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International Journal of Bifurcation and Chaos, Vol. 16, No. 3 (2006) 559–577 c World Scientific Publishing Company


SYMMETRY AND SYNCHRONY IN COUPLED CELL NETWORKS 1: FIXED-POINT SPACES FERNANDO ANTONELI Department of Applied Mathematics, University of S˜ ao Paulo, S˜ ao Paulo SP 05508-090, Brazil IAN STEWART Mathematics Institute, University of Warwick, Coventry CV4 7AL, UK Received January 16, 2006 Equivariant dynamical systems possess canonical flow-invariant subspaces, the fixed-point spaces of subgroups of the symmetry group. These subspaces classify possible types of symmetrybreaking. Coupled cell networks, determined by a symmetry groupoid, also possess canonical flow-invariant subspaces, the balanced polydiagonals. These subspaces classify possible types of synchrony-breaking, and correspond to balanced colorings of the cells. A class of dynamical systems that is common to both theories comprises networks that are symmetric under the action of a group Γ of permutations of the nodes (“cells”). We investigate connections between balanced polydiagonals and fixed-point spaces for such networks, showing that in general they can be different. In particular, we consider rings of ten and twelve cells with both nearest and next-nearest neighbor coupling, showing that exotic balanced polydiagonals — ones that are not fixed-point spaces — can occur for such networks. We also prove the “folk theorem” that in any Γ-equivariant dynamical system on Rk the only flow-invariant subspaces are the fixed-point spaces of subgroups of Γ. Keywords: Network; coupled cell; balance; symmetry.

1. Introduction

Fixed-point spaces have an important property: they are flow-invariant. By this we mean that they are invariant under every smooth equivariant vector field f ; that is,

The theory of symmetric dynamical systems [Golubitsky et al., 1988; Golubitsky & Stewart, 2002] is concerned with ordinary differential equations (ODEs)

f (Fix(Σ)) ⊆ Fix(Σ)

x˙ = f (x) where the vector field f is smooth (C ∞ ) and is equivariant under the action of a group Γ on phase space X. Usually X is assumed to be a vector space, and Γ acts linearly, mapping x ∈ X to γx. The equivariance condition states that

See Lemma XIII.2.1 of [Golubitsky & Stewart, 2002] or Theorem 1.17 of [Golubitsky et al., 1988] for the simple proof and the implications for symmetrybreaking. In fact, the fixed-point spaces of subgroups of Γ are the only flow-invariant subspaces in the above sense. We have been unable to find a proof in the literature, so we state and prove the theorem as Theorem 4.1. It should be understood that specific admissible vector fields may possess other

f (γx) = γf (x) ∀ x ∈ X, γ ∈ Γ A central concept is the fixed-point space of a subgroup Σ ⊆ Γ, defined by Fix(Σ) = {x ∈ X : σx = x ∀ σ ∈ Σ} 559

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flow-invariant subspaces, but these depend on special features of the vector field concerned.

1.1. Synchrony in networks Recently, equivariant dynamics has been modified to provide a formal framework for the dynamics of networks [Stewart et al., 2003; Golubitsky et al., 2004a; Golubitsky et al., 2004b; Golubitsky et al., 2005] with particular attention to patterns of synchrony and associated bifurcations. By a network we mean a directed graph [Tutte, 1984; Wilson, 1985] whose nodes and edges are classified according to associated labels or “types”. The nodes (or “cells”) of a (directed, labeled) network G represent dynamical systems, and the edges (“arrows”) represent couplings. Cells with the same label have “identical” internal dynamics; arrows with the same label correspond to “identical” couplings. When this modification of equivariant dynamics is formalized, it turns out that the natural analog of the symmetry group Γ is the symmetry groupoid BG of G. A groupoid is an algebraic structure similar to a group, except that products of elements may not always be defined: see [Brandt, 1927; Brown, 1987; Higgins, 1971]. Groupoids capture “local” symmetries [Weinstein, 1996], which in this case are label-preserving bijections between “input sets” of cells. Each cell c is equipped with a phase space Pc , and the total phase space
of the network is the cartesian product P = c Pc . Points x ∈ P have coordinates xc . In the network case, equivariance is replaced by admissibility. A vector field f is admissible if its component fc for cell c depends only on variables associated with the input set of c (domain condition), and if its components for cells c, d that have isomorphic input sets are identical up to a suitable permutation of the relevant variables (pullback condition). In this formalism, a pattern of synchrony in G corresponds to an equivalence relation  on the set C of cells. Two cells c, d are related (that is, c  d) if and only if they are synchronous (that is, xc (t) = xd (t) for all times t for a trajectory x(t) of an admissible vector field f ). Note that in general  depends on both f and x: synchrony is a property of a trajectory of an ODE, not a property of the network as such. However, the general network architecture can create universal possibilities for synchrony, in the sense that there exists a canonical class of

flow-invariant subspaces of P . Suppose that  is an equivalence relation on the set of cells. The associated polydiagonal ∆
 = {x ∈ P : c  d ⇒ xc = xd } is a subspace of P . A basic theorem [Stewart et al., 2003; Golubitsky et al., 2005] states that a polydiagonal ∆
 is flow-invariant (under all admissible vector fields) if and only if  is balanced. Roughly speaking, this condition states that if c  d then the input sets of c and d are related by a bijection that preserves -equivalence classes. More vividly: color the cells so that c and d receive the same color if and only if c  d. Then the sets of identically-colored cells are the equivalence classes for . The coloring is balanced if and only if any two cells with the same color have “color-isomorphic” input sets — that is, there is a bijection between those sets that preserves both arrow-type and cell color. If there are k colors in total then we call such a pattern a balanced k-coloring. If  is balanced, then the corresponding subspace ∆
 is said to be a balanced polydiagonal. Just as fixed-point spaces classify possible types of symmetry-breaking, so balanced polydiagonals classify possible types of synchrony-breaking, see [Golubitsky et al., 2004b; Wang & Golubitsky, 2005]. A related combinatorial approach to synchrony in networks can be found in [Field, 2004]. Dynamically, a balanced polydigonal represents a “pattern of synchrony”, in which cells with the same color have identical dynamics. This pattern is “robust” in the sense that it can occur for any admissible vector field, provided that appropriate initial conditions (points in the polydiagonal) are chosen. The dynamics of a synchronized pattern can be steady-state, periodic, or chaotic, depending on the choice of admissible vector field. The balance property completely characterizes robust patterns of synchrony [Stewart et al., 2003; Golubitsky et al., 2005]. The general question of synchronization in networks of dynamical systems (“coupled oscillators”) has attracted considerable attention: see for example [Mosekilde et al., 2002; Wu, 2002; Manrubia et al., 2004].

