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

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

## 284 i2"~ r' i' the

284 i2"~ r' i' the modulus sign in (2.6)b denoting [R~ 2 + R 0 j , where ' and ' refer to real and imaginary part respectively. This equation holds for all z I so that (2.4)b reduces to the nonlinear equation dz I ffi * - ~ R I dt + ~I iRol which must then be solved. In case (B) no simplification is possible and we must solve (2.4) for z 0 and z I . Since the tap weights zj and pre- and post-cursors are complex while t and ~. are real, equations (2.4) and (2,7) constitute J coupled nonlinear complex ordinary differential equations in a real independent variable. We now find the fixed points and determine their stability for each of the cases in (2.5). 2.2 Case (A) ~o >> ~I When ~0 >> ~, we must solve (2.7) for Zl(t ) back into (2.6)b to obtain the behaviour of z0(t ) . r i 0 ffi [R~2 R0 l" ' V - - r 2R0 i2]~ + Ro2] ~ [R o + R o then (2.6)b becomes where z 0 - ± (0 + 19) (2.7) and then substitute If we introduce , (2.8)a (2.8)b

and where 285 r r r i i i r i Ro ~ bo + alxl _ aly I , Ro z bo + aly I + alxl , (2.8)c • bj = b r + ib i a. - a r + ia i zj = xj + lyj , 3 3 ' 3 3 3 Since z o ~ O, the fixed point of (2.7) is (2.8)d z, - - b~ " xe + iye ' (2.9)a rr il rl ir e (b 1 br + blbQ) (b'b° b,b.n) x~ ~ - ~ i 2 ' Yl - r 2 i2 ' b o + b o b o + b o and 'e' denotes evaluation at equilibrium. For convenience we transfer the fixed point (2.9)a to the origin by writing to obtain e z I - g 1 = Z 1 , (2.9)b Z I -- - c~izoboZ I , (2.10) where z o is now suitably modified by the change in variable. At For perturbations = e , the stability of the fixed point is found from A - - c~tbo[Z o + Z, aZtaZl-O Z I -0 (2.11)a

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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

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¢* 1 61 T ........... "--7 y¢, --

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Boundary Conditions as Symmetry Con

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65 We illustrate this point in the

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U T(u) 67 m=2 -~ ................ 0

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69 be the reduced bifurcation equat

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71 3 The Couette-Taylor Experiment

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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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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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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

• 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
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• Page 271 and 272: References 265 Chossat, P. [1Q70]:
• Page 273 and 274: 267 In this paper I consider invari
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• 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
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• 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,
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• 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
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