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

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

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

Lecture Notes in Mathematics Editors: A. Dold, Heidelberg B. Eckmann, Ztirich E Takens, Groningen 1463

• Page 2 and 3: M. Roberts I. Stewart (Eds.) Singul
• Page 4 and 5: Preface A year-long symposium on Si
• Page 6 and 7: I.M. Moroz, Some complex differenti
• Page 8 and 9: X (a)Pitchfork bifurcation Fig 1.1
• Page 10 and 11: kernel ~. Then Proof z(r) (2.1) The
• Page 12 and 13: ambiguity with regard to the group
• Page 14 and 15: 8 If there exists an orthogonal lin
• Page 16 and 17: 10 Thus, hx E X n~. It remains to p
• Page 18 and 19: evaluating it at (Zo, \$o) gives 12
• Page 20 and 21: etc. The non-degeneracy condition t
• Page 22 and 23: 16 The purpose of this analysis has
• Page 24 and 25: 18 X is then an invariant subspace
• Page 26 and 27: II'HI . . . . . . . J~ a 0 __~ '°
• Page 28 and 29: 1. Introduction and background BIFU
• Page 30 and 31: Proposition. For all section s of N
• Page 32 and 33: 26 where G(x,H) ~ L(k,p) = Lin(Rk,~
• Page 34 and 35: 28 Formally, u ~ Z precisely when (
• Page 36 and 37: I k=3 (BI) (S2) 30 Figure 3 (M I )
• Page 38 and 39: standard projection u ~ {u,-u} . 32
• Page 40 and 41: 34 For generic h , the apparent out
• Page 42 and 43: 36 [B 2] Bruce, J.W., The duals of
• Page 44 and 45: Structurally Stable Heteroclinic Cy
• Page 46 and 47: 40 numerical experiments. A heteroc
• Page 48 and 49: 2. The interaction of spherical mod
• Page 50 and 51: (7a) (7b) (8a) (8b) (8c) 44 Ao(x,y
• Page 52 and 53:

3 46 ~.2Yo+bxo2+Cyo+dyo 3=0 o'(12)=

• Page 54 and 55:

3.2. Phase oortraits in Fix(O(2)-)

• Page 56 and 57:

50 It can be shown that the type 3

• Page 58 and 59:

52 equilibria. In P1 the heteroclin

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54 IV + ~ . Typical phase portrait

• Page 62 and 63:

56 Fix(Tj_l)+Fix(Tj). Their real pa

• Page 64 and 65:

n ¢ 58 Figure 7. A "strange attrac

• Page 66 and 67:

~'/,., r,~ I ! I i I J 60 Figure 10

• Page 68 and 69:

62 Aknowledgements. Pictures 6,7,8,

• Page 70 and 71:

64 Couette experiment, Rayleigh-Btn

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66 (b) Next we discuss the expected

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68 Generalizations In this section

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70 (c) Higher-Dimensional Domains T

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72 held fixed and the speed of the

• Page 80 and 81:

74 Hence odd modes in y bifurcate i

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76 It has also been suggested that

• Page 84 and 85:

References 78 C.D.Andereck, S.S.Liu

• Page 86 and 87:

Equivariant Bifurcations and Morsif

• Page 88 and 89:

82 weight parts of these algebras 5

• Page 90 and 91:

84 work on bifurcation theory. In t

• Page 92 and 93:

Lemma 1.1: Xhp = 2"Xp, Xhb = 2"Xb,

• Page 94 and 95:

Now, we can compute the modular cha

• Page 96 and 97:

90 of critical points with ~.' < 0

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92 integer values; hence, a suffici

• Page 100 and 101:

1 ~p 94 1 (abc) 1 1 2 -1 where 1 an

• Page 102 and 103:

permutation representation is given

• Page 104 and 105:

construction, G-invariant. 98 Then,

• Page 106 and 107:

100 which this form is defined (see

• Page 108 and 109:

102 First Claim: If F is semiweight

• Page 110 and 111:

104 form is clearly positive on all

• Page 112 and 113:

