Topology and Bifurcations in Hamiltonian Coupled Cell Systems

To appear in Dynamical Systems: An International Journal Vol. 00, No. 00, Month 20XX, 1–22

a b c B.S. Chan and P.L. Buono and A. Palacios ∗ aDepartment of Mathematics,San Diego State University, San Diego, CA 92182; bFaculty of Science, University of Ontario Institute of Technology, 2000 Simcoe St N, Oshawa, ON L1H 7K4, Canada; cDepartment of Mathematics,San Diego State University, San Diego, CA 92182

(v5.0 released February 2015)

The coupled cell formalism is a systematic way to represent and study coupled nonlinear diﬀerential equations using directed graphs. In this work, we focus on coupled cell systems in which individual cells are also Hamiltonian. We show that some coupled cell systems do not admit Hamiltonian vector ﬁelds because the associated directed graphs are incompatible. In broad terms, we prove that only systems with bidirectionally coupled digraphs can be Hamiltonian. Aside from the topological criteria, we also study the linear theory of regular Hamiltonian coupled cell systems, i.e., systems with only one type of node and one type of coupling. We show that the eigenspace at a codimension one bifurcation from a synchronous equilibrium of a regular Hamiltonian network can be expressed in terms of the eigenspaces of the adjacency matrix of the associated directed graph. We then prove results on steady-state bifurcations and a version of the Hamiltonian Hopf theorem.

Keywords: Hamiltonian systems; coupled cells; bifurcations; nonlinear oscillators 37C80; 37G40; 34C14; 37K05

1. Introduction

The study of coupled systems of differential equations, also known as coupled cell systems, received much attention recently with various theories and approaches being developed con- currently [1–5]. The groupoid formalism approach to studying coupled cell systems developed by M. Golubitsky, I. Stewart, A. Dias and many other collaborators has shown that such types of systems exhibit generically bifurcation phenomena that are not observed in systems without this structure [6–11]; e.g. patterns of synchrony solutions, nilpotent bifurcations, including more than (1/2)th-power growth at Hopf bifurcation. A coupled cell system where all cells in the network are identical is called a homogeneous network and it is regular if all couplings are of the same type and each cell receives the same number of inputs. In this paper, we study regular coupled cell systems in which individual cells are also Hamilto- nian systems. Our main results are the following. We begin by considering a system of coupled Hamiltonian equations with an equilibrium at the origin and show a necessary and sufficient condition for the linear part at the origin to be a Hamiltonian matrix. We link this result with the structure of the adjacency matrix of the coupled cell network. Then we use this result to give a necessary condition on the digraph of a nonlinear Hamiltonian coupled cell system and also a necessary and sufficient condition for linearly coupled Hamiltonian cells. The second part of the paper is concerned with extending results of Golubitsky and Lauterbach [8] on the critical

∗Corresponding author. Email: [email protected] eigenspaces for codimension one families of regular coupled cell systems to the Hamiltonian case. In the non-Hamiltonian case, for one-parameter families of linear systems, generically, bifurcations occur as simple eigenvalues cross transversally the imaginary axis. On the other hand, in the Hamiltonian case, eigenvalues come in quadruplets λ, λ, λ,¯ λ¯ [12] and so bifurcations occur from collisions of eigenvalues on the imaginary axis, thus− leading− generically to having eigenvalues with non-semisimple eigenspaces. We show that for zero eigenvalues of multiplicity two and purely imaginary eigenvalues in 1 : 1 resonance, the generalized eigenspace can be expressed in terms of the eigenspace of the adjacency− matrix of the graph. We obtain bifurcation results leading to steady-states in synchrony subspaces (as a generalization of the Hamiltonian equivariant branching lemma [13]) and a generalization of the Hamiltonian Hopf bifurcation to synchrony subspaces. Coupled cell systems with Hamiltonian structure arise, for instance, in the context of analyz- ing the bifurcation structure in symmetrically coupled ring networks of gyroscopes [14] and of energy harvesters [15]. In these two cases, the Hamiltonian structure of each cell is obtained by setting the linear damping term to zero. This is a reasonable assumption as the damping term coefficient is several orders of magnitude smaller than the other parameters. It is shown in [14] that unidirectionally coupled ring networks of linear Hamiltonian systems do not preserve the Hamiltonian structure while the bidirectionally coupled rings do keep the structure. In the energy harvester case, the network is all-to-all coupled with SN permutation symmetry group and the Hamiltonian structure is also preserved. These finding prompted us to investigate the generalization to arbitrary coupled networks of Hamiltonian cells presented in this paper. Many systems of coupled Hamiltonian systems have been investigated over the years, al- though not necessarily using the graph theoretic formalism described in this paper. For instance, N-body problems in the form of “kinetic+potential” with the kinetic part describing the free motion of each body can be thought of coupled Hamiltonian systems with the potential function acting as the coupling term via the configuration variables. In the case of the Newtonian N-body problem, the potential acts as an all-to-all coupling term (see [16]), while in models of molecules, the potential energy is the coupling term describing the electronic binding between the atoms, see [17]. Other examples of coupled Hamiltonian systems are the Fermi-Pasta-Ulam (FPU) chains [18] modelling an infinite number of particles coupled via a potential function depending only on the positions. The case of finite number of particles is studied given boundary conditions; for instance, fixed endpoints leading to a Dn symmetric coupled cell system [19]. Fi- nally, we mention the study of the free motion of coupled rigid bodies such as described in [20] where the Lagrangian is given as the kinetic energy of each body, with coupling via a hinge constraint. However, in this case the dynamical equations obtained via symplectic reduction do not preserve the coupled cell structure as described in this paper. Recently, Manoel and Roberts [21] have studied a related problem to the one studied here, that is, whether a network can be regarded as a gradient system. They also determine the requirement that the digraph must be symmetric and characterize the form of admissible functions defining a gradient coupled cell system. Their main results are about regular graphs and they show necessary and sufficient conditions for a point to be a critical point of the admissible function in terms of the coupling function. They also mention how their results apply to the Kuramoto model and the Antiferromagnetic XY model. The paper is organized as follows. In Section 2, we introduce a graph theoretic definition of coupled cell systems and establish linear and nonlinear criteria for the coupled cell system to be Hamiltonian, given that each cell is Hamiltonian. Section 3 discusses generalized eigenspaces properties of the Hamiltonian coupled systems in terms of the eigenspaces of the eigenvalues of the adjacency matrix of the graph. This section specializes the results of [8] to the case of Hamil- tonian coupled cell systems. In particular, we obtain the structure of the generalized eigenspace of the Jacobian to be isomorphic to copies of the eigenspace of a given eigenvalue of the adja-

2 cency matrix. Section 4 presents the steady-state bifurcation and Hamiltonian Hopf bifurcations results when restricted to synchrony subspaces. The ﬁnal section presents a summary of our results and a short discussion on future work.

