COUPLED CELL NETWORKS: HOPF BIFURCATION AND INTERIOR SYMMETRY

We consider interior symmetric coupled cell networks where a group of permutations of a subset of cells partially preserves the network structure. In this setup, the full analogue of the Equivariant Hopf Theorem for networks with symmetries was obtained by Antoneli, Dias and Paiva (Hopf bifurcation in coupled cell networks with interior symmetries, SIAM J. Appl. Dynam. Sys. 7 (2008) 220–248). In this work we present an alternative proof of this result using center manifold reduction.

1. Introduction. Coupled cell systems are networks of dynamical systems (the cells) that are coupled together. Relevant aspects in the study of the dynamics of these systems can be encoded by a directed graph (coupled cell network): the nodes represent the cells and the edges indicate which cells are coupled and if the couplings are of the same type. We consider a special class of non-symmetric networks -the interior symmetric coupled cell networks. These networks admit a subset S of the cells such that the cells in S together with all the edges directed to them form a subnetwork which possesses a non-trivial symmetry group Σ S . Here, we follow the theory of Stewart et al. [4,6,9].
The local synchrony-breaking bifurcations in a coupled cell system occur when a synchronous state loses stability and bifurcates to a state with less synchrony. Such bifurcations can be considered to be a generalisation of symmetry-breaking bifurcations in symmetric coupled cell systems. See Golubitsky, Stewart and Schaeffer [5]. An analogue of the Equivariant Hopf Theorem for coupled cell systems with interior symmetries was obtained by Golubitsky, Pivato and Stewart [4] proving the existence of states whose linearizations on certain subsets of cells, near bifurcation, are superpositions of synchronous states with states having spatial symmetries. Antoneli, Dias and Paiva [1] extended this result obtaining states whose linearizations on certain subsets of cells, near bifurcation, are superpositions of synchronous states with states having spatio-temporal symmetries, that is, corresponding to interiorly C-axial subgroups of Σ S × S 1 . The proof of this result uses a modification of the Lyapunov-Schmidt reduction to arrive at a situation where the proof of the Standard Hopf Bifurcation Theorem can be applied. In this work, we present an alternative proof using center manifold reduction. This approach can be useful in the development of normal form theory aiming at the study of the stability of such periodic solutions (see Antoneli, Dias and Paiva [2]).
In Sections 2-5 we recall the definition of interior symmetry and the structure of coupled cell systems associated with interior symmetric networks. In Section 6 we state the Interior Symmetry-Breaking Hopf Bifurcation Theorem and prove it using Center Manifold Reduction.
2. Coupled Cell Networks with Interior Symmetry. Given a coupled cell network G, the associated coupled cell systems are dynamical systems compatible with the architecture of G. More specifically, each cell c is equipped with a phase space P c , and the total phase space of the network is the cartesian product P = c P c . Call the set of edges directed to a cell c by the input set of c. A vector field f is called admissible if its component f c 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 (pull-back condition). See Golubitsky et al. [6] for the formal definitions of coupled cell network and admissible vector fields.
Consider a subset S of the set of cells of G and let G S be the sub-network of G formed by the cells in G and the edges that are directed to cells in S. By Antoneli et al. [1] (Proposition 3.3), the group of interior symmetries of G (on the subset S) can be canonically identified with the group of symmetries of G S . See Golubitsky et al. [4] for the original definition of interior symmetry and Antoneli et al. [1,8] for the details about its identification with the group of symmetries of G S . Example 1. Figure 1 shows three networks: G 1 (left), G 2 (right) and G S (center) where S = {1, 2, 3, 4}. The network G S obtained from G 1 is the same as the one obtained from G 2 . Observe that for the three networks the arrows coming from the set S = {1, 2, 3, 4} and directed to its complement C \ S = {5} are different. 3. Vector Fields with Interior Symmetry. Suppose that G admits a nontrivial group of interior symmetries Σ S on a subset of cells S. We can decompose the phase space P as a cartesian product P = P S × P C\S where P S = s∈S P s and P C\S = c∈C\S P c . For any x ∈ P we write x = (x S , x C\S ) where x S ∈ P S and x C\S ∈ P C\S and we can take the action of Σ S on P given by: Here Σ S acts on x S by permuting the coordinates corresponding to the cells in S.
By Proposition 1 the subspace Fix P (Σ S ) is flow-invariant under any admissible vector field on P . Since Fix P (Σ S ) is Σ S -invariant and Σ S acts trivially on the cells in C \ S we have that P C\S ⊂ Fix P (Σ S ). The action of the group Σ S decomposes the set S as where the sets S i (i = 1, . . . , k) are the orbits of the Σ S -action. Let Since W is a Σ S -invariant subspace of P S and W ∩ Fix P (Σ S ) = {0} we can decompose the phase space P as a direct sum of Σ S -invariant subspaces: Consider a coupled cell system with interior symmetry group Σ S on S. Let U = Fix P (Σ S ). We can choose coordinates (w, u) with w ∈ W and u ∈ U adapted to the decomposition (2) and write the admissible vector field f as since Σ S acts trivially on U . Thus we may write an admissible vector field f as wheref is Σ S -equivariant.