106 V Vanderbauwhede, A. Local Bifu

• Page 114 and 115:

108 [11,12] and Keener [14]. It cor

• Page 116 and 117:

110 To remove the terms associated

• Page 118 and 119:

112 \ 1 / C P + SNs dP dH TB 2 Figu

• Page 120 and 121:

H sNp 2 114 1 2 Figure 4: Stability

• Page 122 and 123:

116 by one of the two saddle loop b

• Page 124 and 125:

I 2 118 5 6 7 9 10 11 13 I/, 16 21

• Page 126 and 127:

120 simplify if the additional assu

• Page 128 and 129:

The Center Manifold for Delay Equat

• Page 130 and 131:

124 Theorem 2.4 Let X be (D-reflexi

• Page 132 and 133:

126 This ends our analysis of the u

• Page 134 and 135:

128 Theorem 4.2 There is a one-to-o

• Page 136 and 137:

130 Applying p_O* and putting t = 0

• Page 138 and 139:

132 Definition 6.6 (Lipschitz cente

• Page 140 and 141:

6.4 A C ~ center manifold 134 So fa

• Page 142 and 143:

136 this critical value, we may exp

• Page 144 and 145:

138 (iii). If q(0+) /s the adjoint

• Page 146 and 147:

140 is described by ((O,p) = H(O -

• Page 148 and 149:

1 Introduction Local Structure of E

• Page 150 and 151:

144 G and Z(H) = C(H) n H denote th

• Page 152 and 153:

(d2) d(gzg -1,gyg-z) = d(z, y). 146

• Page 154 and 155:

148 Proofi With the notation of Pro

• Page 156 and 157:

lemmas. 150 Our proof is very simil

• Page 158 and 159:

152 Lemma 5.2 Let f 6 Dif~(G/H) cor

• Page 160 and 161:

154 Definition 6.1 ([6]) Let f E Di

• Page 162 and 163:

156 2. For all n > 1, spec(An,G/H)

• Page 164 and 165:

158 1. ~X(z,t) = -r(=,t)h(=,t), (z,

• Page 166 and 167:

160 Lemma 7.1 Let H be a closed sub

• Page 168 and 169:

162 Proof: Set II(Gh/H) = C. Thus C

• Page 170 and 171:

164 I. x( P') has Poineard map P',

• Page 172 and 173:

166 [13] M. W. Hirsch, C. C. Pugh a

• Page 174 and 175:

168 We consider the following non-a

• Page 176 and 177:

2.1 Linearisation 170 We know that

• Page 178 and 179:

172 a map Dq-equivariant with respe

• Page 180 and 181:

and and so 174 = ~ + ~(~, A). ~/ is

• Page 182 and 183:

3.1.2 dimA=l 176 We axe now looking

• Page 184 and 185:

elliptic s~-Ix-al q=3 178 Figure 2:

• Page 186 and 187:

q odd q even 180 °T s~llx-all •

• Page 188 and 189:

q=3 q=4 q_>5 182 0o b = --~orsin(30

• Page 190 and 191:

184 of parameters, we define G .fac

• Page 192 and 193:

186 coordinates, but not of the con

• Page 194 and 195:

We define Au~(p, q) = p= qv - P~ q.

• Page 196 and 197:

qo 7"o ----0 ~o ~o Pa " I=° #o Pa

• Page 198 and 199:

192 [9] J-J.Gervals.Bifurcations of

• Page 200 and 201:

194 is equivalent to its versality

• Page 202 and 203:

1 MORSE FAMILIES 196 We begin by re

• Page 204 and 205:

198 A straightforward calculation s

• Page 206 and 207:

200 2 SYMPLECTIC EQUIVALENCE In thi

• Page 208 and 209:

202 For any G-invariant function ge

• Page 210 and 211:

204 G general in Ep. It follows tha

• Page 212 and 213:

206 smallest codimension of a GL(V)

• Page 214 and 215:

208 + + + Associated to each ~N the

• Page 216 and 217:

4 Z2-SYMMEZ Y 210 We consider coran

• Page 218 and 219:

