Patterns in Square Arrays of Coupled Cells

Transcription

1 Ž JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS 08, ARTICLE NO AY Patterns in Square Arrays of Coupled Cells David Gillis Department of Mathematics, Uniersity of Houston, Houston, Texas and Martin Golubitsky Department of Mathematics, Uniersity of Houston, Houston, Texas Communicated by William F Ames Received August 3, INTRODUCTION In recent years it has been observed that reactiondiffusion equations with Neumann boundary conditions Ž as well as other classes of PDEs possess more symmetry than that which may be expected, and these hidden symmetries affect the generic types of bifurcation which occur Žsee Golubitsky et al 6, Field et al 5, Armbruster and Dangelmayr 1, Crawford et al, Gomes and Stewart 8, 9, and others In addition, equilibria with more highly developed patterns may exist for these equations than might otherwise be expected Epstein and Golubitsky 4 show that these symmetries also affect the discretizations of reactiondiffusion equations on an interval In particular, equilibria of such systems may have well-defined patterns, which may be considered as a discrete analog of Turing patterns In this paper, we use an idea similar to the one in 4 to show that the same phenomena occurs in discretizations of reactiondiffusion equations on a square satisfying Neumann boundary conditions Such discretizations lead to n n square arrays of identically coupled cells By embedding the original n n array into a new n n array, we can embed the Neumann boundary condition discretization in a periodic boundary condition discretization and increase the symmetry group of the equations from X97 $500 Copyright 1997 by Academic Press All rights of reproduction in any form reserved

2 488 GILLIS AND GOLUBITSKY square symmetry to the symmetry group D Ž 4 HZ n, which includes the discrete translation symmetries Ž Z n This extra translation symmetry gives rise Ž generically to branches of equilibria which restrict to the original n n array yielding equilibria of the equations with Neumann boundary conditions As shown in Figs 37 of Section 4, these equilibria may take the form of rolls or quiltlike solutions, which are comprised of square blocks of cells symmetrically arranged within the array, with each block containing its own internal symmetries The notation used in the figures is explained in Section 3 This paper is structured as follows In Section we discuss the embedding and the corresponding symmetry group In Section 3 we list the irreducible representations of, and use the equivariant branching lemma to produce our solutions The proofs of the results listed in this section are given in Section 6 The analysis of steady-state bifurcations on the large array of cells is a discretized version of the analysis in Dionne and Golubitsky 3 of bifurcations in planar reactiondiffusion equations satisfying periodic boundary conditions on a square lattice In Section 4 we discuss the types of pattern that arise from these solutions Finally, in Section 5 we show how to compute the eigenvalues of the Jacobian at a trivial equilibrium for a discretized system of differential equations THE EMBEDDING We begin by considering a square system of n identical cells with Ž k identical nearest neighbor coupling Fig 1 Let xl, m R be the state variables of the Ž l, m th cell ŽThe number k is the number of equations in the original PDE system The general form of the cell system may then be written as l, x m fž x l1, m, x l, m1, x l, m, x l, m1, x l1, m, 1l, mn, Ž 1 where f: ŽR k 5 R k satisfies fž u,,w, x, y fž u, x,w,, y fž y,,w, x,u fž x,u,w, y, Ž In Ž 1 we use the convention that the indices l and m satisfy l 1 1 when l 1 and l 1 n when l n This accounts for Neumann boundary conditions The equalities Ž just reflect the fact that the coupling is identical in all directions In general Ž 1 has precisely square symmetry where the symmetries consist of permutations of the cells induced by the eight rotations and reflections of the square We embed Ž 1 in a larger dimensional system by first folding the square over the right and bottom edges, and then folding it over again to

3 PATTERNS IN SQUARE ARRAYS 489 FIG 1 Square array of n identical cells fill out the large square This leads to a n n square array, containing the original system in the upper left quadrant In this larger system we impose periodic boundary conditions by coupling cells on opposite sides of the boundary Ž Fig This leads to the system of equations l, x m fž x l1, m, x l, m1, x l, m, x l, m1, x l1, m, 1l, mn, Ž 3 where l 1 n when l 1 and l 1 1 when l n FIG Large array of 4n identical cells

