Views
5 years ago

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

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

314 §7. Description of

314 §7. Description of bifurcations with codimension < 2 We intend to apply the previous results to bifurcations of linear Hamiltonian systems L(#) with a fixed involution p, near/~ = 0. Thus, we may assume L = L(0) has only purely imaginary eigenvalues, and L(0), w, p are in normal form as in Theorem 1. At zero eigenvalue, consider bifurcations with the origin be fixed to that v codimension = ½ Zj nj(0) + ~ (by Corollary 2). At purely imaginary eigenvalues • bi, allow b as a parameter, so that codimension = Z jnj(A) -1. In order to deal with bifurcations of periodic A =*bi solutions in Hamiltonian systems (with involution), one has to examine the resonance cases so that the count for codimensions needs to be modified in that situation. (A) Ei~envalue zero. Using x-coordinates on Eo, we have = L(/~)x, L(p) = J A, H = 1 ~,jx i ai j xj. Here, A = (aij) = A o + S with A o = - JL(0). J, L(0), p are in normal forms as in Theorem 1, and B is as in Theorem 3.(b). type cod J #(x) A (0) 2 (0) 4 (o)i(o) I 2' J ,•' _ 1 11 E -I 1 -1 -1 1} 6 (x I ,-x 2 ) 6 (x 1,-x2,x3,-x 4) (x l,-x 2) /~0 O0 Ou 01 00-10 01 O0

H i [X22"FILXl 2] ~ [(2x2x4-x32) -Flzx 12-k/~x22 ] 1 2 2 ~'h +'~21 L(/~) 0 I00' 0010 Ou01 -#000 o} 315 eigenvalues z2+/~=0 z4--~z2+/~:0 z2+pv=O Recall that, (0) k = Jordan block of size k with eigenvalue 0. E, ~f = • 1. Z = bifurcation diagram of eigenvalues of L(/~). To obtain H in p-symplectic coordinates, one simply uses the coordinate transforms given in §3. type (0) 2, ~= 1 (0) 2, ~ : - 1 (0) 4, ~= I (o)4,6=-1 (o)I(o) I coordinate transformation x I = E ql'x2 = Pl x I = 6 PI' x2 =-ql x I -- E ql'x2 = P2 x 3 -- E q2'x4 = Pl x I =EPl, X2 =-q2 x 3=6p2, x4 =-ql xl ---- ql'x2 : Pl H(q,p) E 2 2 ~Pl +ml ) E 2 (ql + ~t pl 2) ,,Ix Z E2--[(2 plP2 - q22) q- ~ ql 2 + v p22] 2~(2qlq2 -- p2 2) + ~ pl 2 + v q2 ~ ~(m 1 I 2 +vpl 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 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 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: 313 b'st = r (-1)s-t[; s t' case(c)
  • 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
Page 1 Page 2 D B 5 6 D MOTOR Mo~, lmotowon T R T ...
B E R L I N C H A U S S E E S T R A S S E 5 7 - 6 1 B ... - Glen Leddy
Page 1 7 6 5 | 4 L 3 | 2 | 1 CK T ll ll ll )l KLÖ'ZOJ T ll ll ll ll klözo) A B ...
D e se m b e r 0 5, V olum e 0 6, Issue 11 w w w .ta n go n o tica s.co m
H O T E L - R E S TA U R A N T- B A R S e e s t r a s s e 2 2 5 6 3 ...
P P P NP NP 12 3 4 5 6 7 8 W 9 10 11 12 13 14 W 15 16 0 17 A C B ...
1 5 . T T O - F O R U M 1 2 . T T O - F O R U M O c t o b e r 1 6 ...
DVAA: Flat Galaxies (by R.A.) A B C D E F G H I J K L 1 2 3 4 5 6 7 8 ...
Reading grade 6 2.A.5.b - mdk12
a 1 2 a b b a 3 4 a b b a 5 6 a b b a 7 8 a b b a 9 10 ... - CodeMath.com
8 * 5 * 4 * 3 * 2 * 1 A 8 A 7 A 6 A 5 A 4 A 3 A 2 B 8 B 7 B 6 B 5 B 4 B 3 ...
Lösung 6 Aufgabe 1 a) b) c) Aufgabe 2 Aufgabe 3 a) 5 4 2 6 4 20 9 2 ...
5 b 5 6 10 c 12 a 5 8 A a 13 b Classroom Exercises - flip@mrflip.com
EAS 540 - STABLE ISOTOPE GEOCHEMISTRY Thrs., BUS B-5; 6 ...
Panel B 4:45-5:30 Round 2 Panel A 5:30-6:15 | Panel B 6:15-7 ...
TEST ANSWERS Version A 1. B 2. B 3. E 4. E 5. D 6. B 7. C 8. B 9. A ...
Solutions to Quiz 5 Sample B - Loyola University Chicago
¡' &) (10 , 2 354 6&7 98@ ) AC B$ E DG FH I 6 &Q ... - Université Lille 1
(0 12 3¡) 4(5 6 3 798 9 %@4a7(6 b cedgfhfhiqps rti u)vxwtwty
V 5 1 5 B 6 L 4 X P T S F
1. Nyelvismereti feladatsor 1. C 2. B 3. D 4. D 5. A 6. B 7. B 8. C ... - Itk
1 A 2 B 3 A 4 B 5 6 7 8 A 9 B 10B½ 11 A 12 13 14 B 15 A 16 B 17 A ...
Friday Warm-up A 5:00-5:30 PM, Warm-up B 5:30-6:00 PM Starts at ...
1. NC 2. L 3. NP 4. D 5. F 6. A 7. M a) b) b) 8. T 9 ... - Bangladesh Bank
1. Nyelvismereti feladatsor 1. A 2. B 3. A 4. A 5. D 6. D 7. B 8. C 9 ... - Itk
Midterm 1 KEY MC Answers 1. B 2. B 3. A 4. D 5. A 6. D 7. C 8. A 9 ...
1 2 3 4 5 6 A B C D Last Name FirstName Major Advisor ... - Alumni
F i b r l o kT M 2 6 5 0 -AC K - Connex Telecom