212 ~y ....................... ~y X

• Page 220 and 221:

214 (2) The parameters ctj appearin

• Page 222 and 223:

216 Z 2 (rn~)Z29 denote the ideal i

• Page 224 and 225:

+ Z2 invariant under the action of

• Page 226 and 227:

Symplectic Singularities and Optica

• Page 228 and 229:

(p',q') = ~(p,q). 222 We claim that

• Page 230 and 231:

224 defined at each point p ~ P by

• Page 232 and 233:

226 2 Hamiltonian dynamics of a par

• Page 234 and 235:

228 Example 3.1 Let ¢p : P1 "} P2

• Page 236 and 237:

230 torsion spring of unit strength

• Page 238 and 239:

232 dimX = 4: A1, A2,A3±,A4,D4±,A

• Page 240 and 241:

234 and v is a smooth function in t

• Page 242 and 243:

Definition 236 Let A c_ (M x 1~, ~x

• Page 244 and 245:

238 5 Diffraction on Apertures Cons

• Page 246 and 247:

Let I be the diagonal in 9. By ~ we

• Page 248 and 249:

a) ~ -~ c) aperture 5/,3, " -NN~. 4

• Page 250 and 251:

Proof / obstacle curve 244 Fig.6.1

• Page 252 and 253:

Fig.6.5 Generating family F 2. Fig.

• Page 254 and 255:

d) ,,,,, ,,,,,,,, L 5 2 248 3f \ 2

• Page 256 and 257:

250 Def'mition A generalized mechan

• Page 258 and 259:

252 For notation, see Fig. A.2. The

• Page 260 and 261:

254 dy for both cases. We find that

• Page 262 and 263:

Dynamics near Steady State Bifurcat

• Page 264 and 265:

258 that C().)=c(),)I. where I stan

• Page 266 and 267:

260 Z "G 1U G 2 U G3U G 4, where (3

• Page 268 and 269:

262 Fix(D:) - { z c Fix(~-)l z 1" Z

• Page 270 and 271:

Appendix 264 Let z_ 3 ..... z 3 den

• Page 272 and 273:

Caustics in Time Reversible Hamilto

• Page 274 and 275:

268 isolated singular points (where

• Page 276 and 277:

270 then f is k-TCz~-determined (in

• Page 278 and 279:

272 Proposition 6 Suppose (c, 4-)

• Page 280 and 281:

Figure 2: The caustic of z +, for I

• Page 282 and 283:

276 by six vector fields. In fact w

• Page 284 and 285:

~\$tract Some Complex Differential E

• Page 286 and 287:

280 precursor distortion is removed

• Page 288 and 289:

282 This is known to exhibit limit

• Page 290 and 291:

284 i2"~ r' i' the modulus sign in

• Page 292 and 293:

We can show that w ~z0 ~ Re @Z I -3

• Page 294 and 295:

288 eigenvalue is always real and i

• Page 296 and 297:

§4 AUTO InteEratlons 290 In this s

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292 Hopf bifurcations arise but AUT

• Page 300 and 301:

CLASSIFICATION OF TWO-PARAMETER BIF

• Page 302 and 303:

296 diffeomorphism germ ~3, 0 ~ ~3,

• Page 304 and 305:

t~ II ~L II II II ~O C~O y - IJ

• Page 306 and 307:

300 References [1] Arnold, V. I., W

• Page 308 and 309:

302 Normal forms for a linear Hamil

• Page 310 and 311:

304 symplectic with involution (or

• Page 312 and 313:

306 (B) If, in addition that, L is

• Page 314 and 315:

308 acting on a subspace ¢g of fig

• Page 316 and 317:

310 Thus, we have a block basis on

• Page 318 and 319:

312 l l li] [-I-I l-I Here, a,b are

• Page 320 and 321:

314 §7. Description of bifurcation

• Page 322 and 323:

316 (B) Purely imaginary eigenvalue

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318 type coordinate transformation

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320 b2p22 + 2qlq 3 _ q22) + AlP2ql

• Page 328:

James Montaldi Irene M. Moroz Marti

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