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

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

## 304 symplectic with

304 symplectic with involution (or L E sp-(V,p)) if L E sp(V) and pL =-Lp (or equivalently, Hop = H). (~) Fix an inner product (,) on V. Define the dual L* of a linear transformation , L via (L x,y)= (x,Ly) for all x,y. Introduce a linear transformation J via w(x,y) = (x,Jy) for all x,y (Thus, J + J = 0). One can readily see that L E sp(V) if and only if JL is symmetric. In this situation, the associated Hamiltonian H = ½ (x,Ax) with A = -JL. It is of interest to note that J (p) is orthogonal if and only if j2 = -1 (p* = p). ('r) Let V{: = V~iV be a complexification of V. For convenience in matrix computations, we extend w,p,L, (,),J to V£ so that they become linear or bilinear over {:. When no confusion may arise, we will use the same notation for these extensions. A complex basis of V is a basis of Vf in the form el,...,en, el,...,en. To each x E V, (zt,...,Zn) is called the complex coordinates of x if x = L ~n zie i ÷ L ~n zie i . To each real linear transformation L on V, the i=l i=l associated matrix (L) = (t s t) is given by Lf(et) = ~s/stes ' s,t E {1,...,n,i,...,n}, with ef = El, etc. Thus, Tst = t~ with the convention ~ = s. Using complex coordinates on V, (Lz)i = L~j=I /ijzj + L~j=l gl~z--j, i= 1,...n, where z = (Zl,...,Zn). The associated matrix (Wst) is given by Wst = W(es,et), s,t E {1,...,~} . Thus ~sst = w~ ~ and Wst = - Wts. Using complex coordinates on V, w(z,¢) = 2 Re [L-~i= 1 zi(~jj =1 wij~j + L~j: 1 wi] ~j)]" Now, consider an orthonormal complex basis el,...,e n, el,...,en in which (e I + El) , i(e 1 - el),... , (e n + en) , i(e n- ~n) forms an orthonormal basis of V. The inner product on V can now be expressed as = Re (L~i=l zi~i). Furthermore, (L*) = (Pst) with g*st=g~ (or ~ts ). From w(x,y)= w(x,Jy)= Re[L~i=l zi(L~j=l

305 Jij q + L~j=l Ji]~j )' we have -J'ij = 2wi], Ji] = 2 wij, i,j = 1,...,n. Be aware that the complex multiplication on V given by c(zi) = (czi) through the introduction of complex basis is different from the complex multiplication through complexification of V. For a general discussion of complex basis, please consult Nickerson et al. [8] p. 459. §3. Normal forms Let us describe the normal forms for L ¢ sp(V) or sp-(V,p) found in Wan [9]. A decomposition V = Vl\$...e V r into subspaces Vj is sam to be symplectic if w(vi,vj) = 0 for any v i e Vi, vje Vj, i \$ j. Theorem 1 (A) Suppose _a linear transformation L is infinitesimally symplectic (i.e. L E sp(V)). Then, there exists ~ symplectic decomposition V = VI\$...*V m into L-invariant s_ubspaceS Vj, such that on each Vj, _a suitable real or complex base can be chosen with the matrix representations ~ L,w given below according to eig~nvalues L I Vj. eigenvalues L ±bi ±bi :t:a *A,+~ A=a+bi L 0 Lo@L 0 Lib@L_ib Lib*L._ib L ~L & ---a LAeL ~ eI~L_A E~ 0 EJ o i 2 o o 1~i% t ~iJo~iJo 1 ~'0 Jo~(-1)nj ° J2~)J2 ba.~es p e; n = even P(en) = (-1)n-l& n e,f; n = odd P(en) = e n PCfn) =-f ~,;; n = ewn PC%) = -~ e,e; n = odd e,f e,f; e,f p(%) = P(en) = (-1)n-lf p(e n) = (-1)n-lf

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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

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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

• 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
• 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.
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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,
• 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: 303 form. (2) By dropping the sympl
• 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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