1.2. Symmetric networks This paper discusses an area of common ground in which group-symmetric dynamical systems and groupoid-symmetric networks overlap, namely: networks that possess a group of symmetries Γ. In this context Γ is a group of permutations of the cells

Symmetry and Synchrony in Coupled Cell Networks 1: Fixed-Point Spaces

(and arrows) that preserves the network structure (including cell- and arrow-types) and its action on P is by permutation of cell coordinates. It follows that every admissible vector field is Γ-equivariant, so it is reasonable to expect parallels between the group-theoretic and groupoid-theoretic formalisms. We will show that such parallels do exist, but that the relationship is not entirely straightforward. Indeed, some aspects of it remain unexplained. Others, however, are obvious. For example, it is trivial to prove that every fixed-point space of a subgroup Σ ⊆ Γ is a balanced polydiagonal: the corresponding equivalence relation  is then “in the same Σ-orbit”. The main question addressed in this paper is: does the converse statement hold? In this generality, the answer is easily seen to be “no”. Networks with trivial symmetry group can possess nontrivial balanced equivalence relations [Stewart et al., 2003; Golubitsky et al., 2004a; Golubitsky et al., 2005]. Figure 1 below is a further example. Networks with nontrivial symmetry can also possess “exotic” balanced polydiagonals — ones that do not arise as fixed-point spaces, see Sec. 3 below. One reason why such examples can occur is that although every admissible map is equivariant, the converse can be false. Section 6 studies one of these examples, a ring of twelve cells with nearest- and next-nearest-neighbor coupling, in detail, including a numerical example and its stability analysis. Section 7 studies a related example, a ring of ten cells with nearest- and next-nearest-neighbor coupling, finding nine exotic balanced colorings, up to the action of the symmetry group D10 . In Part 2 of this paper, [Antoneli & Stewart, 2006], we restrict attention to a natural class of group-symmetric networks, and show that even in this restricted context exotic balanced colorings exist. The precise classification of balanced colorings for this class of group-symmetric networks remains mysterious.

1.3. Structure of the paper Section 2 recalls the formal definition of a coupled cell network and the associated dynamical systems, and states some basic features, including the concept of a balanced equivalence relation (coloring). Section 3 discusses the symmetry group (or automorphism group) of a network, and includes the (elementary) proof that if H is a subgroup of the automorphism group then its fixed-point space is

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balanced. For motivation and completeness, Sec. 4 provides a proof of the folk theorem that in a Γequivariant system the only subspaces that are flowinvariant for all equivariant vector fields are the fixed-point spaces of subgroups of Γ. Section 5 introduces examples where admissible maps differ from equivariant maps, and where balanced colorings occur whose corresponding polydiagonals are not fixed-point spaces. Section 6 analyzes a ring of twelve cells, finding an exotic pattern of synchrony numerically and analyzing its stability. Section 7 analyses a ring of 10 cells, finding 9 exotic patterns of synchrony.

2. Network Formalism First, we recall the formal definition of a coupled cell system. For a survey, overview, and examples, see [Golubitsky & Stewart, 2005]. The initial definition of Stewart et al., (2003) was modified in [Golubitsky et al., 2005] to permit multiple arrows and self-connections, which turns out to have major advantages. We will therefore employ this “multiarrow” formalism for consistency with the existing literature. However, it introduces some technical distinctions which complicate the notation.

2.1. Coupled cell networks Definition 2.1.

A coupled cell network G com-

prises: (a) A finite set C = {1, . . . , N } of nodes or cells. (b) An equivalence relation ∼C on cells in C, called cell-equivalence. The type or cell label of cell c is its ∼C equivalence class. (c) A finite set E of edges or arrows. (d) An equivalence relation ∼E on edges in E, called edge-equivalence or arrow-equivalence. The type or coupling label of edge e is its ∼E equivalence class. (e) Two maps H : E → C and T : E → C. For e ∈ E we call H(e) the head of e and T (e) the tail of e. We also require a consistency condition: (f) Equivalent arrows have equivalent tails and heads. That is, if e1 , e2 ∈ E and e1 ∼E e2 , then H(e1 ) ∼C H(e2 )

T (e1 ) ∼C T (e2 )

In this definition, an arrow may be a selfconnection (that is, T (e) may be the same as H(e))

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and multiple arrows may exist (distinct arrows e, e may have T (e) = T (e ) and H(e) = H(e )).

with a cell-based coordinate system x = (xc )c∈C If D ⊆ C then we write

2.2. Input sets and the symmetry groupoid

PD =



Pd

d∈D

Associated with each cell c ∈ C is a canonical set of edges, namely, those that represent couplings into cell c:

For any β ∈ B(c, d) we define the pullback map

Definition 2.2. If c ∈ C then the input set of c is

by

I(c) = {e ∈ E : H(e) = c}

β ∗ : PT (I(d)) → PT (I(c)) (β ∗ z)T (i) = zT (β(i))

(1)

An element of I(c) is called an input edge or input arrow of c.

for all i ∈ I(c) and z ∈ PT (I(d)) . We may now state:

Definition 2.3. The relation ∼I of input equivalence on C is defined by c ∼I d if and only if there exists a bijection

admissible if:

β : I(c) → I(d)

(2)

such that for every i ∈ I(c), i ∼E β(i)

(3)

Any such bijection β is called an input isomorphism from cell c to cell d. The set B(c, d) denotes the collection of all input isomorphisms from cell c to cell d. The union  B(c, d) (4) BG = c,d∈C

is the symmetry groupoid of the network G. The groupoid operation on BG is a composition of maps, and in general the composition βα is defined only when α ∈ B(a, b) and β ∈ B(b, c) for cells a, b, c. This is why BG need not be a group. Note that the union in (4) is disjoint. By the consistency condition (f) of Definition 2.1, c ∼I d implies c ∼C d, but the converse fails in general.

2.3. Admissible vector fields We now define admissible vector fields, which are compatible with the labeled graph structure, or equivalently are symmetric under the groupoid BG . For each cell in C define a cell phase space Pc , which we here assume to be a nonzero finitedimensional real vector space. We require c ∼C d ⇒ Pc = Pd and in this case we employ the same coordinate systems on Pc and Pd . The total phase space is then  Pc P = c∈C

(5)

Definition 2.4. A vector field f : P → P is G-

(a) Domain condition: For all c ∈ C the component fc (x) depends only on the internal phase space variables xc and the coupling phase space variables xT (I(c)) ; that is, there exists fˆc : Pc × PT (I(c)) → Pc such that fc (x) = fˆc (xc , xT (I(c)) )

(6)

(b) Pullback condition: For all c, d ∈ C and β ∈ B(c, d) fˆd (xd , xT (I(d)) ) = fˆc (xd , β ∗ xT (I(d)) )

(7)

for all x ∈ P . An ODE defined by an admissible vector field on a coupled cell network is called a coupled cell system. A subspace V of P is flow-invariant if f (V ) ⊆ V for every admissible vector field f on P .