2. Hamiltonian coupled cell systems

In a coupled cell system, each cell is a system of diﬀerential equations with phase space variable Rki xi , for i 1,..., n . Suppose that cell i receives input from cells j1,..., jmi 1,..., n , then∈ the dynamics∈ { of the }ith component is ∈ { }

dxi = fi(xi, x j ,..., x j ). dt 1 mi Another feature of this formalism is that a coupled cell system can be represented graphically using so-called directed graphs.

Deﬁnition 2.1: A directed graph (or digraph) G consists of a vertex set V(G) and an arc set E(G), where an arc is an ordered pair of distinct vertices.

As an example, the graph at the top of Figure 1 has vertex set V = v , v and arc set E = { 1 2} e1, e2 . In this situation, vertices v1 and v2 represent the internal dynamics of the two cells. {Similarly,} arcs e = 2, 1 and e = 1, 2 represent the coupling dynamics between the two 1 { } 2 { } vertices. Figure 1 (bottom) has vertex set V = v1, v2, v3 and arc set E = e1, e2, e3, e4 with e = 2, 1 , e = 1, 2 , e = 3, 1 and e = 2, 3 .{ } { } 1 { } 2 { } 3 { } 4 { } e1

v1 v2

e4 v1 v2 v3

Figure 1.: Examples of digraphs representing coupled cell systems.

Given this setup, we may view the system as

dxi = gi(xi) + hi(x j ,..., x j ), dt 1 mi where gi represents the dynamics pertaining to cell i and hi is the function of the inputs into cell i.

3 = If we assume each cell dynamics has an equilibrium solution gi(xi∗) 0, then we can translate the equilibrium (x1∗,..., xn∗) to the origin. Without loss of generality, we can assume that the system has an equilibrium at the origin. Then, at the linear level, the matrices for internal and coupling dynamics can be written as

∂gi ∂hi = Qi and = Rij, (1) ∂x = ∂x i x 0 j x=0

where x = (x ,..., x )T , Q Rki ki , and R Rki k j . We can write the linearized internal 1 n i ∈ × ij ∈ × dynamics for the entire system as Q = diag(Q1,..., Qn) and the linearized coupling matrix as

R11 R12 ...... R1n  R21 R22 ...... R2n   . . . .  R =  . . .. .  . (2)    . ..   . . Rn 1,n   −   Rn1 Rn2 ... Rn,n 1 Rn,n   −    Diﬀerent coupling schemes lead to diﬀerent structure in the coupling matrix. For example, the nearest neighbor coupling scheme is represented by

0 R12 0 ... R1n  R21 0 R23 ... 0   . . . .  R =  0 ......  .    . .. ..   . . . Rn 1,n   −   Rn1 0 ... Rn,n 1 0   −    Overall, we may write the linear system of a coupled cell system as

M = Q + R, (3)

n k k where k = = k and M, Q, R R . We now try to seek answers to the following fundamental i 1 i ∈ × issues: GivenP a coupled cell system, can we determine if it is Hamiltonian? Given a directed graph, does there always exist an admissible Hamiltonian vector field corresponding to a coupled cell system? Suppose that each cell in a coupled cell system is already a Hamiltonian system, what are the criteria on hi and the coupling scheme so that the overall system is Hamiltonian? From another perspective, given a Hamiltonian system, what is the associated digraph? These issues are related to coupling topology and coupled Hamiltonian systems and they can be seen in [14]. In the aformentioned work, the internal dynamics of each cell is Hamiltonian, but if linear coupling terms are added, the resulting system is not always Hamiltonian. The authors showed that the Dn case maintains the Hamiltonian structure while the Zn does not. Thus, symmetry is not a sufficient criteria for coupled cell systems to remain Hamiltonian. In this section, we discuss some criteria related to the topology of the system and Hamiltonian vector fields.

2.1. Linear Criteria We first need to define a Hamiltonian coupled cell system. Given a function H(q, p) : R2ℓ R, a Hamiltonian system is a differential system consisting of 2ℓ ordinary differential equations→ of

4 the form ∂H ∂H q˙i = (q, p), p˙i = , for i = 1,...,ℓ, (4) ∂pi − ∂qi

T T where q = (q1,..., qℓ) and p = (p1,..., pℓ) are traditionally called the position and momen- tum vectors, respectively. Let 0ℓ and Iℓ denote the ℓ ℓ zero and identity matrices, respectively. Then the skew symmetric matrix J can be written as×

0ℓ Iℓ Jℓ = . " Iℓ 0ℓ # − Suppose the Jacobian matrix of the system (4) is M, then this matrix satisﬁes

T M Jℓ + Jℓ M = 0.

A Hamiltonian system can be written as

(˙q, p˙)T = J H(q, p). ℓ∇(q,p) The natural setting we adopt is to assume that each cell is Hamiltonian and we want to construct Hamiltonian coupled cell systems. In other words, for each cell i, there must be a Hamil- tonian function corresponding to the internal dynamics and the corresponding diﬀerential equations can be written in the same way as system (4). Furthermore, the dimension of the phase = = n variable of cell i must be even, i.e., ki 2li. Let ℓ i=1 li, then the total dimension of the R2ℓ 2ℓ system is 2ℓ. Suppose that M × is the Jacobian matrixP of the Hamiltonian coupled cell system, then it must satisfy ∈

MT J + JM = 0, (5)

= where J diag(Jl1 ,..., Jln ) is the matrix giving the symplectic structure of phase space. For = T = T each cell, let xi (qi,1,..., qi,li , pi,1,..., pi,li ) and set x (x1,..., xn) . Let the Hamiltonian of the system be H(x) : R2ℓ R. In component form, the system may be written as → x˙ = J H(x). i li ∇xi With these deﬁnitions, we now discuss the relationship between topology and Hamiltonian coupled cell systems. The topological conditions for the linear coupled cell system to be Hamil- tonian are summarized in the following theorem. Theorem 2.2: Suppose we have a connected coupled cell system and that the internal dynamics of each cell is Hamiltonian. Then, the linearized system at the origin is Hamiltonian if and only if

RT J + J R = 0, for 1 i, j n. (6) ji l j li ij ≤ ≤ where Rij is from (2) and Jli is the symplectic structure matrix deﬁned above. Proof. Let us use the notation for linear internal and coupling matrices found in (1), then the linear part of a coupled cell system is shown in (3). Assuming that the system is Hamiltonian,

5 then we can write

MT J + JM = 0.

