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

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

I.M. Moroz, Some complex

I.M. Moroz, Some complex differential equations arising in telecommunications ..... M. Peters, Classification of two-parameter bifurcations . . . . . . . . . . . . . . . Yieh-Hei Wan, Versal deformations of infinitesimally symplectic transformations with antisymplectic involutions . . . . . . . . . . . . . . . . . . . . . Addresses of contributors . . . . . . . . . . . . . . . . . . . . . . . . . VIII 278 294 301 321

Scaling Laws and Bifurcation P.J.Aston ABSTRACT Equations with symmetry often have solution branches which are related by a simple rescaling. This property can be expressed in terms of a scaling law which is similar to the equivarlance condition except that it also involves the parameters of the prob- lem. We derive a natural context for the existence of such scaling laws based on the symmetry of the problem and show how bifurcation points can also be related by a scaling. This leads in some cases, to a proof of existence of bifurcating branches at a mode interaction. The results are illustrated for the Kuramoto-Sivashinsky equation. 1. Introduction Consider the bifurcation problem g(x,A) ---- O, g:XxR--+X (1.1) where X is a real Hilbert space, and suppose that g(0, A) = 0 for all A E R. If g also satisfies the equivaxiance condition Sg(x, A) = g(Sx, A) (1.2) for all x E X, where S E L(X) is an orthogonal transformation for which S # I, S 2 = I, then it is well known that symmetry-breaking bifurcation from the

  • Page 1 and 2: Lecture Notes in Mathematics Editor
  • Page 3 and 4: Editors Mark Roberts Ian Stewart Ma
  • Page 5: Co~e~s P.J. Aston, Scaling laws and
  • 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 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

  • Page 71 and 72:

    65 We illustrate this point in the

  • Page 73 and 74:

    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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    79 M.G.M. Gomes [1989]. Steady-stat

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    81 To describe the results, we supp

  • Page 89 and 90:

    83 The author's work on these quest

  • Page 91 and 92:

    85 certain elements of G may interc

  • Page 93 and 94:

    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

  • Page 99 and 100:

    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

  • Page 109 and 110:

    103 generator for m A, the module o

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    105 D3 On the number of branches fo

  • Page 113 and 114:

    On a Codimension-four Bifurcation O

  • Page 115 and 116:

    109 Setting x = u3 and introducing

  • Page 117 and 118:

    111 not matter. We choose b > 0. Th

  • Page 119 and 120:

    113 SLot x SLs Figure 2: Phase port

  • Page 121 and 122:

    SLs(H} H.(SL s) SNs I SN~ (~6) 115

  • Page 123 and 124:

    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

  • Page 145 and 146:

    and 139 (q,p) = foh d-"~)p(-r) (7.8

  • Page 147 and 148:

    [Cha71] [Die87] [Dui76] [DvG84] [Ha

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    143 Much is already known about the

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    145 We refer to ,g = {Su[y E a} as

  • Page 153 and 154:

    147 ttemark 3.1 It follows by our m

  • Page 155 and 156:

    149 Definition 4.1 Let 11 be a clos

  • Page 157 and 158:

    151 Remark 4.3 It follows from Theo

  • Page 159 and 160:

    (a) f(=) = ~(=)x, au x E GIH. (b) f

  • Page 161 and 162:

    155 Since ] and ~ are smooth, so ar

  • Page 163 and 164:

    157 of X at all points z E a. Neces

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    159 In this section we wish to desc

  • Page 167 and 168:

    161 A straightforward application o

  • Page 169 and 170:

    8.1 Poincard maps 163 We review the

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    165 Proposition 8.3 Let P. be a rel

  • Page 173 and 174:

    On the Bifurcations of Subharmonics

  • Page 175 and 176:

    169 and so we give in a fourth part

  • Page 177 and 178:

    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

  • Page 185 and 186:

    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

  • Page 195 and 196:

    qo r0 =0 =O #o ~o Pa. ~o 1=° top-c

  • Page 197 and 198:

    qo =0 #o References .p~ =0 191 ~o ~

  • Page 199 and 200:

    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

  • Page 207 and 208:

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

  • Page 209 and 210:

    G $~G-versal unfolding of F(.,0) in

  • Page 211 and 212:

    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

  • Page 217 and 218:

    211 where q%(x,y) = Z Vbc(X'y)xc an

  • Page 219 and 220:

    Theorem 4.5 213 (i) If r >_. s then

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    215 (4) In [JR] we show that the ca

  • Page 223 and 224:

    217 In terms of the invariants the

  • Page 225 and 226:

    R~P'ERF~CES [AI [ADI [AGVI [BPW] [D

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    221 instrument in terms of composed

  • Page 229 and 230:

    ~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

  • Page 249 and 250:

    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

  • Page 267 and 268:

    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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    273 Now we return to generating fam

  • Page 281 and 282:

    ~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

  • Page 287 and 288:

    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

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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

  • Page 323 and 324:

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

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    319 unfolding H(g) of a Hamiltonian

  • Page 327 and 328:

    ADDRESSES OF CONTRIBUTORS D.Armbrus

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