Linear Maps with Interior Symmetry.
In the linear case, we may choose a basis of P adapted to the decomposition (2) and then a G-admissible linear vector field L can be written as (4), The spectral properties of L in (5) are summarized in the following proposition.

Proposition 2.
Let G be a network admitting a non-trivial group of interior symmetries Σ S and fix a total phase space P . Let L : P → P be a G-admissible linear vector field and consider the decomposition of L given by (5). Then with interior symmetry group Σ S on S and suppose that it undergoes a codimensionone synchrony-breaking bifurcation at a synchronous equilibrium x 0 ∈ Fix P (Σ S ) when λ = λ 0 . Let L = (df ) (x0,λ0) be written as in (5). As defined in Antoneli et al. [1], we say that f undergoes a codimension-one interior symmetry-breaking Hopf bifurcation if the following conditions hold: Assume that L as in (5) has ±i as eigenvalues that come only from the sub-block A of L and that they are the only critical eigenvalues of L. Consider A c = A| Ei(A) . As A has ±i as eigenvalues there is a natural action of Σ S × S 1 on P , where S 1 acts on E i (A) by exp(s(A c ) t ) and trivially on P \ E i (A). The action of Σ S on P is given by (1).

Remark 1. Observe that, in general, there is no action of Σ
6. Interior Symmetry-breaking Hopf Bifurcation Theorem. In this section we shall give an alternative proof the main result of Antoneli et al. [1] (Theorem 4.8) using a center manifold reduction approach.
The original proof of this theorem in [1] is through a modified version of the Lyapunov-Schmidt reduction. The new proof presented below combines an extra condition that the vector field is in "interior normal form up to all orders" with a "hidden symmetry" center manifold reduction. Let us start by giving a precise definition of the notion of "interior normal form".

Definition 1.
We say that f is in interior normal form (to all orders), with respect to L = (df ) (x0,0) near λ 0 , iff (·, λ) is in normal form (to all orders) with respect toL = (df ) (x0,0) near λ 0 , that is,f commutes with the action of Σ S × S 1 on P defined above, for λ near λ 0 .
It is well known that the definition of normal form as used above is equivalent to the following definition.

Definition 2. A smooth family of vector fields
where the mappings f NF and r satisfy: In (i) we use the Lie bracket of two vector fields: Therefore, iff is in normal form (up to all orders) thenf =f NF . Theorem 1. Let G be a coupled cell network admitting a non-trivial group of interior symmetries Σ S relative to a subset S of cells and fix a phase space P . Consider a smooth 1-parameter family of G-admissible vector fields f : P × R → P in interior normal form up to all orders. Suppose that the bifurcation problem undergoes a codimension-one interior symmetry-breaking Hopf bifurcation at an equilibrium point x 0 ∈ Fix P (Σ S ) when λ = 0. Let L = (df ) (x0,0) be written as in (5) and Δ ⊂ Σ S × S 1 be an interiorly C-axial subgroup (on E c (A)). Then generically there exists a family of small amplitude periodic solutions of (7) bifurcating from (x 0 , 0) and having period near 2π. Moreover, to lowest order in the bifurcation parameter λ, the solution x(t) is of the form where w(t) = exp(tL)w 0 (w 0 ∈ Fix W (Δ)) has exact spatio-temporal symmetry Δ on the cells in S and u(t) = exp(tL)u 0 (u 0 ∈ Fix P (Σ S )) is synchronous on the Σ S -orbits of cells in S.
In the proof of Theorem 1 we will need two lemmas and one proposition that we introduce next.

