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

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

Structurally Stable

Structurally Stable Heteroclinic Cycles in a System with 0(3) Symmetry P.Chossat and D.Armbruster Abstract The existence and stability of structurally stable heteroclinic cycles are discussed in a codimension 2 bifurcation problem with O(3)-symmetry, when the critical spherical modes 1=1 and 1=2 occur at the same time. Several types of heteroclinic cycles are found, which may explain aperiodic attractors found in numerical simulations of the onset of convection in a self-gravitating fluid spherical shell (Friedrich, Haken [1986]). Introduction Seemingly chaotic attractors have recently been observed in the numerical study performed by Friedrich and Haken for a problem of steady-state mode interaction in an O(3)-symmetric system, in relation with the hydrodynamical problem of the onset of convection in a fluid sphere (Friedrich, Haken [1986]). The characteristics of the chaotic trajectories suggest that they are associated with a heteroclinic loop connecting equilibria of the system which belong to different group orbits: indeed these trajectories starting from a neighborhood of an equilibrium seem to explore successively regions of the phase space close to different other equilibria before eventually coming back to the fn'st one. We shall see in this paper that such heteroclinic cycles can indeed be found by applying the ideas developed recently for systems with symmetry (Guckenheimer, Holmes [1988], Armbruster, Guckenheimer, Holmes [1988], Melbourne, Chossat, Golubitsky [1988]). These heteroclinic cycles are structurally stable, by which we mean that they persist under

39 small equivariant perturbations (i.e. perturbations preserving the symmetry of the problem). The proof of most results stated here are given in a paper of Armbruster and Chossat [1989]. The purpose of this paper is to show how group-theoretic ideas can explain the numerical observations of Friedrich and Haken and to illustrate these ideas with computer simulations. Let us first set up the problem. Consider the differential equation dX _ F(X,~t), F: Vx ~ 2 > V smooth map at (0,0), (1) dt where V--VI~V2 and VI (1=1,2)is real and invariant by the absolutely irreducible natural representation of the group O(3) of dimension 21+1 (therefore dimV=8). Assume that F(.,~t) commutes with the representation of 0(3) in V defined as the sum of the representations in V1 and V2. This already implies that 0 is an equilibrium of (1) whatever ~t (trivial solution). Thanks to the absolute irreducibility hypothesis we can decompose the linear operator DxF(0,lx) in two blocks (rl(p.)Idv] (1=1,2). We finally assume that ~1(0)=~2(0)=0, so that (0,0) is a steady state-steady state interaction bifurcation point. Note that absolute irreducibility is a generic hypothesis for steady-state bifurcation with symmetry (Golubitsky, Stewart, Schaeffer [ 1988]). A first approach to this problem was made by Chossat [1983]. In the absence of higher order degeneracies, it was shown that there are at most three kinds of equilibria which may occur in the local bifurcation diagram. In particular there exist axisymmetric steady-state equilibria corresponding to pure 1=2 modes. In the framework of Friedrich and Haken these solutions bifurcate transcritically but with a turning point on one branch, so that two different solutions of this type (i.e. belonging to different group orbits) coexist beyond the bifurcation point. They are called (x and ~-cells and play a crucial role in the organisation of the chaotic dynamics observed by Friedrich and Haken. The numerical simulations of these authors exhibited a chaotic, intermittent-type behavior, consisting of trajectories exploring sequentially the 0~ and ~ ceils (and their conjugates under certain group transformations). Our analysis has shown the existence of heteroclinic cycles connecting o~ and I] cells in a way which is coherent with these

  • 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: 37 [GG] Golubitsky, M. and Guillemi
  • 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
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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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    143 Much is already known about the

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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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    273 Now we return to generating fam

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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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    ADDRESSES OF CONTRIBUTORS D.Armbrus

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