariants the

- Page 225 and 226:
R~P'ERF~CES [AI [ADI [AGVI [BPW] [D

- Page 227 and 228:
221 instrument in terms of composed

- Page 229 and 230:
~p'_-sin0' ~0' sin0' 3p -sin0 30 si

- Page 231 and 232:
225 perfect gas with V = volume, S

- Page 233 and 234:
227 according to co and all maximal

- Page 235 and 236:
a) a submersion p : X -~ Y, 229 b)

- Page 237 and 238:
231 Then the reduced symplectic spa

- Page 239 and 240:
233 Definition The phase space of a

- Page 241 and 242:
where m ~t-- 1 +,~2 = a9 '2 - 2 9'

- Page 243 and 244:
237 The Billiard Map as an Optical

- Page 245 and 246:
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

- Page 251 and 252:
245 Now taking an inflection point

- Page 253 and 254:
247 1 5 2 3 q2)+kl(ql~.2+q2)2 H 4 :

- Page 255 and 256:
249 A2(k+l) singularities by specif

- Page 257 and 258:
251 Assume that the surfaces have t

- Page 259 and 260:
253 -- tsin @ )+(t-'+~in~>t_~os2q>

- Page 261 and 262:
255 Poston, T. and Stewart, I. [197

- Page 263 and 264:
257 0ii) A(),)-A 0 + B(X) is a hoto

- Page 265 and 266:
259 their lists. The correct lists

- Page 267 and 268:
261 denote the generator of the Lie

- Page 269 and 270:
263 4 ÷ D6 d O(3), O(2)-, O- O(2)-

- Page 271 and 272:
References 265 Chossat, P. [1Q70]:

- Page 273 and 274:
267 In this paper I consider invari

- Page 275 and 276:
269 be the set of critical points i

- Page 277 and 278:
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

- Page 287 and 288:
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,

- Page 297 and 298:
291 I. The fixed point (2.9) remain

- Page 299 and 300:
References 293 [i] R. W. Lucky (196

- Page 301 and 302:
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

- Page 311 and 312:
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

- Page 319 and 320:
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

- Page 327 and 328:
ADDRESSES OF CONTRIBUTORS D.Armbrus