Views
5 years ago

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

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

CLASSIFICATION OF

CLASSIFICATION OF TWO-PARAMETER BIFURCATIONS Martin Peters 0 Introduction The problem treated in this paper is the classification of two-parameter bifurcations in one state variable up to codimension one, using a two-parameter version of parametrised contact equivalence. This notion was introduced by Golubitsky and Schaeffer [5] in order to study bifurcations using methods from singularity theory. In [6] the same authors classify one-parameter bifurcations up to codimension four. The result described below consists of the following components: 1 A list of normal forms for the germs having codimension less than or equal to one. 2 Recognition conditions for each normal form in the list, i. e. conditions that characterise the equivalence class of the normal form. These conditions are equations and inequalities for the Taylor coefficients of the germs. 3 Universal unfoldings for each normal form and their geometrical description. There is a result due to Izumiya [7], who classified a restricted class of two-parameter bifurcations, namely germs of the form x 2 + q0(X 1,~-2), ~'1 and ~2 being the parameters. As we shall show, even at codimension zero there are germs which are not of the above form, e. g. x3 + X~.l + ~.2. The normal forms given in this paper also appear as part of another classification by Arnold etal. [1], which arises in a related but different context. This coincidence is not obvious in advance and does not continue at higher codimension. I thank my PhD supervisor Dr. Ian Stewart for his support -- mathematical and otherwise. Furthermore, I thank Dr. Mark Roberts, Dr. Ian Melbourne, Dr. Ton Marar and Prof. Jim Damon for some very helpful discussions and advice. 1 Notation We denote coordinates in ~ x ~2 by x, ~'1, ~.2- Putting ~, := (~1, 7~2) we define ~x,;~ tobe the ring of all C**- function germs ~ x ~2-----~IR at (0,0) ~ ~ x ~2. k4x) " denotes the maximal ideal in ~xA" Let h be a germ in ~x;k- We denote its Taylor coefficients as follows:

295 -- ba+~+~t h For small values of 0t, ~ and T we write h x, hxx, h~.t~.t, hx~.l~q etc., instead. It will always be clear from the context, whether h = 0 means h(0) = 0. The symbol sg denotes the sign function. 2 Parametrised contact equivalence In this section we define parametrised contact equivalence for two-parameter bifurcations. This definition is analogous to the one introduced by Golubitsky and Schaeffer in the one-parameter case (See [5] and [6]. ). We introduce another slightly modified version of this equivalence relation. Each equivalence relation corresponds to a group. The following definition incorporates both equivalence relations. For notational convenience we use the term E-equivalence for parametrised contact equivalence. Compare [1], [3], [4] and [8] for the concept of ordinary contact equivalence. 2.1 Definition. Two germs f, g E ~4 x,z are E-equivalent if there exist smooth germs S, X: ~3, 0---, lrl, and A 1, A2: ~2 0 ~ ~l such that g (x, Z 1, Z p = s (x, Z l , Z p f (x (x, Z I , Z e), A I (;~ t , Z e), A 2(Z ~ , Z p ) and the following conditions are satisfied: X(O, O, O) = 0 A~(O, O) -- 0 A2(O, O) = 0 (2. I) S(O, O, 0)>0 xx(o, o, o) > o (ap z (A2)}~ I (A2)~ 2 ~0. Furthermore, if the germs X, A I, A 2 and S satisfy the conditions (2.1) and additionally s(o)= I (Apx I = I (2.3) (ap~ 1 = 0 then f and g are U-equivalent. (aPxe = I Let E be the set of all quadruples (S, X, A 1, A2) satisfying the conditions (2.1) and (2.2). E acts on k4x) " in the followihg way: Let f ~ k4xA and e = (S, R) ~ E, where R = (X, A 1, A2). The conditions in the previous definition imply that R is a (2.2)

  • 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 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

  • Page 75 and 76:

    69 be the reduced bifurcation equat

  • Page 77 and 78:

    71 3 The Couette-Taylor Experiment

  • Page 79 and 80:

    73 00 u(x,y) = v(x,y) = ~-x (x,y) =

  • Page 81 and 82:

    75 (dS) F of S at F has an eigenval

  • Page 83 and 84:

    77 experimental geometry. 'Upper bo

  • Page 85 and 86:

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

  • Page 87 and 88:

    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

  • Page 95 and 96:

    in particular, Zp~ = 1 + ~ iii) Vp

  • Page 97 and 98:

    91 Similarly we let X(G) denote the

  • Page 99 and 100:

    93 F I Fix(G') x ~ : Fix(G') x $t -

  • Page 101 and 102:

    Example 4.5: 95 Let G = Z/mE act on

  • Page 103 and 104:

    97 Thus, we see that X q = 1 + 4V.

  • Page 105 and 106:

    99 Q(F1) = Cx,~/I(F1) = Cx,~/I(F 0)

  • Page 107 and 108:

    Corollary 7 101 For i) we know by t

  • Page 109 and 110:

    103 generator for m A, the module o

  • Page 111 and 112:

    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 .

  • Page 125 and 126:

    119 extent the dynamics is influenc

  • Page 127 and 128:

    121 [12] J. Guckenhelmer, SIAM J. M

  • Page 129 and 130:

    123 produces a continuous function.

  • Page 131 and 132:

    is generated by 125 V(A~) = {f : f(

  • Page 133 and 134:

    and the abstract integral equation

  • Page 135 and 136:

    129 (iii). X® and X+ are finite di

  • Page 137 and 138:

    131 Definition 6.1 Let E and F be B

  • Page 139 and 140:

    133 6.3 Contractions on embedded Ba

  • Page 141 and 142:

    135 Theorem 6.13 (Center Manifold)

  • Page 143 and 144:

    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

  • Page 149 and 150:

    143 Much is already known about the

  • Page 151 and 152:

    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

  • Page 165 and 166:

    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

  • Page 171 and 172:

    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

  • Page 179 and 180:

    173 From the fixed-point subspaces

  • Page 181 and 182:

    175 and the following curve is the

  • Page 183 and 184:

    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 ~

  • Page 187 and 188:

    3.2 dimB--1 181 We are now in the s

  • Page 189 and 190:

    183 Puting them back into (14) and

  • Page 191 and 192:

    4.2 The recognition problem 185 Thi

  • Page 193 and 194:

    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

  • Page 201 and 202:

    195 internal variables p and extern

  • Page 203 and 204:

    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

  • Page 213 and 214:

    3 FINITE DETERMINACY 207 Good deter

  • Page 215 and 216:

    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

  • Page 221 and 222:

    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

  • Page 227 and 228:

    221 instrument in terms of composed

  • Page 229 and 230:

    ~p'_-sin0' ~0' sin0' 3p -sin0 30 si

  • Page 231 and 232:

    225 perfect gas with V = volume, S

  • Page 233 and 234:

    227 according to co and all maximal

  • Page 235 and 236:

    a) a submersion p : X -~ Y, 229 b)

  • Page 237 and 238:

    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
  • Page 269 and 270: 263 4 ÷ D6 d O(3), O(2)-, O- O(2)-
  • 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.
  • Page 291 and 292: and where 285 r r r i i i r i Ro ~
  • 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: References 293 [i] R. W. Lucky (196
  • 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
Reading grade 6 2.A.5.b - mdk12
V 5 1 5 B 6 L 4 X P T S F