4 490 GILLIS AND GOLUBITSKY The new system still has square symmetry consisting of permutations on the cells induced from the rotations and reflections of the large square array We denote this symmetry group by D 4 Due to periodic boundary conditions this system is also symmetric with respect to translations of the form Ž r, s, which shifts the Ž l, m th cell r cells down and s cells to the right These translations form the group Ž Z n which does not commute with the action of D Ž 4 The group D4 does, however, normalize Zn so that the new system has the semidirect product D Ž 4 Zn as its group of symmetries Ž Ž 4n Note that any solution x R k to Ž 3 that satisfies l, m xl, m xl,n1m x n1l,m, 1l, mn, Ž 4 restricts to a solution of Ž 1 In terms of group theory, let D4 be the subgroup generated by the horizontal flip about the midline and the vertical flip about the midline, where Define the fixed-point subspace Ž xl, m Ž x n1l, m, Ž 5 Ž xl, m Ž x l,n1m Ž 6 4 FixŽ Ž x : Ž x Ž x Ž x l, m l, m l, m l, m Points in FixŽ are precisely those points that satisfy Ž 4 Thus Ž 4 implies that equilibrium solutions to Ž 1 can be found by finding equilibria to Ž 3 in FixŽ This is the observation that allows us to determine patterned solutions to Ž 1, as solutions to Ž 3 inside FixŽ may have larger symmetry groups than would be apparent by solving Ž 1 for equilibria directly 3 BIFURCATION THEORY In this section we use equivariant bifurcation theory to explain how patterned solutions to the original n n cell system may be expected to appear by bifurcation from a trivial group invariant equilibrium From the point of view of pattern, we may assume that only one equation determines the dynamics of each individual cell; the general case reduces to this one after application of center manifold or LiapunovSchmidt reduction tech- Ž 4n niques That is, we may assume that k 1 and that x R l, m Suppose that f in Ž 1 depends on a bifurcation parameter and that n X R is a group invariant equilibrium for Ž 1 0 for all Without loss of generality, we may assume that X 0 and hence that fž 0, 0 0

5 PATTERNS IN SQUARE ARRAYS 491 These assumptions then apply to the larger system Ž 3 and we discuss our bifurcation analysis in terms of this larger system We will use the equivariant branching lemma 7 to produce branches of equilibria to Ž 3 which restrict to equilibria of Ž 1 We assume that the trivial equilibrium changes stability at a fixed value ; we may assume, without loss of generality, that 0 Let ˆ 0 0 L and L be the linearizations of Ž 1 and Ž 3 at the trivial equilibrium 0 and at 0 Let Vˆ be the kernel of ˆL and let V be the kernel of L Our assumption on change of stability of the trivial solution implies that V ˆ 0 4 The same trick that allows us to extend the system of ODEs Ž 1 to the system Ž 3 allows us to embed Vˆ as a subspace of V, and hence to 4 4n note that V 0 Indeed, V is the smallest -invariant subspace of R that contains the image of V ˆ We make the genericity assumption that V is -irreducible Žsee 7 Our approach to finding patterned equilibria of Ž 1 is to determine all -irreducible subspaces V of R 4 n which are generated by a nonzero ˆ n subspace V of R We then assume that ker L V The equivariant branching lemma 7 guarantees that for each axial subgroup of acting on V there exists a branch of equilibria of Ž 3 bifurcating from a trivial equilibrium By an axial subgroup we mean an isotropy subgroup of with a one-dimensional fixed-point subspace in V Finally, we determine which of these branches of equilibria restrict to equilibria of Ž 1 In the remainder of this section we list the distinct absolutely irreducible -representations of R 4 n, and, for each of these representations, we list the axial subgroups which contain the subgroup generated by the elements and By Ž 4 these equilibria satisfy Neumann boundary conditions when restricted to Ž 1 The proofs of these statements, which are summarized in Lemma 31, Lemma 3, and Theorem 33, are given in Section 6 The Ž Z -Irreducible Subspaces n Note that any -irreducible representation must be a direct sum of Ž Ž Z -irreducible subspaces Let k a, b Z and consider the space n ž ž // 5 i 4 n Vk ½ Re z exp Ž l, m k R : z C and 1 l, m n n Ž 31 Ž ŽŽ Let I be the set of indices k a, b listed in 1 8 of Table 1