By (3), we rewrite the left hand side in terms of Q and R as

(QT J + JQ) + (RT J + JR) = 0.

Recall that Q = diag(Q1,..., Qn). By our assumptions on internal dynamics, each Qi is Hamil- T + = T + = tonian with respect to Jli , so Q J JQ 0 holds. Thus, we must show that R J JR 0 implies (6). A direct calculation shows that

T + T + T + R11 Jl1 Jl1 R11 R21 Jl2 Jl1 R12 ... Rn1 Jln Jl1 R1n RT J + J R RT J + J R ... RT J + J R T  12 l1 l2 21 22 l2 l2 22 n2 ln l2 2n  R J + JR =  . . . .  = 0. (7)  . . .. .   . . .   T + T + T +   R Jl1 Jln Rn1 R Jl2 Jln Rn2 ... Rnn Jln Jln Rnn   1n 2n    We see that the condition in (6) holds true if and only if (7) holds.

Algebraically, if Rij , 0, then R ji , 0 must be true to satisfy the condition in (6). Topologi- cally, the criteria in (6) implies that if there is a connection from cell i to cell j, there must also be a reciprocal connection from cell j to cell i. On the digraph representing the Hamiltonian coupled cell system, if there is an arc from vertex i to j, there must be another arc from vertex j to i. The necessary and suﬃcient conditions in Theorem 2.2 are now specialized to Hamiltonian matrices on coupled cell systems with identical nodes and identical edges. Deﬁnition 2.3: A coupled cell system is called homogeneous if all the nodes are identical, which means the state spaces of the cells all have the same dimension. A homogeneous coupled cell system with identical couplings is called a regular system.

In particular, a regular homogeneous coupled cell system has the same number of inputs to each cell, that is, its “valency” is constant. For a regular Hamiltonian coupled cell system, the phase space variables for all cells are set to be of dimension 2l. Since the coupling functions are all identical, we may simplify the notation by writing = Rij and this matrix must be a k k matrix with k = 2l. Thus, we may view this necessaryR and suﬃcient condition as requiring× to be a Hamiltonian matrix as per the standard deﬁnition. This means the total phase space hasR dimension 2ℓ = kn = (2l)n. Furthermore, for a regular Hamiltonian coupled cell system, Theorem 2.2 has a direct analog that can be express in graph theoretic terms. The linear component of a homogeneous system with identical coupling functions can be conveniently expressed using the adjacency matrix of the digraph.

Deﬁnition 2.4: The adjacency matrix A(G) of a directed graph G is the integer matrix with rows and columns indexed by vertices of G, such that the A(G)[i, j] is equal to the number of arcs from cell i to cell j.

6 In the examples of Figure 1, the adjacency matrices are

0 1 1 0 1 and 1 0 0 , 1 0 !    0 1 1      respectively.

α β γ δ Deﬁnition 2.5: Let A R × and B R × . Then the Kronecker or tensor product of A and B is deﬁned as ∈ ∈

a11B a1βB . ···. . A B =  . .. .  . ⊗    aα1B aαβB   ···    Note that (A B)T = AT BT and that if A and B are square matrices, then (A B)T = (A B) implies⊗A and B are symmetric.⊗ Under the homogeneous assumption, we may⊗ express ⊗ the identical linearized internal dynamics of each cell as = Qi. Suppose that the digraph of a homogeneous coupled cell system has adjacency matrixQ A, then the Jacobian matrix of a homogeneous system with identical coupling terms can be written as

= I + A , (8) M n ⊗ Q ⊗ R see [22] for a proof, but note that in this paper, the opposite convention for Kronecker products is used.

Definition 2.6: A digraph is symmetric if for every edge (i, j) there is also an edge ( j, i). Based on these definitions, the adjacency matrix of a symmetric digraph is symmetric (i.e. A = AT ). With these definitions in mind, we can express Theorem 2.2 for a homogeneous system with identical coupling in the following corollary.

Corollary 2.7: Suppose we have a connected regular coupled cell system and that the internal dynamics for each cell is Hamiltonian. Then, the linearized system at the origin is Hamiltonian if and only if the coupling matrix is Hamiltonian and the adjacency matrix of the digraph associated with the coupled cell systemR is symmetric.

Proof. Note that J = In Jl and because the underlying digraph is connected Jl is nonzero. Using the notation in (8)⊗ then R

T T J + J = (In Q + A R) (In Jl) + (In Jl)(In Q + A R) M M ⊗ T ⊗ ⊗ T ⊗ T ⊗ ⊗ = (In Q Jl) + (In JlQ) + (A R Jl) + (A JlR) ⊗ T ⊗ T ⊗ T ⊗ = In (Q Jl + JlQ) + (A ( JlR) ) + (A JlR) = (⊗A J R)T + (A J R) ⊗ − ⊗ − ⊗ l ⊗ l where QT J +J Q = 0 because Q is Hamiltonian. Therefore, T J+J = 0 if and only if A J R l l M M ⊗ l is symmetric and because A and JlR are square matrices, this is equivalent to A and JlR being T symmetric matrices. But JlR symmetric implies R Jl + JlR = 0 and so R is Hamiltonian.

7 Example 2.8: Consider the n n cyclic permutation matrix × 01 0 0 ... 0  00 1 0 ... 0   . . . . .   ......  C =   . n  . . .   ...... 0     0 ...... 0 1     1 0 ...... 0      D = + T For the examples given by [14], the adjacency matrix for the n case is ADn Cn Cn . Similarly Z = Z for the n case, we have AZn Cn. Clearly, ADn is symmetric while AZn is not. The n symmetric network is unidirectionally coupled, but the Dn symmetric one is bidirectionally coupled. Finding a characterization of the spectrum of a symmetric adjacency matrix (with trace zero) is an important problem investigated in graph theory [23]. In particular, ﬁnding out if an eigenvalue has multiplicity greater than 1. We now present examples of networks with symmetric adjacency matrices and multiple eigenvalues. Those are studied throughout this paper to illustrate the dif- ference in information obtained using symmetry methods as opposed to the one extracted from the adjacency matrix.

Example 2.9: Consider the six cell network Γ1 in Figure 2.

2 5 Γ1

1 4

3 6

Figure 2.: Network Γ1.