2.4. Balanced equivalence relations An equivalence relation  on C determines a unique partition of C into -equivalence classes, which can be interpreted as a coloring of C in which equivalent cells receive the same color. Conversely, any partition (coloring) determines a unique equivalence relation. Formal arguments are generally phrased in the language of equivalence relations, but informally can often be more easily understood by using colors. If  is an equivalence relation on C then the corresponding polydiagonal is ∆
 = {x ∈ P : c  d ⇒ xc = xd }

(8)

Polydiagonals are subspaces of P determining possible patterns of synchrony, as we now explain.

Symmetry and Synchrony in Coupled Cell Networks 1: Fixed-Point Spaces

Fig. 1.

1

2

3

4

Example of a balanced equivalence relation.

Definition 2.5. An equivalence relation  on C is

balanced if for every c, d ∈ C with c  d, there exists β ∈ B(c, d) such that T (i)  T (β(i)) for all i ∈ I(c). The associated coloring is called a balanced coloring. As an example, Fig. 1 shows a coupled cell network with four cells and trivial symmetry group. Here, as usual, the cell type is indicated by the shape of the cell symbol (circle, square, triangle) and the arrow type is indicated by the form of the arrow (solid, dotted, shape and color of arrowhead). An equivalence relation  with equivalence classes {1}, {2, 3}, {4} is indicated by colors. The only distinct -related cells are 2 and 3; their input sets comprise only the arrows entering from cell 1, so  is obviously balanced. A crucial property of a balanced equivalence relation is that it defines a flow-invariant polydiagonal, and conversely:

Theorem 2.6. Let  be an equivalence relation on

a coupled cell network. Then ∆
 is flow-invariant if and only if  is balanced. The proof is given in [Stewart et al., 2003; Golubitsky et al., 2005]. The dynamical implication of such flow-invariance is that  determines a pattern of synchrony: there exist trajectories x(t) of the ODE such that c  d ⇒ xc (t) = xd (t) ∀ t ∈ R Such trajectories arise when initial conditions x(0) lie in ∆
 . Then the entire trajectory, for all positive and negative times, lies in ∆
 and is a trajectory of the restriction f |∆
 . The associated dynamics can be steady-state, periodic, even chaotic, depending on f and its restriction to ∆
 . An example of synchronized chaos generated by this mechanism can be found in [Golubitsky & Stewart 2005]. For applications see [Golubitsky et al., 2004a; Antoneli et al., 2005; Wang & Golubitsky, 2005]. Any balanced coloring  on a network G determines a quotient network G
 in which all cells

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of any given color are identified, while input set topology is preserved. The dynamics of G
 corresponds to synchronous dynamics of G; that is, dynamics restricted to ∆
 . See [Stewart et al., 2003; Golubitsky et al., 2005].

3. Symmetry Groups of Networks We now consider symmetries of networks in the group-theoretic (“global”) sense. Self-connections and multiple arrows lead to complications (separate but related actions of the group on C and E). Definition 3.1. Let G be a coupled cell network with cells C and arrows E. An automorphism of G is a pair

σ = (σC , σE ) of bijections σC : C → C σE : E → E satisfying the following conditions: (a)

σC (H(e)) = H(σE (e)) ∀ e ∈ E σC (T (e)) = T (σE (e)) ∀ e ∈ E

(b)

i ∼C σC (i) ∀ i ∈ C

(c)

e ∼E σE (e)

∀e ∈ E

The set of all automorphisms of G forms a group under composition, called the automorphism group or symmetry group of G and denoted by Aut(G). Remark 3.2

(a) If |C| = N and |E| = M then we can consider Aut(G) as a subgroup of SN × SM . There are then canonical projections of Aut(G) onto SN and SM respectively, whose images are denoted by AutC (G) and AutE (G). They consist, respectively, of the bijections σC and σE as σ runs through Aut(G). (b) The need to consider both σC and σE is a complication arising from the multiarrow formalism. (c) However, σC is uniquely determined by σE provided the consistency conditions H(e) = H(e ) ⇒ H(σE (e)) = H(σE (e )) T (e) = T (e ) ⇒ T (σE (e)) = T (σE (e )) hold. Therefore we can consider Aut(G) as a subgroup of SM .

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(d) If G has no multiple arrows, then σE is determined by σC since each arrow e can be identified with a pair of cells (T (e), H(e)). Now we can consider Aut(G) as a subgroup of SN . (e) Automorphisms preserve input sets in a natural sense. Because of the way input sets are defined in the multiarrow formalism, the precise relation is: σE (I(c)) = I(σC (c))

∀c ∈ C

where σ ∈ Aut(G).

vector fields) are the fixed-point spaces of (isotropy) subgroups of Γ. Since we have been unable to locate a published proof, we give one here. It is a theorem in equivariant dynamics, and does not involve any explicit network structure. Theorem 4.1. Let Γ be a finite group acting lin-

early on a finite-dimensional real vector space X. The only subspaces of X that are flow-invariant for all equivariant vector fields are the fixed-point spaces of isotropy subgroups of Γ.

We can construct (some) balanced equivalence relations on a network G from subgroups of the automorphism group Aut(G) of G. Namely, suppose that Ω ⊆ Aut(G). Let ΩC be the projection of Ω into AutC (G). Define the relation Ω by

In order to give a self-contained proof we start recalling some definitions. Let Σ be a subgroup of Γ. The normalizer NΓ (Σ) = N (Σ) (of Σ in Γ) is the largest subgroup of Γ that contains Σ as a normal subgroup, and is given by

c Ω d ⇔ ∃ω ∈ Ω : ωC (c) = d

N (Σ) = {γ ∈ Γ : γ −1 Σγ = Σ}

Then the Ω -classes are the ΩC -orbits of cells, and the corresponding polydiagonal

The relation between the normalizer of Σx and its fixed-point subspace is given by the following. Let Σx be the isotropy subgroup of x ∈ X then

∆Ω = ∆
Ω = Fix(Ω) where Fix(Ω) is the fixed-point space of Ω acting on the total phase space P . This polydiagonal is balanced: Proposition 3.3. Let G be a network and let Ω be any subgroup of Aut(G). Then Fix(Ω) is a balanced polydiagonal.

Let f be an admissible vector field on G. Then it is easy to prove that f is equivariant under the natural action of Aut(G). By Theorem 1.17 of [Golubitsky & Stewart, 2002], Fix(Ω) is flowinvariant for f . By Theorem 2.6, Fix(Ω) is a balanced polydiagonal.


Proof.

(A more direct proof is possible by routine computations.) This result motivates the following terminology: Definition 3.4. Let  be a balanced equivalence relation on a network G with automorphism group Aut(G). We say that  is a fixed-point coloring if ∆
 = Fix(H) for some subgroup H ⊆ Aut(G), and  is exotic otherwise.