6 49 GILLIS AND GOLUBITSKY Types of Z n TABLE 1 -irreducible representations in V Type dim k Restrictions Ž 1 1 Ž 0, 0 Ž 1 Ž n, n Ž 3 1 Ž 0, n Ž 4 1 Ž n,0 Ž 5 Ž a, a 1an1 Ž 6 Ž a,0 1an1 Ž 7 Ž 0, b 1bn1 Ž 8 Ž a, b 1ban1 LEMMA 31 Ž a Each V Ž k is Z n -irreducible Ž b The subspaces V Ž k for k I are all distinct Z n -irreducible representations Ž c The subspaces Vk satisfy V R 4 n k Ž 3 ki Hence, eery Ž Z n -irreducible representation is equal to Vk for some k I The -Irreducible Subspaces We now enumerate the -irreducible subspaces of R 4 n The action of D Ž Ž 4 permutes the Zn -irreducible representations Vk since Zn is a normal subgroup of D Ž 4 Z n Hence the -irreducible representations are just sums of the Ž Z n -irreducibles V k, where k is taken over a D4 group orbit That is, each -irreducible has the form Ý W V Ž 33 k D 4 Žk This sum is not direct as some of these spaces are redundant LEMMA 3 The distinct -irreducible representations of R 4 n are those listed in Table

7 PATTERNS IN SQUARE ARRAYS 493 TABLE -irreducible subspaces Type dim W Restrictions Ž a, b T 1 V Ž0, 0 Ia Ž 1 V Žn,n Ib Ž 4 VŽa,a VŽa,na 1an1 IIŽ a VŽ n,0 VŽ0, n IIŽ b 4 VŽ a,0 VŽ0, a 1 a n 1 III 8 VŽ a, b VŽb, a VŽa,nb VŽb,na 1 b n, b a, a n, a n The Axial Subgroups Our last step is to enumerate the axial subgroups for the irreducible representations given in Table Note that bifurcations where the kernel of the linearization has a trivial action cannot lead to patterned equilibria Hence we do not need to consider type T bifurcations Also note that any representation where FixŽ 0 cannot support axial solutions which restrict to the small square array satisfying Neumann boundary conditions This happens in representations of Types IŽ a and IIŽ a ; see Table 11 in Section 6 Let d be the greatest common divisor gcdž a,n, for types IŽ b and IIŽ b, d½ Ž 34 gcdž a, b,n, for type III, and define the subgroups 1 Ž d Ž K Z, 35 ž /; n n K Ž Z d,,, Ž 36 d d where ², : denotes the group generated by the elements within Let ²,,K : 1 1 ²,,K : ž / ; n ž / ; n n 1,,,K1 d d Ž 37,0,,K d ² : R,, 0,1,K Ž 1

8 494 GILLIS AND GOLUBITSKY THEOREM 33 Up to conjugacy, the axial subgroups of which contain the subgroup are those gien in Table 3 4 PATTERN FORMATION As discussed in Section 1, the branches of equilibria found in Section 3 lead to interesting patterns appearing in the original system Each equilibrium of the original discretized Neumann boundary condition cell system corresponds to an n n array of vectors x k, l The isotropy subgroup of the equilibrium in the extended n n array forces certain of these vectors to be equal and hence forces a pattern to appear in the n n array If we represent each of the Ž possibly different vectors by a different color, then the pattern of equalities in the xk, l will be immediately transparent In our analysis in Section 3 we showed that the kernel K always contains Z ; see Ž 35 and Ž 36 d Consequently, each axial subgroup contains the translations Ž nd, 0 and Ž 0, nd It follows that in each equilibrium corresponding to an axial subgroup, the vector xk, l has to be equal to the vector obtained by translating right, left, up, or down by nd cells; that is, x x x k, l knd, l k, lnd TABLE 3 Axial subgroups Type Parity Restrictions Axials Ib Ž dn 1 dn IIŽ b d n 1 dn 1 1 R III d n 1 dn a b, even d d a b, odd d d a b mod d d 1 1