The network Γ1 has Abelian symmetry group D2 and its adjacency matrix is

011110  102000   120000  A =   . 1  100011     000102     000120      1 √ D Γ ﬃ The eigenvalues of A1 are 3, 2 (1 17), 2, 2, 0. The 2 symmetry of 1 is not su cient to guarantee the double eigenvalue ±2 because− irreducible− representations of Abelian groups are one-dimensional. Thus, the 2 eigenvalue− of multiplicity 2 is a consequence of the particular network structure. −

Example 2.10: Consider now the network Γ2 given in Figure 3.

8 Γ2 1 2 5 6

3 4 7 8

Figure 3.: Network Γ2.

The adjacency matrix of Γ2 is

01110000  10100010   11010000    =  10101000  A2    00010110   00001002     01001001     00000210      Note that this network has only a Z2 reﬂection symmetry given by the permutation (1 3)(2 4)(5 7)(6 8). The eigenvalues of A are 3, 1 √3, 1, 1 and three more simple eigen- 2 − ± − − values quite too long to express here and we refer to them as µ1, µ2 and µ3. Again, we see here an eigenvalue of multiplicity 2 arising from the topology of the network. In particular, let T T V1 = (1, 1, 1, 1, , 1, 1, 1, 1) and V2 = (1, 1, 1, 1, 1, 1, 1, 1) , then A2Vi = Vi for i = 1, 2. − − − − − − − − −

2.2. Nonlinear Criteria In Section 2.1, we deﬁne Hamiltonian coupled cell system and we show the necessary and suﬃcient topological condition for a linear coupled cell system to be Hamiltonian. For general nonlinear Hamiltonian coupled cell systems, the result is stated in the following theorem.

Theorem 2.11: Suppose that a Hamiltonian system of diﬀerential equations can be represented as a coupled system of Hamiltonian cells and the digraph of its coupled cell representation is connected. Then the digraph must be bidirectionally coupled. i.e., if there is an arc from cell i to j, then there must be a reciprocal connection from cell j to i.

Proof. Given that the system is Hamiltonian, there is a corresponding Hamiltonian function which can be written as

n H = Hi(xi) + Hc, Xi=1

n From the perspective of the coupled cell system, i=1 Hi represent the internal dynamics within the subsystems. Thus, Hc must capture all couplingP dynamics. If the number of nodes in the system is n = 1, then Hc = 0. The system is trivially connected by the set of empty arcs.

9 Suppose n 2. Now, the diﬀerential equations of each x can be written as ≥ i x˙ = J H = J (H + H ). i li ∇xi li ∇xi i c , Since we have a connected system, xi Hc 0 and xi Hc must be a function of x j, for some , ∇ ∇ j i. If xi Hc is not a function of any other variable, then the corresponding digraph is not connected.∇ Correspondingly on the graph G, we show now that there must be an arc from cell j to i representing this coupling. Given the equality of the mixed partial derivatives of Hc with respect to x and x , then for cell j, we know that H , 0 and thus H must be a function i j ∇x j c ∇x j c of xi. Thus, a reciprocal arc from cell i to cell j. We may repeat the above argument for each cell in the system and each connection.

We have a stronger result if we assume only linear coupling. This is the case studied for gyroscopes in [14] and for energy harvesters in [15]. Theorem 2.12: Consider a regular coupled cell system with linear coupling functions and for which each cell is Hamiltonian. Then, the coupled cell system is Hamiltonian if and only if the coupling matrix is Hamiltonian and the adjacency matrix is symmetric. R Proof. The ﬁrst implication is a consequenceof Theorem 2.11. The second implication is proved as follows. By Taylor expanding near the origin, we can write the Hamiltonian for each cell as Hcell(xi) = H2(xi) + Hr(xi) where H2 is a homogeneous quadratic polynomial and Hr contains terms of degree three and higher. If the linear coupling is bidirectional then the adjacency matrix is symmetric. Therefore, we know that the linear system is Hamiltonian by Corollary 2.7 and the = 1 T = Hamiltonian function is Hlin(x) 2 x J x.Then,for a n cell network, letting x (x1,..., xn), the coupled cell system has Hamiltonian− M function

n H(x) = Hlin(x) + Hr(xi). Xi=1

Remark 2.13: Here is an example with nonlinear coupling such that is Hamiltonian and its adjacency matrix is symmetric, but the full system is not Hamiltonian.R Consider the two cell system deﬁned as follows. Let u = (x , y ) R2 i i i ∈

u˙1 = F(u1) + h(u2, u1), u˙2 = F(u2) + h(u1, u2)

T where F is a Hamiltonian vector ﬁeld such that F(0) = 0 and h(u2, u1) = (0, x1 x2) . The linearized system at the origin is of the form I + A where = dF(0) and = 0 are = 0 1 Q ⊗ R ⊗ Q R Hamiltonian with respect to J2 1 0 . However, if H is a Hamiltonian function for this sys- − tem (with respect to J = I J ), it must contain a term H˜ such that ⊗ 2 ∂H˜ ∂H˜ = x1x2 = . ∂x2 − ∂x1

Integrating with respect to x , we obtain H˜ = 1 x x2 + K(x ) and taking the derivative with 2 − 2 1 2 1 respect to x leads to 1 x2 + K (x ) = x x and this equation has no solution. 1 − 2 2 ′ 1 − 1 2

10 3. Linear Structure in Regular Hamiltonian Coupled Cell Systems

Using results established in Section 2, we can construct coupled cell systems by coupling Hamil- tonian cells with the proper coupling functions and topology. In this section, we focus on the linear part of Hamiltonian coupled cell systems. We begin with a review of linear structure of general regular coupled cell systems. Then we describe generic codimension-one bifurcations for general Hamiltonian systems.

3.1. Eigenvalues of Hamiltonian Coupled Cell Systems The Jacobian matrix of a regular coupled cell system can be written as

= I + A , M n ⊗ Q × R where and are the linearized internal and coupling dynamics as previously stated. Let Q R µ1,...,µs be the distinct eigenvalues of A. Furthermore, let