4. Folk Theorem The initial motivation for this paper and its sequel [Antoneli & Stewart, 2006] was the “folk theorem” that in any Γ-equivariant dynamical system, the only flow-invariant subspaces (for all equivariant

N (Σx ) = {γ ∈ Γ : γ(Fix(Σx )) = Fix(Σx )} Since Σx acts trivially on Fix(Σx ) and is normal in N (Σx ) we can factor it out and consider the group W (Σx ) =

N (Σx ) Σx

which acts naturally on Fix(Σx ). More generally, for any subgroup Σ of Γ the group W (Σ) = N (Σ)/Σ is the Weyl group of Σ and acts naturally on Fix(Σ). Elements in W (Σ) (or in N (Σ)\Σ) are apparent symmetries of Fix(Σ). Definition 4.2. Let Σ ⊆ Γ. An element γ ∈ Γ\ N (Σ) is a hidden symmetry of Fix(Σ) if

γ(Fix(Σ) ) ∩ Fix(Σ)
Fix(Γ)

(9)

Recall that Fix(Γ) is contained in each fixedpoint space, and all symmetries fix vectors in Fix(Γ). Thus (9) states that the intersection on the left-hand side of (9) contains a nonzero vector. Moreover, since γ ∈ / N (Σ) the intersection on the left-hand side of (9) is a proper subspace of Fix(Γ). Now let f : V → V be a Γ-equivariant mapping and Σx be the isotropy subgroup of x ∈ X. Equivariance f (Fix(Σx )) ⊂ Fix(Σx ) implies that the restriction f of f to Fix(Σ) is equivariant under the action of W (Σ). If γ ∈ Γ is a hidden symmetry of Fix(Σ) then there exists a nonzero x ∈ Fix(Σ) such that γx ∈ Fix(Σ). Hence both f (x) and

Symmetry and Synchrony in Coupled Cell Networks 1: Fixed-Point Spaces

f (γx) ∈ Fix(Σ). But these are just f (x) and f (γx), respectively. Since f is Γ-equivariant it follows that f (γx) = γf (x) Therefore the hidden symmetry γ of Fix(Σ) places an extra condition on f , in addition to those conditions imposed by the apparent symmetries in N (Σ). Definition 4.3. Let Σ ⊆ Γ be an isotropy subgroup

and let γ ∈ Γ be a hidden symmetry of Fix(Σ). A W (Σ)-equivariant mapping f0 : Fix(Σ) → Fix(Σ) is said to satisfy the hidden symmetry condition if for all γ ∈ Γ\N (Σ) such that x, γx ∈ Fix(Σ) we have f0 (γx) = γf0 (x)

The next proposition says that if we do not require smoothness then the hidden symmetries are the only obstructions to extending a W (Σ)equivariant mapping on Fix(Σ) to a continuous Γequivariant mapping on X. This is Problem 4.12 on page 49 of [Golubitsky et al., 1988], but we give a proof for completeness. Proposition 4.4. Let Γ be a compact Lie group act-

ing linearly on a finite-dimensional real vector space X, and let Σ ⊆ Γ be an isotropy subgroup of Γ. Suppose that f0 : Fix(Σ) → Fix(Σ) is a W (Σ)equivariant mapping. Then f0 extends to a continuous Γ-equivariant mapping f : X → X if and only if f0 satisfies the hidden symmetry condition. Necessity is obvious. To prove sufficiency consider the set  γ(Fix(Σ)) L=

Next we prove that if Γ is finite then there are no nontrivial hidden symmetries. First we observe that if γ is a hidden symmetry on Fix(Σ) then the subspace (9) is the fixed-point space of the subgroup Σ, γΣγ −1 . Now the following proposition proves the claim. Proposition 4.5. Let Γ be a finite group acting linearly on a finite-dimensional real vector space X, and let V ⊂ X be the fixed-point space of a subgroup of Γ. Then V = Fix(Σx0 ) for some isotropy subgroup Σx0 of Γ. Proof. First, note that V is the union of fixed-point spaces of isotropy subgroups of Γ. Indeed, if x ∈ V then Fix(Σx ) ⊂ V . Since x ∈ V it follows that  Fix(Σx ) V = x∈V

Now since Γ is finite there is only a finite number of isotropy subgroups conjugate to Σx . Thus there exists x0 ∈ V such that V = Fix(Σx0 )

Without loss of generality, we may assume that Γ acts orthogonally on X. If Γ is connected then L is a subspace of X. In general, Γ has finitely many connected components, so L is a finite union of subspaces and thus is closed. The hidden symmetry condition implies that f0 extends uniquely to a Γequivariant map f1 : L → X. By the Tietze extension theorem [Kelley, 1955], f1 admits a continuous extension to a map f2 : X → X. By averaging over the group we define  γ −1 f2 (γx) dµ(γ) f (x) = Γ

where µ is the normalized bi-invariant measure on Γ. Then f : X → X is Γ-equivariant, and f |Fix(Σ) = f0 .





We need three technical lemmas to complete the proof of Theorem 4.1, and we take these in turn. Lemma 4.6. Let f : Rr → Rs be a continuous

map. Let x1 , . . . , xn be n distinct points of Rr . Then for a given a compact set K ⊂ Rr containing x1 , . . . , xn , f is uniformly approximable on K by polynomial maps p satisfying

Proof.

γ∈Γ

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p(xi ) = f (xi )

(1 ≤ i ≤ n)

First we prove the statement of the lemma for s = 1. Given ε > 0, the Weierstrass Approximation Theorem [Apostol, 1957] states that there exists a polynomial function p0 such that

Proof.

|f (x) − p0 (x)| ≤ ε ∀ x ∈ K Write x = (x1 , . . . , xr ) and consider xi for fixed i = 1, . . . , n. Since x1 , . . . , xn are distinct points  we can find coordinates xjj of each xj such that 



xi j = xjj , where 1 ≤ j ≤ n and j = i. Define li (x) = li (x1 , . . . , xr )

i − xi i ) · · · (xn − xnn ) (x1 − x11 ) · · · (x = i (xi 1 − x11 ) · · · (xi − xi i ) · · · (xi n − xnn )

where · indicates an omitted factor. Then li is a polynomial function satisfying li (xj ) = δij

(1 ≤ i, j ≤ n)

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Also, define

Proof.

p1 (x) =

n  (f (xi ) − p0 (xi )) li (x)

γ∈Γ

i=1

We claim that x ∈ / L ∩ Fix(Σx ). Suppose not. Then there exists γ ∈ Γ\Nx such that

Then p1 is a polynomial function with p − 1(xi ) = f (xi ) − p0 (xi ) (1 ≤ i ≤ n)

γ(Fix(Σx )) ∩ Fix(Σx )
Fix(Γ)