9 PATTERNS IN SQUARE ARRAYS 495 Thus, the n n array of cells divides into nd nd blocks of cells, with corresponding cells in each block having identical values We call the block B consisting of the upper left hand nd nd block of cells the primary block If Ž ndn, then the original n n array also divides into blocks identical to B, while if Ž nd n, then there will be an extra half block along the right and bottom of the n n array Only a half block can occur since nd always divides n Neumann boundary conditions also imply that the symmetries and are in each axial subgroup Žsee Ž 4 These symmetries imply that the primary block B is symmetric when reflected across the horizontal and vertical midlines As an example, suppose nd 4 If we imagine identical cells as being the same color, we get the initial pattern for cells in a block as given in Table 4 In this table we represent colors as R for red, G for green, Y for yellow, and B for blue Suppose we let L be the subgroup generated by,, and Z d Then each of the five axial subgroups listed in Ž 37 is generated by just one or two symmetries modulo L Indeed ² : L, 1 ž /; n n 1 ž d d/ ; n n n ž / ž /; n n L,,, d d L,, L,0,,, d d d RL² Ž 0,1 : Ž 41 We now consider the axial subgroup The symmetry implies that 1 the primary block is symmetric, that is, B B t Thus B has square D 4 TABLE 4 Initial pattern R Y Y R G B B G G B B G R Y Y R

10 496 GILLIS AND GOLUBITSKY TABLE 5 Final pattern for equilibria with symmetry 1 R G G R G B B G G B B G R G G R symmetry, and the primary block has the pattern shown in Table 5 Figure 3 is an example of the pattern arising in an equilibrium with 1 symmetry when n 8, a 4, and nd 4 On the R branch, the symmetry Ž 1, 0 forces cells in the same row to be identical As shown in Fig 4, these solutions lead to a roll-like pattern, comprised of horizontal stripes All the remaining branches of equilibria occur only when dn, in which case the n n array is exactly filled by copies of the primary block Moreover, nd is even and the primary block B has an even number of rows and columns We can let A be the upper left hand Ž nd Ž nd subblock of B Once the subblock A is determined, the block B is then determined by the horizontal and vertical flips and Next we consider the pattern associated with the axial subgroup The subgroup is generated by the symmetries and Ž nd, nd over L The symmetry forces the subblock A to be symmetric, that is, A A t The symmetry Ž nd Ž nd forces A to be invariant under a 180 rotation It follows that when nd the subblock ž / R G A, G R FIG 3 Solution with symmetry: n 8, a 4, d 4 1

11 PATTERNS IN SQUARE ARRAYS 497 FIG 4 Solution with R symmetry: n 14, a 4, d 4 and when nd 3 the subblock ž / R Y G A Y B Y G Y R In Table 6 we picture the primary block when nd and in Fig 5 we picture the entire pattern when nd 3, n 1, a 4, and d 4 On the 1 branch, as on the branch, we may again divide the primary block B into four subblocks determined by the upper left hand subblock A The Ž nd, nd symmetry of 1 forces A to equal the 180 rotation of itself It follows that when nd the subblock ž / R G A, Y R TABLE 6 Final pattern for equilibria with symmetry R G G R G R R G G R R G R G G R

12 498 GILLIS AND GOLUBITSKY FIG 5 Solution with symmetry: n 1, a 4, d 4 and when nd 3 the subblock ž / R Y G A W B Y P W R In Table 7 we picture the primary block when nd, and in Fig 6 we picture the entire pattern when nd 3, n 15, a 5, and d 5 The Ž nd, nd symmetry of the branch forces the subblock A to equal a 180 rotation of itself; that is, aij a nd1i,nd1j The Ž nd, 0 symmetry forces aij a nd1j,i Note that when nd these conditions imply that all entries in A have the same color, thus yielding the constant pattern in Table 8 This is an TABLE 7 Final pattern for equilibria with symmetry R G G R Y R R Y Y R R Y R G G R 1