= + µ . (9) Mµi Q iR We adapt two results from [8] to our context where A is a symmetric matrix by Corollary 2.7 and so its eigenvalues and eigenvectors are real. The ﬁrst one states that the eigenvalues of the matrices are the eigenvalues of . Mµi M Lemma 3.1 ([8]): The 2ℓ eigenvalues of the Jacobian are the union of the eigenvalues of the 2l 2l matrices for 1 i s. Speciﬁcally, supposeM v Rn is an eigenvector of A. Then × Mµi ≤ ≤ ∈ (v u) = v u . M ⊗ ⊗ Mµ h i Therefore, if u C2l is an eigenvector of , then v u is an eigenvector of . ∈ Mµ ⊗ M The next lemma shows that the generalized eigenspace associated to an eigenvalue of µ of A is invariant under . M n Lemma 3.2 ([8]): Let µ R be an eigenvalue of A and let EA(µ) R denote the associated eigenspace. Then E (µ) ∈C2l is invariant under . ⊂ A ⊗ M Example 3.3: Consider again the network Γ1 of Example 2.9. From (9), the eigenvalues of are obtained by computing the eigenvalues of the 2l 2l matrices 3 = + 3 , 0 = , M = + = 1 √ = × M DQ R M Q µ µ where µ 2 (1 17) and 2 2 . Considernow the 2 isotypic decom- M ± Q R±6R ±R6 = 2± 2 2 M− Q− R position of given by T A1 A where T represents trivial representations generated T ⊕ ⊕ T by the vectors (1, 0, 0, 1, 0, 0) and (0, 1, 1, 0, 1, 1) , A1 represents alternating representations T T generated by (0, 1, 1, 0, 1, 1) and (1, 0, 0, 1, 0, 0) while A2 represents also an alternating − − − T T representations (non-isomorphic to A1) generated by (0, 1, 1, 0, 1, 1) , (0, 1, 1, 0, 1, 1) . In this basis, = diag(Q , Q , 2 , 2 ) where − − − − M 1 2 Q − R Q − R + + 2 Q1 = Q R R and Q2 = Q R R . 2 + 2 ! ! R Q R Q−R R We thus see the advantage of using the adjacency matrix to compute the spectrum because the symmetry decomposition requires the computation of the eigenvalues of 4l 4l matrices instead of 2l 2l matrices. × × 11 3.2. Codimension-One Bifurcations of Hamiltonian Systems Given that we are studying real coupled cell systems, and must both be real Hamiltonian matrices. Consequently, is also a real Hamiltonian matrix.Q R For real Hamiltonian matrices, if Mµ σ is an eigenvalue, then so is σ [16]. Let σR denote the set of real eigenvalues of a Hamiltonian − matrix that are non-zero, i.e., σR = σ : σ R, σ , 0 . Similarly, let σ R denote the set of purely { ∈ } i imaginary eigenvalues, i.e., σ R = σ : σ C, Re (σ) = 0 . If σ σR or σ σ R, then clearly i { ∈ } ∈ ∈ i σ and σ form an eigenvalue pair on the real or the imaginary axis, respectively. Let σC denote − the set of truly complex eigenvalues, i.e., σC = σ : σ C and Im(σ) , 0 . Since the system { ∈ } is real and complex eigenvalues for real matrices must come in conjugate pairs, if σ σC is an eigenvalue of , σ, σ, and σ are also eigenvalues. Together, these four eigenvalues∈ Mµ − − form a Krein quartet. Let σ0 denote the set of zero eigenvalues of µ. Based on our previous discussion, one can see that the multiplicity of the zero eigenvalue mustM be even. As outlined by Dellnitz et al [24], in generic one-parameter Hamiltonian systems, there are two types of bifurcations from equilibrium. Either, (1) a steady-state bifurcation with a zero eigenvalue having multiplicity two, or (2) a Hamiltonian Hopf bifurcation with a pair of purely imaginary eigenvalues having multiplicity two. Let σ C be an eigenvalue of a Hamiltonian matrix . We denote by (σ) the generalized eigenspace∈ associated with σ. For steady-state bifurcations,M the zero eigenvalueGM of a Hamil- tonian matrix has multiplicity two and therefore dim( (0)) = 2. For a Hamiltonian Hopf bifurcation, the dimension of the generalized eigenspaceGM associated with the eigenvalue iω is dim( (iω)) = 4. ± GM

3.2.1. Codimension one pairs of Hamiltonian Systems In order to study codimension one bifurcations in regular coupled cell systems, the concept of “codimension one pairs” is introduced in [8] which is defined as pairs of k k matrices α,β such that = α + µβ and the real parts of the eigenvalues of the set of matrices× ,..., are Mµ Mµ1 Mµn distinct where µi (i = 1,..., n) are the eigenvalues of the adjacency matrix of the graph. This definition focuses on eigenvalues with distinct real parts and this is sufficient for studying bifurcations in dissipative systems. Bifurcation in these systems generically occur if the bifurcating eigenvalue is simple and crosses the imaginary axis transversely. On the other hand, bifurcations in Hamiltonian systems occur through collisions of eigenvalues at the origin of the complex plane or on the imaginary axis. These collisions do not always result in a loss of stability and emergence of new solutions. We can use Krein signature, [12], to predict the collision behaviour of the eigenvalues. = 1 T Definition 3.4: Let K be a Hamiltonian matrix and H(ξ) 2 ξ JKξ be the corresponding quadratic Hamiltonian. If K has a nonzero eigenvalue pair iω with− eigenvectors v = u iw and ± ± let ζ E iω = span(u, w) be any vector in the invariant subspace for iω. The Krein signature ∈ ± ± of E iω is ± sgn (H (ζ)) where sgn is the signature of the quadratic form H. Note that the Krein signature is well-defined for simple eigenvalues. For a purely imaginary eigenvalue with multiplicity two, if the double multiplicity arises from a pair of simple purely imaginary eigenvalues colliding as a parameter is varied, then two situations may arise. The

12 eigenvalues have same Krein signature and this is called a 1 : 1 resonance, or they have different Krein signature and this is called a 1 : 1 resonance, see [25]. The 1 : 1 resonance case does not lead to bifurcations, see [12, Theorem− 9.18]. Due to differences in eigenvalue movement between the dissipative and Hamiltonian cases, we introduce an alternative definition of codimension one pairs for Hamiltonian coupled cell systems. We begin by defining the following sets. Let L(2l) be the space of real 2l 2l Hamiltonian N 2 × matrices. Let 1 denote the pairs ( , ) L(2l) such that all the eigenvalues of µ1 ,..., µs are distinct. Let N be similarly definedQ R as∈ N except one of ,..., has eitherM one non-M 2 1 Mµ1 Mµs semisimple zero eigenvalue of multiplicity two or one eigenvalue in σ R in 1 : 1 resonance. i − Definition 3.5: Suppose , L(2l) and is defined as in equation (9). The pair ( , ) is Q R ∈ Mµ Q R a Hamiltonian codimension one pair if ( , ) N1 or ( , ) N2. The set of all Hamiltonian codimension one pairs is denoted by M (2Ql)R= N∈ N . Q R ∈ 2 1 ∪ 2 Remark 3.6: The complement of M2(2l) is not of codimension two in the space of Hamiltonian matrices as it also contains the case of real eigenvalues of multiplicity two at nonzero values and this is a codimension 1 phenomenon. But this is not relevant for the bifurcations we are interested to study since we want to identify the form of the Jacobian J corresponding to the generic codimension one bifurcations in Hamiltonian systems

2 If M2(2l) is dense in L(2l) , then the generic codimension-one bifurcation corresponds to one pair of non-semisimple eigenvalues in a single µ block. We now ﬁrst show that N1 is open and 2 M 2 dense in L(2l) . Using this, we show that M2(2l) is an open and dense subset of L(2l) .