Now sup |p1 (x)| ≤

x∈K

n 

|f (xi ) − p0 (xi )| sup |li (x)| ≤ εM x∈K

i=1

where M=

k 

sup |li (x)|

i=1 x∈K

Note that M depends only upon K and x1 , . . . , xn . Set p(x) = p0 (x) + p1 (x) Then p(xi ) = p0 (xi ) + p1 (xi ) = f (xi )

|f (x) − p(x)| ≤ |f (z) − p0 (x)| + |p1 (x)| ≤ ε(1 + M ) ∀ x ∈ K This inequality proves the lemma for s = 1. Now assume s > 1. Represent f by its s coordinate functions f (x) = (f 1 (x), . . . , f s (x)) Since f is continuous on K, each coordinate function f k is continuous as a function from K to R. Hence the coordinate functions f k can be uni√ formly approximated on K within ε/ s by polynok mial functions p satisfying (1 ≤ i ≤ n, 1 ≤ k ≤ s)

Define p(x) = (p1 (x), . . . , ps (x)) This yields a polynomial function p : Rr → Rs which provides the required approximation on K to f , and satisfies p(xi ) = f (xi )

(1 ≤ i ≤ n)

As observed before, this implies that x ∈ Fix(Σx , γΣx γ −1 ). By Proposition 4.5 the vector x is fixed by a subgroup strictly larger than Σx , which is a contradiction. Therefore γ with the required properties does not exist. Now let B(x) be a ball contained in Fix(Σx ), centered at x and not containing 0. Note that, for all σ ∈ Nx the set σ(B(x)) is a ball of the same size as B(x) and centered at σx. Let g0 : Fix(Σx ) → Fix(Σx ) be a smooth map with support contained in B(x) (that is, g0 vanishes outside B(x)) and g0 (x) = y. We also assume that B(X) is small enough that for any σ1 , σ2 ∈ Nx σ1 (B(x)) ∩ σ2 (B(x)) = ∅

(1 ≤ i ≤ n)

Moreover,

pk (xi ) = f k (xi )

Write Nx = N (Σx ). Consider the closed set  γ(Fix(Σx )) L=




Lemma 4.7. Let x ∈ X and let Σx be the isotropy subgroup of x. Then for all y ∈ Fix(Σx ) there exists an equivariant polynomial mapping f : X → X such that f (x) = y.

Since x ∈ / L ∩ Fix(Σx ) then we may assume that B(x) does not intersect L ∩ Fix(Σx ). Define g1 : Fix(Σx ) → Fix(Σx ) by 1  −1 σ g0 (σz) g1 (z) = |Nx | σ∈Nx

Then g1 is Nx -equivariant and g1 (x) = y, since for all σ ∈ Nx , except σ = id, we have g0 (σx) = 0. It is easy to see that g1 satisfies the hidden symmetry condition. Indeed, since x ∈ / L ∩ Fix(Σx ) it follows that g1 vanishes on   σ(B(x)) Fix(Σx ) σ∈Nx

By Proposition 4.4, g1 : Fix(Σx ) → X admits a continuous extension g2 : X → X and a continuous Γ-equivariant extension g : X → X given by 1  −1 γ g2 (γz) g(z) = |Γ| γ∈Γ

By Lemma 4.6, we can choose a compact set K ⊂ X containing Γx and a polynomial mapping f0 : X → X uniformly approximating g2 on K and satisfying f0 (γx) = g2 (γx) ∀ γ ∈ Γ Now the polynomial mapping 1  −1 γ f0 (γz) f (z) = |Γ| γ∈Γ

Symmetry and Synchrony in Coupled Cell Networks 1: Fixed-Point Spaces

is Γ-equivariant, uniformly approximates g, and satisfies 1  −1 γ f0 (γx) f (x) = |Γ| γ∈Γ

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Fix(Σx ) is the smallest flow-invariant subspace of X that contains x. Proof.

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Fig. 2. Patterns of synchrony in a six-cell Z3 -symmetric network.

networks in Part 2 of this paper [Antoneli & Stewart, 2006].

P(x) = {f (x) : f is Γ-equivariant}

Example 5.1. Consider the 6-cell network G of

Observe that P(x) is a flow-invariant subspace of X. To show this, let y ∈ P(x). Then there is an equivariant map f such that f (x) = y, so for all equivariant mapping g we have

Fig. 2. The automorphism group of G is isomorphic to the cyclic group Z3 . Its action on cells is generated by the permutation (135)(024), and this induces a unique action on edges since there are no multiple arrows. (To prove this, observe that the three-cycle (135) is the unique 3-cycle in which no pair of cells (a, b) exists with arrows from a to b and from b to a. Therefore the cell component σC of any automorphism σ fixes {1, 3, 5} setwise, and the rest is easy.) There are precisely two balanced polydiagonals coming from fixed-point spaces of subgroups of Z3 : one is P itself and the other is {(x, y, z, x, y, z)}. However, this network has several other balanced polydiagonals: any of the pairs {0, 1}, {2, 3}, {4, 5} can be independently identified and the resulting polydiagonal is balanced.

g(y) = g(f (x)) = (g ◦ f )(x) ∈ P(x) Hence g(P(x)) ⊆ P(x). Now suppose that V ⊆ P(x) is a flow-invariant subspace. If y ∈ V then f (y) ∈ V for all equivariant mappings f , so P(x) ⊆ V . Therefore P(x) is the smallest flow-invariant subspace of X that contains x. Also, P(x) is contained in Fix(Σx ). Indeed, let f (x) ∈ P(x), where f is some Γ-equivariant map. Then σf (x) = f (σx) = f (x) for all σ ∈ Σx . Finally, Lemma 4.7 implies that Fix(Σx ) = P(x).
Suppose that V ⊂ X is flow-invariant. For all x ∈ V , Lemma 4.8 implies that Fix(Σx ) ⊆ V . Hence  Fix(Σx ) V =

Proof of Theorem 4.1.

x∈V

Since Γ is finite there is only a finite number of isotropy subgroups conjugate to Σx , so there exists x0 ∈ V such that V = Fix(Σx0 )


5. Exotic Balanced Colorings We now consider some simple examples of symmetric networks, possessing exotic balanced colorings; that is, balanced colorings that are not fixed-point colorings. These examples also motivate the introduction of a more restrictive class of symmetric

Example 5.2. Consider a bidirectional ring of twelve cells with nearest neighbor and next nearest neighbor identical coupling. See Fig. 3 (centre). Here each line segment between cells c, d represents a pair of identical arrows: one from c to d and one from d to c. If we label the cells by elements of Z12 , then each cell i is coupled to cells i − 2, i − 1, i + 1, i + 2, with all arrows identical. It is proved in [Golubitsky et al., 2004a] that both rings in Fig. 3 (left, centre) have the same symmetry group D12 . This implies that they both have several balanced polydiagonals determined by fixed-point spaces. However, the 12-cell ring with nearest and next nearest neighbor coupling supports an exotic balanced polydiagonal, namely, the one shown in Fig. 3 (right). It is balanced because every dark cell has two dark and two light inputs, and every light cell has two dark and two light inputs. We call this the three-block

