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

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

## 18 X is then an

18 X is then an invariant subspace of Y and this action induces representations of 0(2) on both X and Y which are orthogonal relative to the inner products < , >4 and < , >0 defined by (5.2) respectively. Also g is equivariant with respect to these representations. The finite cyclic groups Zk, k E Z + (generated by R2,~/k) are all normal sub- groups of 0(2) and are the kernels of the epimorphisms flk : 0(2) ---* 0(2) defined by The mapping hk defined by Z (S) = S. = = (ks) maps Y onto yzk (and X onto X zh) and is an orthogonal linear homeomorphism which satisfies R,~hk = hkR~k, Shk = hkS. Thus the hk's are suitable scaling functions as they satisfy (3.5) for each k E Z +. It is then straightforward to show that g satisfies the scaling law kShkg(x, .~) = g(khkx, k2\$) (5.3) as discussed in Section 1. We note that the epimorphisms k and the homeomor- phlsms h~ satisfy hjhk = hkhj = hjk and so they have the structure of the free abelian monoid Z + (with multiplication). The first bifurcation from the O(2)-symmetric trivial solution occurs at \$ = 4 and is a pitchfork. The Equivariant Branching Lemma holds at this point resulting

]9 in a bifurcating branch of solutions with isotropy subgroup ~ --- Z2 - {I, S}. By Theorem 4.7 and the scaling law (5.3)~ there must also be branches bifurcating from the trivial solution at A = 4k 2, k E Z + which have isotropy subgroups Dk (the dihedral groups generated by R2,,/k and S), since the trivial solution is invariant under the scaling transformations hk. These are the only branches which bifurcate from the trivial solution and so all of these primary solution branches are related by the scaling. Numerical results show that primary branch k (ie. x E X Dk) intersects primary branch 2k at a pitchfork bifurcation point. Thus, primary branch k consists of a loop of solutions intersecting the trivial solution and primary branch 2k. If secondary bifurcation from primary branch k occurs resulting in a branch of solutions with isotropy subgroup Dt say, with k = lm for some l, m E Z +, then this bifurcation can only be scaled back to a bifurcation on primary branch m using hl. The KS equation has many branches of solutions (see for example Scovel et al (1988)) and so we show only a few to illustrate these ideas in Fig 5.1 where branches labelled by "n" have isotropy subgroup D, and rescaled branches are shown using dotted lines. By using a logarithmic scale for A, the scaling takes the form of a simple translation (in A) which makes the results clearer. This idea has previously been used by Duncan and Eilbeck (1987). Finally, we note that (periodic) travelling wave solutions of the time dependent KS equation occur when the reflectional symmetry of a Dk-symmetric branch is broken. These travelling wave solutions satisfy a "steady-state" equation and the scaling law (5.3) can be extended to include this equation also (see Aston, Spence and Wu (1989)). Acknowledgements The author acknowledges the support of the Science and Engineering Research Council throughout the course of this work.

• 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: 17 and from Lemma 4.2, hj : X ~k --
• 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

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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

• Page 85 and 86:

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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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• Page 151 and 152:

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

• Page 157 and 158:

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

• Page 205 and 206:

Remark 1.3 More generally, the orga

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(W*~V,0) (v,o) (where n2 is the nat

• Page 209 and 210:

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

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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>

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255 Poston, T. and Stewart, I. [197

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257 0ii) A(),)-A 0 + B(X) is a hoto

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259 their lists. The correct lists

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261 denote the generator of the Lie

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263 4 ÷ D6 d O(3), O(2)-, O- O(2)-

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References 265 Chossat, P. [1Q70]:

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267 In this paper I consider invari

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269 be the set of critical points i

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271 Figure 1: Two trajectories in a

• Page 279 and 280:

• Page 281 and 282:

~LLL I 275 ~::'.'~'.C': • ,',': :

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277 Let L = £(J20) and m E M. Work

• Page 285 and 286:

§1. Introduction 279 In this paper

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where 281 Im(ei~R0) - 0 , (l.4)b M

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283 b 0 to avoid negative suffices.

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and where 285 r r r i i i r i Ro ~

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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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291 I. The fixed point (2.9) remain

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References 293 [i] R. W. Lucky (196

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295 -- ba+~+~t h For small values o

• Page 303 and 304:

h =0 XX ~0 (h~ 1, h~L 2) H = (0, O)

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299 4 Description of the proof of t

• Page 307 and 308:

Versal Deformations of Infinitesima

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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

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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

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