2 Lemma 3.7: The set of matrices N1 is open and dense in L(2l) .

Proof. Recall that the adjacency matrix A has s distinct eigenvalues. Let ( , ) N1, then the 2ls eigenvalues of ,..., are distinct. Thus, there exists a disk of radiusQ R ∈δ > 0 around Mµ1 Mµs i each of the 2ls eigenvalues of µ1 ,..., µs such that no other eigenvalue is within that disk. Let M M

mini 1,...,2ls δi δ∗ = ∈{ } 2 denote half of the minimum value of the smallest δ . Because the eigenvalues of ,..., i Mµ1 Mµs continuously depend on the entries of the matrices, we can always find ε > 0 and †, † Q R ∈ 2 + L(2l) such that if † † <ε, the eigenvalues of µ†1 ,..., µ†s are inside the 2ls Q − Q R − R N M M δ∗ disks around the eigenvalues of µ1 ,..., µs . Thus, 1 is open. M M 2 Similar to the proof by [8], we show that N1 is dense in L(2l) by showing that the intersection of a finite number of sets involving ( , ) is dense. The relevant sets are Q R D = ( , ) L(2l)2 : eigenvalues of M are distinct and a Q R ∈ µa = n o ff Ea,b ( , ) Da Db : eigenvalues of Mµa and Mµb are di erent . n Q R ∈ ∩ o Given that there are a finite number of cells in the system, there is a finite number of eigenvalues in the adjacency matrix of the system. As a result, there is a finite number of Da sets and 2 finite number of intersections of these sets. Consequently, if Da is dense in L(2l) for all possible µa and Ea,b is dense in Da Db for all possible combinations of µa and µb, then Ea,b is dense 2 ∩ N = in L(2l) . Since by definition 1 µa,µb σ(A)Ea,b, we have the desired result if Ea,b is dense in L(2l)2 holds. ∩ ∈ The following argument on is useful in showing D and E are dense in D D . Since Mµ a a,b a ∩ b

13 A is a real symmetric matrix, its eigenvalues, µ, are all real-valued. Because µ R, we may write ∈ = + µ = + µ + (c c) = ( + c ) + (µ c) . (10) Mµ Q R Q R − R Q R − R

The series of identities in (10) show that we may shift the values of µi by a constant to simplify the proof. 2 We first show that Da is dense in L(2l) . Through the series of identities in (10), we may, = without loss of generality, assume µ 0 for convenience. Then Mµa has distinct eigenvalues if has distinct eigenvalues. Assume that is in Williamson normal form, [16]. We can find smallQ Q Hamiltonian perturbations applied to such that the eigenvalues are distinct. Thus, Da is dense in L(2l)2. Q To show that E is dense, let ( , ) be a pair in D D . If the eigenvalues of are a,b Q R a ∩ b Mµa distinct from the eigenvalues of µb , then ( , ) is in Ea,b. Conversely, if the eigenvalues of are not distinct from those ofM , thenQ weR need to show that there are members of E Mµa Mµb a,b in any neighborhood of ( , ). By the series of identities in (10), we may assume that µa = 0 , Q R and µb 0. Furthermore, we may assume that µa and µb are in Williamson normal form. M M = Let ε L(2l) be a small Hamiltonian perturbation. We fix and perturb so that µa and =∈ + + Q R M Q µb µb ( ε). Clearly, we can choose a small perturbation so that the eigenvalue of µa M Q R = + + N M are distinct from µb µb ( ε). Therefore, Ea,b is dense in Da Db and 1 is dense in L(2l)2. M Q R ∩

Proposition 3.8: The set of Hamiltonian codimension one pairs M2(2l) is open and dense in L(2l)2.

2 2 Proof. As shown in Lemma 3.7, N1 is a dense subset of L(2l) . Since N1 M2(2l) L(2l) , N1 dense in L(2l)2 implies it is dense in M (2l). All that remains to be shown⊂ is that if⊂ ( , ) N , 2 Q R ∈ 2 then there exists a small open ball around this point entirely contained in M2(2l). Let ǫ > 0. Suppose ﬁrst that = + µ has a non-semisimple zero eigenvalue of multiplicity two. Let Mµ Q R ( , ) be in the ǫ ball near ( , ), then either † still has a non-semisimple zero eigenvalue Q† R† Q R Mµ of multiplicity two and then ( †, †) N2, or the zero eigenvalue splits into a conjugate pair of purely imaginary eigenvalues,Q a pairR of∈ real eigenvalues or a quadruplet of complex eigenvalues. Because the other eigenvalues of the ( , ) pair are distinct, by choosing ǫ small enough, all Q R the nonzero eigenvalues of µ† are distinct from the remaining ones and ( †, †) N1. Thus, ( , ) M (2l). Suppose insteadM that ( , ) has a non-semisimple purely imaginaryQ R ∈ eigenvalue Q R ∈ 2 Q R of multiplicity two in 1 : 1 resonance. Then, for ǫ small enough, for ( †, †) in the ǫ ball, either the 1 : 1 resonance− persists and all other eigenvalues are still distinct,Q R thus ( , ) − Q† R† ∈ N2. Otherwise, the purely imaginary eigenvalue of multiplicity two splits into a pair of purely imaginary eigenvalues (with diﬀerent signatures) or a quadruplet of complex eigenvalues. For ǫ small enough, ( †, †) has distinct eigenvalues and thus belongs to N . Q R 1

3.3. Critical Generalized Eigenspaces of Hamiltonian Coupled Cell Networks

Let σ be an eigenvalue of µ and let E (σ) be the eigenspace of restricted to the invariant subspace E (µ) C2l. If theM eigenvaluesM are simple and under someM mild assumptions, we show A ⊗ there exists an isomorphism η between the eigenspaces EA(µ) and E (σ) (similar to [8, Theorem 2.7]). We begin with the ﬁrst theorem in the case of simple eigenvalues.M Theorem 3.9: Suppose ( , ) N . Let µ R be an eigenvalue of A, and let σ C be a simple Q R ∈ 1 ∈ ∈

14 eigenvalue of Then there exists an isomorphism Mµ

η : EA(µ) E (σ). (11) → M

Proof. If ( , ) N1, the eigenvalues are simple. Because Aµ is symmetric, for each eigenvalue µ, we haveQ aR complete∈ set of eigenvectors generating E (µ). Thus, for each eigenvector V A µ ∈ EA(µ), we deﬁne η(Vµ) = Vµ U where U is an eigenvector of the simple eigenvalue σ of µ. Therefore, η(V ) is automatically⊗ an element of the kernel of σ by Lemma 3.1. M µ M If ( , ) N2, there is one µ block that has an eigenvalue with algebraic multiplicity two. In theQ nextR theorem,∈ we treat thisM case separately to construct the isomorphism.