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coloring. The three-block coloring is not balanced for the 12-cell ring with just nearest neighbor coupling. It is easy to prove that the three-block coloring is exotic, that is, it does not arise as the fixed-point space of a subgroup of D12 . To do so, let the associated polydiagonal be W = {(x, x, x, y, y, y, x, x, x, y, y, y)}

(10)

Then the subgroup H of D12 that fixes a generic point of W (a point of the form (x, x, x, y, y, y, x, x, x, y, y, y) with x = y) has order 4, and is generated by two reflections in orthogonal diameters of the ring. However, the fixed-point space of this subgroup is Fix(H) = {(x1 , x2 , x1 , y1 , y2 , y1 , x1 , x2 , x1 , y1 , y2 , y1 )} which has dimension four, not two. In particular, in that fixed-point space, the central cell in each block of three does not have the same color as its two neighbors. A similar coloring with polydiagonal (x, x, x, y, y, y) also occurs in a simpler network, a ring of six cells with nearest and next-nearest neighbor coupling, Fig. 6 below. However, that coloring is a fixed-point space, because now Aut(G) > D6 . In fact Aut(G) has order 48 and is isomorphic to the octahedral group O ⊕ Z2 of symmetries of the regular octahedron (or its dual, the cube). This is most easily seen by considering the complementary network with the same six cells, joining them by arrows if and only if they are not so joined in the original network.

6. Stability Analysis for the 12-Cell Ring In this section we outline the stability analysis of the three-block coloring in the 12-cell ring, in the case when the dynamics is steady-state. Label the cells by elements of Z12 as in Fig. 3 (centre), so that for each i ∈ Z12 , cell i is coupled to cells i − 2, i − 1, i + 1, i + 2, with all arrows identical. The admissible differential equations corresponding to this network are those of the form x˙ i = f (xi , xi+1 , xi+2 , xi−1 , xi−2 )

(11)

for 0 ≤ i ≤ 11, where the overline indicates that f is invariant under permutation of the last four arguments. Simulations show that the three-block coloring can be realized as a stable equilibrium state in the case when f (xi , xi+1 , xi+2 , xi−1 , xi−2 ) = −x3i + px2i + qxi + b(xi+1 + xi+2 + xi−1 + xi−2 ) Now (11) becomes x˙ i = −x3i + px2i + qxi + b(xi+1 + xi+2 + xi−1 + xi−2 ) (i ∈ Z12 )

(12)

In fact, when p = −0.1, q = 0.35, b = −0.15, we find a three-block coloring of equilibria by taking initial conditions near W . For example, initial conditions x0 = (−0.51, −0.513, −0.517, 0.112, 0.113, 0.115, −0.53, −0.51, −0.52, 0.104, 0.102, 0.108) lead to Figs. 4 and 5.

Symmetry and Synchrony in Coupled Cell Networks 1: Fixed-Point Spaces 1

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12-cell ring: Simulation.

for the network in Fig. 3 (centre), at any three-block steady state.

6.1. Six-cell quotient

Fig. 5.

12-cell ring: simulation. Superposition of time-series.

We begin by analyzing the analogous problem for a six-cell ring, which is a quotient of the network in Fig. 3 (centre). This calculation provides six of the twelve eigenvalues for the 12-cell case, and also serves as a useful warm-up problem. To form the quotient network, observe that the automorphism group D12 of the 12-cell ring is generated by the permutations 

We now describe the computation of the linear stability of the three-block coloring in the system (12). The same calculations apply with minimal modifications to any admissible system of equations

0 1 2 3 σ= 1 2 3 4  0 1 2 κ= 0 11 10

4 5 6 5 6 7 3 4 5 9 8 7

8 9 10 11 9 10 11 0 6 7 8 9 10 11 6 5 4 3 2 1 7 8

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2

The Jacobian J  U b   b J = 0   b b

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at x∗ is b U b b 0 b

b b U b b 0

0 b b V b b

b 0 b b V b

 b b   0  b   b V

where U = −3X 2 + 2pX + q V = −3Y 2 + 2pY + q

Fig. 6. Six-cell bidirectional ring. Colors show quotient of three-block coloring.

The 12-cell ring admits a balanced equivalence relation  with classes

We seek eigenvalues of J. Let ω be a primitive cube root of unity, and define xk = [1, ω k , ω 2k , 0, 0, 0]T yk = [0, 0, 0, 1, ω k , ω 2k ]T

{0, 6}, {1, 7}, {2, 8}, {3, 9}, {4, 10}, {5, 11} That is, i  i + 6 for i ∈ Z12 . The corresponding polydiagonal is

for k = 0, 1, 2, where T is the transpose. The subspaces Xk = R{xk , yk } are invariant under J. In fact, computation shows that

∆ = {(u, v, w, x, y, z, u, v, w, x, y, z)}

Jxk = U xk + b(ω k + ω 2k )(xk + yk ) Jyk = V yk + b(ω k + ω 2k )(xk + yk )

which is the fixed-point space of the subgroup of D12 generated by σ 6 , which geometrically is rotation through π. Identifying cells in the same -class leads to the quotient network of Fig. 6. This is a six-cell ring, again with next nearest neighbor coupling. The vector field (12) reduces to the same equations, but now i ∈ Z6 , and cell j ∈ Z12 corresponds to cell j (mod 6) in Z6 . The three-block coloring in the 12-cell ring is a lift of an analogous coloring in the six-cell ring, indicated by the colors in Fig. 6, which corresponds to an equilibrium of the form x∗ = (X, X, X, Y, Y, Y ) Such equilibria are given by the intersection of two cubic curves in the (X, Y )-plane, namely −X 3 + pX 2 + qX + 2b(X + Y ) = 0 −Y 3 + pY 2 + qY + 2b(X + Y ) = 0

(13)

For suitable choices of parameters there exist intersection points off the diagonal X = Y . We assume such parameters have been chosen and ignore the details of existence of such solutions. Note that there is a “parameter symmetry” that interchanges X and Y .

