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

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

## 40 numerical

40 numerical experiments. A heteroclinic cycle involving Sirnikov-type connexions has also been found. Some of these heteroclinic cycles can be asymptotically stable for an open set of parameter values. For the values taken by Friedrich, Haken [1986] these objects are unstable, but still, they locally govern the dynamics. Let us now set up the "group theoretic" framework for the study of HC's in symmetric systems. Given a point x~ V, the isotropy subgroup of x is Z={g~O(3) / gx=x} (here we abuse notation in identifying the representation of O(3) with 0(3) itself). We define Fix(Z)={y~ V/Zy=y}. This is a subspace of V with the remarkable property that, because F(.,~t) commutes with the group representation, it is flow-invariant for equation (1). Points in the same group orbit have conjugate isotropy subgroups. Conjugacy classes of isotropy subgroups are called isotropy types. They are partially ordered by group inclusion, which forms the so-called isotropy lattice. If Y, DT, then Fix(Z)C Fix(T). The following proposition is just the formalization of a very simple idea. Proposition 1. Assume that there exists two sequences of isotropy subgroups: Z: ...... Z k with Z k conjugate to Z 1, and T 1 ...... Tk_l, such that for every j=l ..... k-l, one has: i) TjC ~'.j and Zj+I; ii) dimFix(Ej)=l and dimFix(Tj)=2; iii) there exist saddles Xj with isotropy Zj, attracting trajectories in Fix(Z j); (iv) in each Fix(Tj) there exists a connection Xj ---> Xj+ 1, with X k standing for the copy of X 1 in Fix(2:k). Then there exists a structurally stable heteroclinic cycle connecting the equilibria X 1 ..... X k. Point (iv) is the only non-algebraic condition to check in this proposition. The fact that the connection Xj ---> Xj+ 1 is required in the flow-invariant plane Fix(Tj) makes life simpler, thanks to the Poincar6-Bendixon theorem. In Melbourne et al. [1988] more precise conditions were derived for the existence of these connexions when the equilibria bifurcate from the origin. These conditions am based upon the assumptions of existence of a Lyapounov function (leading to forward bounded trajectories) and of non-existence of equilibria with isotropy Tj (j=l ..... k-l).

41 It may happen that an obstruction to the connexion Xj ~ Xj+ 1 in Fix(Tj) exists for some value of j, due to the presence of a sink Yj with isotropy Tj, the stable manifold of which contains the unstable manifold of Xj. In this case it may happen in addition that Yj is connected to Xj+ 2 in a way which realizes a new, structurally stable heteroclinic cycle, bypassing Xj+ 1 but involving Yj. Take j=l for simplicity and suppose that T 1 and T 2 contain an isotropy group A such that dimFix(A)=3. If there exists a connexion in Fix(A) between Y1 and X2CFix(T2), this connexion is structurally stable since X 2 is a sink in Fix(A). Since the connexion between X 1 and Y1 in Fix(T 1) is also structurally stable, we have now a heteroclinic cycle involving the connexions X1---~ Y1---~ X 2, the second one running in the 3-dimensional space Fix(A). A variant of this situation is when the sink Y1 has bifurcated to an attracting limit cycle in Fix(T1), which connects itself to X 2. We shall encounter these two kinds of heteroclinic cycles in section 3. The main result of this paper is that heteroclinic cycles of the various types described above can bifurcate from the zero state under generic conditions. They are described in section 4.1 (table 2). The asymptotic stability (local attractivity) of a heteroclinic cycle depends on the relative rates of contraction-expansion at the involved equilibria. When there exist continuous group-orbits of equilibria, as is the case in problems with O(3)-symmetry, the stability is of orbital type (the orbits, rather than individual equilibria, are attracting). Asymptotic stability is discussed in section 4.2. It turns out that the heteroclinic cycles listed above do not bifurcate as attracting invariant sets, but stability conditions can be satisfied at finite distance on the bifurcating branches.

• 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: 39 small equivariant perturbations
• 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
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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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