Theorem 3.10: Suppose that ( , ) N2. Let µ R be an eigenvalue of A and σ C be a non-simple eigenvalue of . WeQ haveR ∈ the two cases.∈ ∈ Mµ 2l (1) If σ = 0, then (σ) EA(µ) EA(µ) and there exists a basis of EA(µ) C such that GM ≃ ⊕ ⊗ 0 1 (0)= Ip M |GM 0 0 ! ⊗

where p = dim EA(µ). (2) If σ = iω, then (σ) EA(µ) EA(µ) EA(µ) EA(µ) and there exists a basis of E (µ) C2l suchG thatM ≃ ⊕ ⊕ ⊕ A ⊗ 0 ω 1 0 ω −0 0 1 (iω)=   Ip. (12) M |GM  0 0 0 ω  ⊗  −   0 0 ω 0      Proof. Suppose σ = 0. Let U be in the kernel of and U is a generalized eigenvec- 1 Mµ 2 tor; µU2 = U1. We know from Lemma 3.1 that for all (linearly independent) eigenvectors V ,...,M V of the eigenvalue µ of A, then V U is in the kernel of for j = 1,..., p. Now, 1 p j ⊗ 1 M (V j U2) = V j µU2 = V j U1 which implies that span V j U1, V j U2 (0) for each M ⊗ ⊗M ⊗ 2l { ⊗ ⊗ } ⊂ GM j = 1,..., p. Suppose that W1 W2 (0) EA(µ) C where W1 = α1V1 αpVp EA(µ). Then, there exists r N such⊗ that ∈ GM ⊂ ⊗ ⊕· · ·⊕ ∈ ∈ p r(W W ) = α V r W = 0 M 1 ⊗ 2 j j ⊗Mµ 2 Xj=1

r = = forces µW2 0 by linear independence of V1,..., Vp. But this means r 2 because the eigenvalueM 0 of has algebraic multiplicity 2 and so W span U , U . Thus, Mµ 2 ∈ { 1 2} p

(0) = span V j U1, V j U2 GM { ⊗ ⊗ } Mj=1 which shows the isomorphism. The proof proceeds similarly in the case of σ = iω. Let U be an eigenvector of iω and U a generalized eigenvector. The matrix restricted to 1 2 Mµ Im(U1), Re(U1), Im(U2), Re(U2) has the form given by the 4 4 matrix in (12). Again, let{ V ,..., V be a basis of E (µ}) and consider the vectors V ×Im(U ), V Re(U ), V 1 p A { j ⊗ 1 j ⊗ 1 j ⊗

15 (a) λ < 0 (b) λ = 0 (c) λ > 0

Figure 4.: Movement of the eigenvalues in the complex plane at a zero eigenvalue. (a) - (c) show the eigenvalues splitting from the imaginary axis as λ transitions from less than zero to greater than zero.

Im(U ), V Re(U ) for j = 1,..., p. Then, a similar computation as above guarantees that 2 j ⊗ 2 } p

(iω) = span V j Im(U1), V j Re(U1), V j Im(U2), V j Re(U2) (13) GM { ⊗ ⊗ ⊗ ⊗ } Mj=1 and by construction (12) holds.

Remark 3.11: The correspondance with the symmetric case as described in Golubitsky and Stewart [13] and Dellnitz et al [24] can be seen as follows. In the zero eigenvalue case, the decomposition into the sum of two copies of EA(µ) is analogous to the decomposition of the zero eigenspace as two absolutely irreducible representations of the symmetry group. There is no analog to the non-absolutely irreducible case. The 1 : 1 resonance case is analogous to the case described in Theorem 4.4(b) of [24] where the sum− of the generalized eigenspaces of the eigenvalues iω is the direct sum of four isomorphic absolutely irreducible representations. ± Remark 3.12: Note that there are no restrictions on the cell dimension for the zero eigenvalue case since the generalized eigenspace obtained from µ is two-dimensional and the cells must be of dimension at least 2. However, for purely imaginaryM eigenvalues, the cell dimension must be at least 4 in order to have the 1 : 1 resonance in . − Mµ The generic movement of eigenvalues for the two cases of Theorem 3.10 follow directly from the existing versal unfoldings for the 2 2and4 4 matrices, see for instance [24]. The unfolding of the zero eigenvalue is × ×

0 1 Ip λ 0 ! ⊗ which means that, generically, the eigenvalues move from the real axis to the imaginary axis. The eigenvalues split as shown in Figure 4. For the 1 : 1 resonance, the versal unfolding is − 0 ω 1 0 ω −0 0 1   I .  λ 0 0 ω  ⊗ p  −   0 λ ω 0      16 (a) λ < 0 (b) λ = 0 (c) λ > 0

Figure 5.: Movement of the eigenvalues in the complex plane ofa1: 1 resonance. (a) - (c) show the eigenvalues splitting from the imaginary axis as λ transitions− from less than zero to greater than zero. and so generically, the eigenvalues split from the imaginary axis to form a Krein quartet. See Figure 5.