Thus J decomposes into three 2×2 blocks Jk , where   U + 2b 2b J0 = 2b V + 2b and

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U −b −b

−b V −b



The eigenvalues of J0 are q 1 (U + V + 4b ± (U − V )2 + 16b2 ) 2 whereas those of J1 = J2 are  1 (U + V − 2b ± (U − V )2 + 4b2 ) 2 and so occur as eigenvalues of K with multiplicity 2. Thus the eigenvalues of J are:  1 (U + V + 4b + (U − V )2 + 16b2 ) simple 2  1 (U + V + 4b − (U − V )2 + 16b2 ) simple 2  1 (U + V − 2b + (U − V )2 + 4b2 ) double 2  1 (U + V − 2b − (U − V )2 + 4b2 ) double 2

Symmetry and Synchrony in Coupled Cell Networks 1: Fixed-Point Spaces

{0, 1, 2, 6, 7, 8} {3, 4, 5, 9, 10, 11}

6.2. 12-cell ring We now consider the full 12-cell ring. Let W be the polydigonal corresponding to the threeblock coloring, as defined in (10). The 12-cell ring has the same ODE as the six-cell ring, but with indices considered (mod 12). It also has the same conditions for an equilibrium, but now lying in W , hence of the form (X, X, X, Y, Y, Y, X, X, X, Y, Y, Y ). So the equilibrium under consideration is (x∗ , x∗ ) where x∗ = (X, X, X, Y, Y, Y ) as before. Let  be the balanced equivalence relation with classes 

U  b  b  0   0  0  J = 0  0   0  0   b b

b U b b 0 0 0 0 0 0 0 b

b b U b b 0 0 0 0 0 0 0

0 b b V b b 0 0 0 0 0 0

0 0 b b V b b 0 0 0 0 0

so that W = ∆
 . Observe that ∆
 ⊆ ∆ so that x∗ ∈ ∆ . Restricting the vector field to ∆ yields precisely the six-cell quotient studied in the previous section. Thus we have found six of the twelve eigenvalues of the Jacobian (which we also call J because its restriction to ∆ is our previous J). It remains to find the other six eigenvalues. Explicitly, we compute: 0 0 0 b b V b b 0 0 0 0

0 0 0 0 b b U b b 0 0 0

0 0 0 0 0 b b U b b 0 0

0 0 0 0 0 0 b b U b b 0

0 0 0 0 0 0 0 b b V b b

b 0 0 0 0 0 0 0 b b V b

D = diag[U, U, U, V, V, V, U, U, U, V, V, V ] so that B = J − D is the circulant matrix  0 b b  b 0 b  b b 0  0 b b   0 0 b  0 0 0  B= 0 0 0  0 0 0   0 0 0  0 0 0   b 0 0 b b 0

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Now J = B + D, and we use the eigenvectors of B to determine those of J.

 b  b  0  0   0  0   0  0   0  b   b V

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We can find the eigenvectors and eigenvalues for B by analogy with the six-cell ring. Namely, introduce vectors in C12 defined by zj = [1, ζ j , ζ 2j , . . . , ζ 11j ]T where ζ = e2πi/12 is a primitive twelfth root of unity. Then Bzj = b(ζ j + ζ 2j + ζ −j + ζ −2j )zj so the zj are eigenvectors of B with eigenvalues bµj where µj = b(ζ j + ζ 2j + ζ −j + ζ −2j ) Consider the two subspaces (or rather, their real parts) Z = C{z0 , z2 , z4 , z6 , z8 z10 } Z O = C{z1 , z3 , z5 , z7 , z9 z11 } E

and therefore leaves all isotypic components of H invariant. (The isotypic components of a representation are the subspaces formed by adding together all invariant subspaces that are isomorphic to a given irreducible representation, and these are invariant under any equivariant linear map, see [Golubitsky et al., 1988].) We already know Bzj = bµj zj . Direct computation yields formulas for Dzj . However, it is more informative to consider the isotypic components of κσ −1 on Z O , which must be J-invariant (hence D-invariant). The fixed-point space (trivial isotypic component) of κσ −1 on Z O is Z0O = {(u, v, u, x, 0, −x, −u, −v, −u, −x, 0, x)T } which is spanned by h0 = (1, 0, 1, 0, 0, 0, −1, 0, −1, 0, 0, 0)T h1 = (0, 1, 0, 0, 0, 0, 0, −1, 0, 0, 0, 0)T h2 = (0, 0, 0, 1, 0, −1, 0, 0, 0, −1, 0, 1)T

Since ζ 2 is a primitive sixth root of unity, it is easy to see that ∆ = Z E so our computations for the six-cell ring are the restriction of the twelve-cell ring onto Z E . We claim:

Since D is diagonal, Dh0 = U h0 Dh1 = U h1 Dh2 = V h2

Lemma 6.1. The space Z O is J-invariant.

This statement can be verified by direct computation. However, it is possible to understand why this invariance occurs by appealing to group representation theory. Introduce the transformation

Proof.

ρ = −I12 R12

C12 ,

and Bh0 = b(h0 + 2h1 + h2 ) Bh1 = b(h0 + h2 ) Bh2 = b(h0 + 2h1 − h2 )

acting on and where I12 is the 12×12 identity matrix. Clearly every 12 × 12 matrix commutes with ρ. Therefore the Jacobian J has extra symmetry: it commutes with ρ as well as with any elements of D12 that fix the equilibrium x∗ . One such element is σ 6 , so J must commute with ζ 6 and, more importantly at this stage, with ρζ 6 . Therefore J leaves invariant the fixed-point space of ρζ 6 , which is the space

Therefore the matrix  U +b   2b b

{(u, v, w, x, y, z, −u, −v, −w, − x, −y, −z) : u, v, w, x, y, x ∈ R}

which is spanned by

Because ζ 6 = −1, this space is identical to Z O .




Observe for later use that there is another element of D12 that fixes x∗ , namely the reflection κσ −1 . So J is (group)-equivariant under the subgroup H = Z 2 × Z 2 × Z 2 = ζ 6 , κσ −1 , ρ

of J on Z0O is  b b  U 2b  b V −b

(14)

In the same manner, the minus-action space (nontrivial isotypic component) of κσ −1 on Z O is Z1O = {(u, 0, −u, x, y, x, −u, 0, u, −x, −y, −x)T } h3 = (1, 0, −1, 0, 0, 0, −1, 0, 1, 0, 0, 0)T h4 = (0, 0, 0, 1, 0, 1, 0, 0, 0, −1, 0, −1)T h5 = (0, 0, 0, 0, 1, 0, 0, 0, 0, 0, −1, 0)T Since D is diagonal, Dh3 = U h3 Dh4 = V h4 Dh5 = V h5

Symmetry and Synchrony in Coupled Cell Networks 1: Fixed-Point Spaces

Recall the parameter values p = −0.1, q = 0.35, b = −0.15. The numerical solution of the ODEs leads to an equilibrium of three-block type at

and Bh3 = b(−h3 − h4 − 2h5 ) Bh4 = b(−h3 + h4 + 2h5 ) Bh5 = b(−h3 + h4 ) Therefore the matrix of J on  U −b −b  V +b  −b −2b 2b

Z01 is  −b  b  V

U = 0.567307

(15)