Lemma 1. Let f : P × R l → P be a smooth l-parameter family of G-admissible vector fields in interior normal form (up to all orders). Then for every subgroup
Proof. Let Δ ⊆ Σ S × S 1 be a subgroup. Since f is in interior normal form up to all orders,f (·, λ) is Σ S × S 1 -equivariant for all λ ∈ R l and sof (Fix P (Δ) × R l ) ⊆ Fix P (Δ). Since h : P × R l → P maps into P C\S and P C\S ⊆ Fix P (Δ), we arrive at (8).
Finally, let us recall a result from Leite and Golubitsky [7], which may be seen as a center manifold reduction for vector fields with "hidden" symmetries.

then the center manifold reduction f c may be chosen so that σ|
Proof. See Leite and Golubitsky [7, p. 2346].
We start the proof of Theorem 1 with the following lemma.
Proof. Consider x = (w, u) ∈ P where w ∈ W, u ∈ Fix P (Σ S ). Assume dim W = k and dim Fix P (Σ S ) = l. As AND INTERIOR SYMMETRY   7 and B does not have ±i as eigenvalues, we get In particular, it follows that dim(E i (A)) = dim(E i (L)). As Fix P (Δ) = Fix W (Δ) ⊕ Fix P (Σ S ), we have Proof of Theorem 1. Consider f written in the coordinates (w, u) adapted to the decomposition (2) as in (3), that isf (w, u, λ) is Σ S -equivariant. Now let us assume that f is in interior normal form (up to all orders) near λ = 0, that is,f =f NF andf comutes with the action of Σ S × S 1 . Then by Lemma 1 we have In our case, E c (L) = E i (L) since the only critical eigenvalues of L are ±i and these come only from the sub-block A of L. Then, under the condition (9), Proposition 3 grants that a center manifold reduction f c : Then, by Lemma 2, it follows that dim(E i (L) ∩ Fix P (Δ)) = 2. Now we may apply the standard Hopf Theorem to get the result.
The argument that the Theorem holds in the case where f is in interior normal form may be regarded as the statement that the branch of periodic solutions is persistent under a certain kind of perturbation. The Equivariant Hopf Theorem applied tof NF provides the existence of a branch of periodic solutions emanating from (x 0 , 0), with period near 2π, such that, to lowest order in the bifurcation parameter λ, the solution x(t) is of the form x(t) ≈ exp(tL)w 0 where w 0 ∈ Fix W (Δ). Now our argument shows that the addition of the term h tof NF does not destroy the branch of periodic solutions, it just breaks the exact spatio-temporal symmetry of the periodic solution. On the other hand, Field [3, p. 163] shows that a branch of periodic solutions with maximal isotropy symmetry group of a generic Σ S × S 1equivariant vector field persists under perturbations that breaks the S 1 symmetry. Now, in the general case, the vector field can be written as where,f NF is Σ S × S 1 -equivariant andr(x, λ) is Σ S -equivariant. Thus, one is tempted to argue that since the periodic solution independently persists under both types of perturbation, namelyr and h, it will persist under their sum as well. Unfortunately, this argument does not hold in general, since a G S -admissible vector field may never be a generic Σ S -equivariant vector field, that is, the structure of network forces the vector field to be more degenerate than it would be expected if it were a Σ S -equivariant vector field. Nevertheless, it is true if one can show that, for a particular network under consideration, the set of admissible vector fields and the equivariant vector fields are equal.