13 PATTERNS IN SQUARE ARRAYS 499 FIG 6 Solution with symmetry: n 15, a 5, d 5 1 interesting example where symmetry breaking from an equilibrium yields a new equilibrium with the same symmetry as the original equilibrium When nd 3 the pattern that appears is more complicated An example of such a pattern is given in Fig 7 with n 15, a 3, b 9, d 3, and nd 5 5 LINEAR THEORY In Sections 3 and 4 we discussed which types of pattern are possible in a general square array The particular type of pattern one may expect to see in a particular bifurcation problem of course depends on the form of the kernel in the problem In this section we discuss which irreducible repre- TABLE 8 Final pattern for equilibria with symmetry R R R R R R R R R R R R R R R R

14 500 GILLIS AND GOLUBITSKY FIG 7 Solution with symmetry: n 15, a 3, b 9, d 3 sentations can be expected to arise for a given system of equations In particular we compute the eigenvalues of the linearization of Ž 3 along the trivial solution Ž x l, m 0, and show that the eigenspaces at a zero eigenvalue correspond to the spaces W defined in Ž 33 k We also show that the eigenvalues of the linearization may be computed by computing the eigenvalues of 4n k k matrices, a substantial reduction Let fl, m fž x l1, m, x l, m1, x l, m, x l, m1, x l1, m, and set F Ž f 1,1,, f 1,n,, f n,1,, f n,n Note that xi, j and xl, m are nearest neighbors on the square array if il jm1 with appropriate exceptions made for indices on the boundary Along the trivial branch we may use Ž to write d f p, xl, m l, m x0 d f q, il jm1, xi, j l, m x0 d f 0, i l j m 1, xi, j l, m x0 Ž 51 for certain k k matrices p and q

15 PATTERNS IN SQUARE ARRAYS 501 Let L Ž df 0 We claim that the eigenvalues of L are eigenvalues of certain linear combinations of the matrices p and q Recall that the spaces Wa, b are distinct irreducible representations of the group Indeed, when k 1, the state space R 4 n decomposes into the direct sum of the Wa, b and each of these is an invariant subspace for L For general k, the state space R 4 n k is the direct sum of the subspaces Wa, k b and each of these is L-invariant Let a l cos n a ž / THEOREM 51 The eigenalues of L restricted to W k are the eigenalues a, b of the k k matrix p Ž l l q a b each with multiplicity equal to dim W a, b Proof Let Using Ž 51 we can compute the Jacobian matrix L as follows yl Ž x l,1,, xl,n denote the states of the lth row of cells, and let x Ž x 1,1,, x 1,n,, x n,1,, xn,n denote the states of all cells arranged one row at a time Then P Q Q Q P Q Ln, Ž 5 Q P Q Q Q P

16 50 GILLIS AND GOLUBITSKY where the nk nk matrices P and Q are defined by the y-coordinates In the x-coordinates these matrices have the structure: P p q q q p q q p q q q p q q Q The circular structure of the matrices L and P come from the nearest neighbor coupling of our array The eigenvalue analysis for matrices of the form Ž 5 is discussed in 7 i n Let e be a primitive Ž n th root of unity, and for a 0,,n1 define the complex space q q and 4 Ž a a Ž n1a nk Ya,,,, : R 53 A direct calculation shows a a Ž n1a L,,,, Ž a a Žn1a PlaQ,,,,, Ž 54 so that the eigenvalues of L are the same as the eigenvalues of the nk nk matrices P laq There are n of these matrices Each of these matrices can be written in terms of the k k matrices p and q, as follows: p laq q q q plaq q Pla Q q pl q q a q q pl q a a0,,n1,