Neither of the 3 × 3 matrices (14) and (15) is especially tractable. The characteristic equation is cubic and there are no obvious eigenvalues. Mathematica yields the following formulas, derived by Cardano’s method. As we see below, they suffer from the usual defect of Cardano’s method, namely the occurrence of complex numbers in the formula when the roots are all real, see for example [Stewart, 2003]. Define F1 G1 F2 G2

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Then the eigenvalues of (14) are 21/3 F1 2U + V q − 3 3(G1 + 4F13 + G21 )1/3  q 1  3 2 1/3 + + 4F + G ) G 1 1 1 3.21/3 √ (1 + i 3)F1 2U + V q + λ2 = 3 3.22/3 (G1 + 4F13 + G21 )1/3  q √ 1  3 2 1/3 − 3)G + 4F + G ) (1 − i 1 1 1 6.21/3 √ (1 − i 3)F1 2U + V q + λ3 = 3 3.22/3 (G1 + 4F13 + G21 )1/3  q √ 1  3 2 1/3 − 3)G + 4F + G ) (1 + i 1 1 1 6.21/3 λ1 =

and those of (15) are obtained from these by interchanging U and V and replacing F1 , G1 by F2 , G2 . This follows since the rotation σ 3 has the effect of interchanging U and V .

6.3. Numerical example We now relate the above analysis to the specific numerical example described earlier.

V = −0.621329

We seek numerical solutions of (13). By B´ezout’s Theorem we expect nine complex solutions, possibly fewer real solutions. Numerical solution of Eqs. (13) produces the following zeros: (X, Y ) = (−0.621329, 0.567307), (−0.440603 − 0.379831i, −0.440603 + 0.379831i) (−0.440603 + 0.379831i, −0.440603 − 0.379831i) (−0.05 − 0.497494i, −0.05 − 0.497494i) (−0.05 + 0.497494i, −0.05 + 0.497494i) (0, 0) (0.367614 − 0.359085i, 0.367614 + 0.359085i) (0.367614 + 0.359085i, 0.367614 − 0.359085i) (0.567307, −0.621329) and we recognize the last one as the one found by the ODE solver. We therefore take (U, V ) = (−0.728972, −0.683882) The eigenvalues of J are, in the listed order, −0.705581 −1.307270 −0.404742 −0.708112

[double] [double]

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[double] [double]

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The eigenvalues of (14) are

Table 1. Conjugacy classes of subgroups of D10 and their fixed-point spaces.

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Z2

and those of (15) are −1.101650 −0.582776 −0.412308 in exact agreement with the analytic calculations. Since all eigenvalues are real and negative, the three-block equilibrium is stable.

7. Ten-Cell Ring An instructive example occurs in a ring of ten cells (or, more generally, 5n where n ≥ 2). Here a large number of exotic colorings exists. We outline how these colorings arise, omitting the stability analysis which is similar to the twelve-cell case above. Let G be a ring of ten identical cells, in which each cell is coupled identically to its nearest and next-nearest neighbors. That is, the cells are 0, 1, 2, . . . , 9 ∈ Z10 and there are edges from node i to nodes i−2, i−1, i+1, i+2 for all i ∈ Z10 . All cells are cell-equivalent and all edges are edge-equivalent. See Fig. 7. It is easy to prove that the automorphism group is Aut(G) ∼ = D10 which is the dihedral group determined by the obvious rotational and reflectional symmetries of the ring, generated by σ : i → i + 1 (mod 10) κ : i → −i (mod 10)

3

2

σ 

Next, we classify the subgroups of D10 (up to conjugacy) and their fixed-point spaces; the result is shown in Table 1. Note that some fixed-point spaces are listed more than once. Any balanced equivalence relation that does not correspond to a fixed-point space listed in the table yields an exotic coloring. We now show that there are nine of these up to action by D10 . Let 5 be the equivalence relation on the cells of G for which i 5 j ⇔ i ≡ j

(mod 5)

Then the quotient network G5 = G/ 5 is a five-cell ring with nearest and next-nearest neighbor coupling, as in Fig. 8. Any G-admissible ODE, restricted to ∆
5 , is a G5 -admissible ODE. Any balanced polydiagonal of G5 therefore lifts to a balanced polydiagonal of G that is contained in ∆
5 = {(x, y, z, u, v, x, y, z, u, v}, hence forms a five-periodic coloring.

1

4

1

5

0 6

2

0

9 7

8

Fig. 7. Bidirectional ring of ten cells with nearest and nextnearest neighbor coupling.

3

4

Fig. 8. Bidirectional ring of five cells with nearest and nextnearest neighbor coupling.

Symmetry and Synchrony in Coupled Cell Networks 1: Fixed-Point Spaces Table 2. Five-periodic colorings on Z10 . Pattern (x, x, x, x, x, x, x, x, x, x) (x, x, x, x, y, x, x, x, x, y) (x, x, x, y, y, x, x, x, y, y) (x, x, y, x, y, x, x, y, x, y) (x, x, x, y, z, x, x, x, y, z) (x, x, y, x, z, x, x, y, x, z) (x, x, y, y, z, x, x, y, y, z) (x, y, x, y, z, x, y, x, y, z) (x, y, y, x, z, x, y, y, x, z) (x, x, y, z, u, x, x, y, z, u) (x, y, x, z, u, x, y, x, z, u) (x, y, z, u, v, x, y, z, u, v)

575

[Golubitsky & Stewart, 2002], any equivalence relation on {0, 1, 2, 3, 4} is balanced in G5 . Lifting back to G we conclude that any equivalence relation  on Z10 with the property i 5 j ⇒ i  j is balanced for G. In other words, any five-periodic coloring on G is balanced. Up to equivalence under the action of D10 , the five-periodic colorings on Z10 are shown in Table 2. Figure 9 shows the balanced colorings determined by fixed-point spaces for D10 . Figure 10 shows the exotic balanced colorings obtained by lifting from G5 .

Acknowledgments It so happens (this is why 5 is the crucial number here) that G5 is the complete graph on five nodes [Tutte, 1984; Wilson, 1985] which has automorphism group S5 . Therefore, by Example 1.12 of

Fig. 9.

The work of INS was supported in part by NSF Grant DMS-0244529 and a grant from EPSRC. It was mainly carried out at the University of Hong Kong, under the auspices of a Royal Society Kan Tong Po visiting professorship. We thank

Balanced colorings determined by fixed-point spaces for D10 .

576

F. Antoneli & I. Stewart

Fig. 10.

Balanced colorings, lifted from G5 , that are not fixed-point spaces for D10 .

the Isaac Newton Institute, the University of Houston, and the University of Hong Kong for hospitality and additional financial support. We also thank Martin Golubitsky and Andreij T¨ or¨ ok for helpful discussions, and Steve Donkin for pointing out some obscurities and omissions in a previous version of this paper.

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