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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

  • Page 60 and 61:

    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

  • Page 72 and 73:

    66 (b) Next we discuss the expected

  • Page 74 and 75:

    68 Generalizations In this section

  • Page 76 and 77:

    70 (c) Higher-Dimensional Domains T

  • Page 78 and 79:

    72 held fixed and the speed of the

  • Page 80 and 81:

    74 Hence odd modes in y bifurcate i

  • Page 82 and 83:

    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

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    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

  • Page 98 and 99:

    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

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    268 isolated singular points (where

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    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

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    ~$tract Some Complex Differential E

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    280 precursor distortion is removed

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    282 This is known to exhibit limit

  • Page 290 and 291:

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

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

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    288 eigenvalue is always real and i

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    §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

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    296 diffeomorphism germ ~3, 0 ~ ~3,

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    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

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    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

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    314 §7. Description of bifurcation

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    316 (B) Purely imaginary eigenvalue

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    318 type coordinate transformation

  • Page 326 and 327:

    320 b2p22 + 2qlq 3 _ q22) + AlP2ql

  • Page 328:

    James Montaldi Irene M. Moroz Marti

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