17 PATTERNS IN SQUARE ARRAYS 503 Note that these matrices have the same structure as Ž 5 Define the nk-vector 4 b b Ž n1b k b w, w, w,, w : wr A similar calculation shows that Ž Ž Pl Q p Ž l l q Ž 55 a b a b Substituting for in Ž 54 b, it follows that the eigenvalues of L are the same as those of the 4n k k matrices L, where La, b p Ž la l b q, a, b 0,,n1, with the corresponding eigenspace a, b 4 a a Ž n1a Ya, b b, b, b,, b Ž Ž Note that r, s Z acts on Y as rotations: n a, b a a Žn1a Ž r, s b, b, b,, b i a a Žn1a exp Ž r, s Ž a, b b, b, b,, b n k It follows that, as representations, the spaces Y and W defined in Ž 33 a, b a, b are isomorphic Note that many of the L s are equal In particular, since cosž a, b cosž, we have la l na and l b l nb Furthermore, la l b l b l a Thus L L L L L L a, b na, b a,nb na,nb b, a nb, a Lb,naL nb,na Generically, each La, b has only simple zero eigenvalues It then follows that the dimension of ker L is given as in Table 9 It now follows that we may determine which of the representations listed in Table may be expected to appear by calculating the eigenvalues of the k k matrices L a, b The Discretized Brusselator Following 4, we consider as an example a system of coupled Brusselators As a notational convenience to indicate nearest neighbor coupling in a square array, let u u u 4u u u Ž l, m l1, m l, m1 l, m l, m1 l1, m b

18 504 GILLIS AND GOLUBITSKY TABLE 9 Dimensions of kernel L dimž ker L Applicable Ž a, b 1 Ž a, b Ž 0, 0 or Ž a, b Ž n, n Ž a, b Ž 0, n or Ž a, b Ž n,0 4 0abn or 0 a b n or 0 b a n 8 0abn or 0 b a n Ž For the Brusselator k, x u,, and f is defined as follows: u 1 Ž 1 u u D Ž u l, m l, m l, m l, m u l, m ž / ž / l, m ul, m ul, ml, m DŽ l, m, Ž 56 where,, D, and D are positive constants Equation Ž 56 u has the trivial equilibrium ul, m 1 and l, m Rescale by setting D rd and D Ž 4 l l D Then u a, b a b 1rDa, b La, b D a, b It follows that by varying parameters we can force each La, b to have a zero eigenvalue, which then implies that each of the representations listed in Section 3 will occur for this example 6 PROOFS In this section we verify the results stated in Section 3 We begin by showing that the spaces V Ž k listed in Table 1 give all the Z n irreducible representations in V

19 PATTERNS IN SQUARE ARRAYS 505 Ž Proof of Lemma 31 a The translations act on V as k ž ž // i Re z exp Ž l r, m s k ž ž n // i i ž ž n n // i Ž r, s Re z exp Ž l, m k n Re z exp Ž r, s k exp Ž l, m k Vk so that V Ž k is Z n -invariant and clearly irreducible Ž b In coordinates, the translations Ž r, s act on V as rotations: Note that and i Ž r, s z exp Ž r, s k z Ž 61 n k V V Ž 6 k k V V Ž 63 k kž pn,qn for p, q Z To enumerate the distinct Ž Z -irreducibles V, begin by using Ž 63 n k to see that we may assume 0 a, b n Note that Ž 6 and Ž 63 combine to yield V V, Ž 64 Ža, b Žna,nb V V, Ž 65 Ža,0 Žna,0 VŽ0, b V Ž0,nb Ž 66 It follows from Ž 65 that the distinct V are of types ŽŽ 1, 3, and Ž Ž a,0 6 and from Ž 66 that the distinct V are of types ŽŽ 1, 4, and Ž Ž0, b 7 Similarly, it follows from Ž 64 that the distinct V are of types ŽŽ 1,, and Ž Ž a, a 5 Now we assume that 1 a, b n 1 and a b Finally, we use Ž 64 to reduce to case Ž 8 We claim that the Ž Z n representations V for k I are distinct irreducible representations This k