4. Codimension One Bifurcations from Equilibrium

We now present some bifurcation results for one-parameter families of Hamiltonian regular homogeneous coupled cell systems. The ﬁrst one generalizes the Hamiltonian version of the Equiv- ariant Branching Lemma for symmetric systems [13] to the coupled cell case just as Theorem 6.3 of [8] does in the non-Hamiltonian case. Our second result is a version of the Hamiltonian Hopf bifurcation theorem. Recall that cells of a graph G are given by the vertex set V(G) where V(G) = n and we label the elements of V(G) from 1,..., n. Because we are assuming homogeneity,| | the state space of all cells is of the same dimension. The total phase space is P = (R2l)n and recall that we describe an element of P as x = (x1,..., xn) where x j is an element of cell j. Let ⊲⊳ be an equivalence relation on V(G). A polydiagonal subspace associated with ⊲⊳ is deﬁned as

∆ = x P x = x whenever i ⊲⊳ j, i, j 1,..., n . ⊲⊳ { ∈ | i j ∀ ∈{ }}

If ⊲⊳ is a so-called “balanced” equivalence relation then ∆⊲⊳ is flow-invariant for all admissible vector fields, see [26] for details. If ⊲⊳ is balanced, ∆⊲⊳ is called a synchrony subspace. Proposition 4.1: Suppose that H is the Hamiltonian function of a coupled Hamiltonian cell network with phase space P given by the digraph G having a symmetric adjacency matrix. If ∆ is a synchrony subspace, then ∆ is a symplectic subspace. In particular, H ∆ is a Hamiltonian | function for the vector field XH restricted to ∆. Proof. The Hamiltonian structure of our coupled cell network implies the existence of a symplectic form ω : P P R. That is, ω is bilinear, skew-symmetric and nondegenerate. There × → exists a quotient map π : P P⊲⊳, see [26] for details, such that P⊲⊳ and ∆⊲⊳ are bijectively related. Suppose that u, v, p, q→ P are such that π(u) = π(p) and π(v) = π(q). We claim that ω(u, v) = ω(p, q), that is, ω is constant∈ on the equivalence classes of ⊲⊳. Therefore, we can define ω : P P R as the restriction of ω to the equivalence classes and ω is still bilinear, skew- ⊲⊳ × ⊲⊳ → symmetric and nondegenerate. This means ω is well-defined and a symplectic form on ∆⊲⊳. We

17 now show the claim. The symplectic form on P is deﬁned as

n ω(x, y) = x jJky j Xj=1

= 0 Ik where Jk I 0 with Ik the k k identity matrix. Let i,..., r be the partition of indices − k × I I corresponding to ⊲⊳; that is, if i, ℓ s for some s 1,..., r then xi = xℓ. In particular, we write ∈ I ∈{ th } π(x) = (x1,..., xr) where x j is a representative of the j equivalence class. Consider u, v, p, q deﬁned as above, then it is straightforward to verify that

r r = = = ω(u, v) niui Jkvi ni pi Jkqi ω(p, q) Xi=1 Xi=1 where n = is the number of elements in . We deﬁne H = H ∆, the Hamiltonian function i |Ii| Ii | for XH = XH ∆ using the symplectic form ω(v, XH(u)) = dH(u) v for all u ∆ and v Tu∆. We write the| vector ﬁeld X asx ˙ = F(x, λ) and assume that F· (0, 0) = 0.∈ Let ∆ ∈= x x = H sync { | i x j for all i, j be the total synchrony subspace. Suppose that dF(0, 0) has a non-semisimple zero eigenvalue with} multiplicity 2. We want to consider bifurcations which break synchrony and so we assume that, K, the generalized eigenspace of 0 is such that K ∆sync = 0 . This means we can assume that (0, λ) to be a trivial equilibrium for λ close to 0, see∩ [8] for details.{ }

Theorem 4.2: Let ∆ be a synchrony subspace such that dim(∆ K) = 2. Then, ∆ K is ∩ ∩ symplectic and the vector ﬁeld XH restricted to ∆ K is Hamiltonian. In particular, being a one-degree of freedom Hamiltonian, the bifurcation∩ of equilibria can be obtained in a direct way, see [27].

Proof. The generalized eigenspace of the eigenvalue of a Hamiltonian matrix is a symplectic space and the intersection of symplectic subspaces is also a symplectic subspace. The statement about XH follows as described at the end of the proof of Proposition 4.1.

Example 4.3: Consider the network Γ2. Synchrony subspaces ∆1 and ∆2 are illustrated in Fig- ure 6.

∆1 ∆2

Figure 6.: Synchrony subspaces ∆1 and ∆2 of network Γ2.

k In coordinates, we have ∆1 = (a, b, b, a, b, b, a, a) a, b R and ∆2 = (a, a, b, b, a, a, b, b) k { | ∈ } { | a, b R . Suppose that 1 has a zero eigenvalue of multiplicity 2 with generalized eigenspace spanned∈ } by U and U .M Then,− K = span U V , U V , U V , U V where V and V 1 2 { 1 ⊗ 1 2 ⊗ 1 1 ⊗ 2 2 ⊗ 2} 1 2 are the eigenvectors of A2, see Example 2.10. Changing basis, one can see that V1 Ui = (U , U , U , U , U , U , U , U )T ∆ and V U = (U , U , U , U , U , U , U , ⊗U )T i − i − i i − i − i i i ∈ 1 2 ⊗ i i i − i − i i i − i − i ∈

18 ∆ for i = 1, 2. Thus, in both cases dim(K ∆ ) = 2 for j = 1, 2 and Theorem 4.2 applies, that 2 ∩ j is, we have bifurcation in each subspace. Note that the Z2 symmetry of the network fixes ∆sync. We can obtain a result analogous to Theorem 4.2 in the case of purely imaginary eigenvalues and so obtain a version of the Hamiltonian Hopf theorem for Hamiltonian coupled cell networks. The Hamiltonian Hopf theorem was established in complete generality by van der Meer [28]. For a Hamiltonian system in R4 with a 1 : 1 resonance, the periodic solutions can be obtained − 2 by studying the Hamiltonian system given by H(x, y) = S + N +µP+aP where S = x1y2 x2y1, = 1 2 + 2 = 1 2 + 2 ffi − N 2 (x1 x2), P 2 (y1 y2). In particular, the sign of the coe cient a determines the bifurcation scenario: if a > 0 the periodic solutions collapse to the origin as the parameter µ 0− and for a < 0 two families of distinct periodic solutions intersecting at the origin pull away→ from the origin as µ becomes negative. Suppose that dF(0, 0) has iω eigenvalues in 1 : 1 resonance and E is the generalized eigenspace. ± − Theorem 4.4: Consider a 1-parameter family of Hamiltonian network of coupled Hamiltonian cells with an equilibrium at the origin. Suppose the linearization at the origin has a 1 : 1 resonance with eigenspace E. Let ∆ be a synchrony subspace such that dim(∆ E) = 4. Let− ∩ a∆ be the coefficient of the normal form of H0 on ∆ E. Then, provided a∆ , 0, the same two scenarios occur as for the ordinary Hamiltonian Hopf∩ bifurcation theorem.

Proof. The subspace ∆ E is a ﬂow-invariant symplectic subspace and the vector ﬁeld XH ∆ E is four-dimensional and∩ the linearization at the origin has a 1 : 1 resonance. Therefore,| the∩ − Hamiltonian function H ