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

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

## etc. The non-degeneracy

etc. The non-degeneracy condition to be satisfied at (Xo, Ao) is 14 o o (4.3) < ¢o,g~¢o + gxx¢oV ># 0 where ¢0 E .hf0 M X fi, ¢o E Af~ M X fi and v E X n is the unique solution of g°v + = o. (4.4) Similarly, the non-degeneracy condition to be satisfied at (bhxo, IAo) is < Ch,gx~¢h h + g~:~¢hvh ># 0 (4.5) where Ca E Afh V} X fi', Ch E A/'~ M X h~ and vh E X u~ is the unique solution of h h gxvh + gx = O. (4.6) By differentiating the scaling law (3.6), it is easily shown that v E X u is a solution of (4.4) if and only if vh = ~hv E X ~ is a solution of (4.6) since b, c, l # 0. Also, we saw in (ii) that h(Afo M X fi) = Afh M X ~ and so we can write Ch = h¢o. As h is an orthogonal transformation~ we have for all xl,x2 E X using (4.2). Thus 0 0 < xl,gxx2 > = < hxl,hgxz2 > b = - < hxl,g~hx2 > C o, b h. < (g~) xl,x2 >= - < (g~) hxl,hz2 > C and so x~ E Af~ if and only if hxl E .Aft, that is hAf~ = Af~ M X ~'. Hence, h(Af~ M X fi) = Af~ M X fi~ by Lemma 4.2 and the fact that X ~z C_ X ~', and so we can write Ca = h¢o. It is then a matter of calculation to verify that h h v h b h V 0 0 = < he0, ~h(g~x¢0 + g~¢0v) > e o o = -- + g;~¢oV > bl < ¢o, g~x¢o

]5 since h is orthogonal. As b, e, 1 ¢ 0, we conclude that (4.3) holds if and only if (4.5) holds. [] As a consequence of Theorem 4.7, it is clear that every bifurcation point on the primary branch can be re-scaled to produce a bifurcation point on the scaled branch. However, a bifurcation point on the scaled branch can be scaled back onto the primary branch if and only if the isotropy subgroup (~ of the secondary branch has ~ as a subgroup in which case the branch which bifurcates from the primary branch has isotropy subgroup/3((~). It is possible that the symmetry of ~ can be lost at a bifurcation point on the scaled branch so that the bifurcation point cannot be scaled back to a bifurcation point of the primary branch. The situation may occur where ~# = fl (eg. fl = F) in which case the primary and scaled branches could coincide. However, Theorem 4.7 still holds in this sit- uation and the two secondary branches then bifurcate from the same branch at different values of A (provided that l # 1). Throughout our analysis, we have not been able to say that A/h = hA/0, only that hA/0 C_ A/h. In the generic situation, both A/0 and A/h are irreducible (Golu- bitsky, Stewart and Schaeffer (1988, p82)) in which case equality holds by Lernma 4.4. However, in a two parameter problem, it is possible that mode interaction could occur on the scaled branch so that A/h # hA/0 at a critical value of the second parameter. In this case, Theorem 4.7 still applies ensuring the existence of a sec- ondary branch bifurcating from the scaled branch, thus providing-some information about the bifurcating branches at a point of mode interaction. We note that it is not possible for a scaled branch to be the continuation of the primary branch since they have different isotropy subgroups and the isotropy subgroup of a branch of solutions is preserved globally along the branch (Healey (1988)). However, the primary and scaled branches could intersect at a bifurcation point (as in the KS equation - see Section 5).

• 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: 13 such that the Equlvariant Branch
• Page 23 and 24: 17 and from Lemma 4.2, hj : X ~k --
• Page 25 and 26: ]9 in a bifurcating branch of solut
• 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
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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

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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

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~LLL I 275 ~::'.'~'.C': • ,',': :

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277 Let L = £(J20) and m E M. Work

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§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

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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

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

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299 4 Description of the proof of t

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Versal Deformations of Infinitesima

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303 form. (2) By dropping the sympl

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305 Jij q + L~j=l Ji]~j )' we have

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307 For (0) n, n--even,~= 1, set Ix

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D% I -% ,,~ Fig. 1 309 Each oblique

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311 I -- T/ Fig. 2 7~-form Now, def

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313 b'st = r (-1)s-t[; s t' case(c)

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H i [X22"FILXl 2] ~ [(2x2x4-x32) -F

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H 2Re[+2AZlZ2+Z2Z2 +#zlT" l ] 2E--R

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319 unfolding H(g) of a Hamiltonian

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