20 506 GILLIS AND GOLUBITSKY claim is verified by noting that a direct calculation using Ž 61 implies that for k, k I, V V k k if and only if i i expž Ž r, s k/ expž Ž r, s k n n / for each 1 r, s n, which then implies that k k Ž c Note that dim V 1 for k of types ŽŽ k 1 4 and dim Vk for k of types ŽŽ 5 8 A calculation shows that dim V 4n Ž 67 Ý ki Since the V are all distinct irreducible representations, Ž 3 k follows from Ž 67 Proof of Lemma 3 The lemma follows from Ž 33 by a calculation using Ž 6 Ž 66 Next we calculate the kernels for each of the remaining representations LEMMA 61 Let d, K, and K be defined as in Ž 34, Ž 35, and Ž 36 1, respectiely Then the kernel K is one of the two subgroups K 1, K Proof A calculation using Table 10 shows that no element of having a nontrivial component in D is in the kernel Thus K Ž Z Let r be a k 4 n TABLE 10 Action of : expž a in, expž b in, ˆ Ž r, s ˆ IŽ b IIŽ b III ˆ Ž Ž Ž rs rs r s r s s r 1 z 1, a z z 1, z z 1, z, r s s r z, z 3 4 Ž rs1 rs1 Ž r1 s Ž r1 s s r1 ˆ z, rz1 z 1, z z 3, z 4, r1 s s r1 z, z 1 ˆ Ž rs rs Ž s r Ž s r r s z 1, z z, z1 z, z 1, s r r s eta z, z 4 3 Ž rs1 rs1 Ž s1 r Ž s1 r r s1 z, z1 z, z1 z 4, z 3, s1 r r s1 z, z 1 Ž rs1 rs1 Ž r s1 Ž r s1 s1 r ˆ z, z1 z 1, z z 3, z 4, r s1 s1 r z, z 1 Ž rs rs Ž r1 s1 Ž r1 s1 s1 r1 z 1, z z 1, z z 1, z, r1 s1 s1 r1 z, z 3 4 Ž rs1 rs1 Ž s r1 Ž s r1 r1 s z, z1 z, z1 z 4, z 3, s r1 r1 s1 z, z 1 Ž rs rs Ž r1 s1 Ž s1 r1 r1 s1 z 1, z z, z1 z, z 1, s1 r1 r1 s1 z, z 4 3

21 PATTERNS IN SQUARE ARRAYS 507 real number A calculation shows that Ž a if arn is even, then r must be an integer multiple of nd, and Ž b if ad is even, then d n It follows from Table 10 that the kernel contains the translations Ž nd,0 and Ž 0, nd ; the kernel may also contain Ž nd, nd, depending on the particular representation Indeed, using Ž a and Ž b, we derive the kernels listed in Ž 35 Ž 36 Proof of Theorem 33 We begin by using Ž 61 to compute FixŽ for each of the representations listed in Table ; we list our results in Table 11 Note that the subgroup fixes only the point 0 for representations of types IŽ a and IIŽ a ; hence representations of this type cannot have any axial subgroups which contain Next we explicitly compute the action of in coordinates If we let be the diagonal flip in D, defined on R 4 n by 4 Ž xl, m Ž x m, l, then,, and the translations Ž r, s generate We list the results of the calculations in Table 10 Our last step is to verify that the subgroups presented in Table 3 are axial and that, up to conjugacy, each axial subgroup is in the table It is a straightforward exercise to check that each of the subgroups listed in Table 3 has a one-dimensional fixed-point subspace, and to verify that these subgroups are isotropy subgroups This verification is made easier by noting that the subgroup K acts trivially on all points in FixŽ and that K is of index in each of the proposed isotropy subgroups TABLE 11 Ž Ž Ž Computation of Fix : exp a in, exp b in Ž Type action Fix dim Ia Ž x x Ib Ž Ž z,z Ž x,x Ž z, z 1 IIŽ a Ž x, x Ž x,x 1 Ž Ž Ž II b z, z x, x Ž z, z 1 III Ž ŽŽ Ž Ž Ž z, z, z, z x, x, B x, x Ž z, z, Bz, z 3 4 1

22 508 GILLIS AND GOLUBITSKY Hence the verification that, 1,, and R have the appropriate fixed-point subspaces relies on the verification that one element in each of these subgroups fixes the listed points These distinguished elements are, Ž nd, nd, Ž nd, 0, and Ž 0, 1, respectively We list the fixed-point subspaces for each of the axial subgroups in Table 1 All that remains is to verify that up to conjugacy these are the only axial subgroups For type IŽ b representations, FixŽ is one-dimensional, and no other choices are possible For the remaining representations, let K be the isotropy subgroup of a point FixŽ, and assume K We claim that is in one of,, orr To see this, recall that we may write uniquely as Ž r, s, where D Ž Ž 4 and r, s Z n Note that, for 1 1 D Ž Ž Ž 4, we have that r, s for some r, s Z n Recalling that, generate D 4, it follows that, up to conjugacy, we may Ž Ž Ž assume equals r, s, r, s,or r, s Since and 1, it follows that Ž r, s if and only if Ž r, s We may therefore assume either Ž r, s or Ž r, s Assume Ž r, s By assumption Ž r, s K For type IIŽ b representations, a quick calculation shows that a nonkernel translation Ž r, s is in the isotropy subgroup of a point ŽexpŽ a in x, expž a in y if and if x 0or y0 Points of this type lie on the same orbit, and hence have conjugate isotropy subgroups Using Table 3, it follows that R For type III representations, we may use coordinates to see that all nonkernel translations act fixed-point freely on FixŽ Next assume that Ž r, s If Ž r, s K then, so suppose Ž r, s K A calculation in coordinates shows, for both type IIŽ b and type III representations, that a point FixŽ is fixed by Ž r, s if and only if Ž r, s acts as minus the identity on that representation A straightforward calculation shows that this will TABLE 1 Ž Ž Axial subgroups: exp a in, exp b in Ž Type Axials Fix Axial Ib Ž RŽ,14 1 RŽ,14 Ž Ž II b R, 4 1 Ž R, 4 1 R R,04 III 1 ŽŽ Ž Ž Ž R,,, 4 ŽŽ Ž Ž Ž R,,, 4 ŽŽ Ž Ž Ž R,,, 4 ŽŽ Ž Ž Ž R,,, 4 1

23 PATTERNS IN SQUARE ARRAYS 509 occur if and only if dn It follows that R, 4 Žfor type IIŽ b ŽŽ Ž Ž Ž representations or R,,, 4 Žfor type III representations and hence Note for type III representations care must be taken when deciding which translations act as 1; the results will depend on the parity of the numbers ad and bd REFERENCES 1 D Armbruster and G Dangelmayr, Coupled stationary bifurcations in non-flux boundary value problems, Math Proc Cambridge Philos Soc 101 Ž 1987, J D Crawford, M Golubitsky, M G M Gomes, E Knobloch, and I N Stewart, Boundary conditions as symmetry constraints, in Singularity Theory and Its Applications, Warwick 1989, Part II, Ž M Roberts and I Stewart, Eds, Lecture Notes in Mathematics, Vol 1463, pp 6379, Springer-Verlag, Heidelberg, B Dionne and M Golubitsky, Planforms in two and three dimensions, Z Angew Math Phys 43 Ž 199, I R Epstein and M Golubitsky, Symmetric patterns in linear arrays of coupled cells, Chaos 3ŽŽ , 15 5 M Field, M Golubitsky, and I Stewart, Bifurcations on hemispheres, J Nonlinear Sci 1 Ž 1991, 03 6 M Golubitsky, J Marsden, and D Schaeffer, Bifurcation problems with hidden symmetries, in Partial Differential Equations and Dynamical Systems, ŽW E Fitzgibbon III, Ed, pp 1810, Research Notes in Mathematics, Vol 101, Pitman, London, M Golubitsky, I N Stewart, and D G Schaeffer, Singularities and Groups in Bifurcation Theory, Vol II, Appl Math Sci, Vol 69, Springer-Verlag, New York, M G M Gomes and I N Stewart, Hopf bifurcations in generalized rectangles with Neumann boundary conditions, in Dynamics, Bifurcation and Symmetry ŽP Chossat, Ed, pp , NATO ARW Series, Kluwer Academic, Amsterdam, M G M Gomes and I N Stewart, Steady PDEs on generalized rectangles: A change of genericity in mode interactions, Nonlinearity 7ŽŽ , 537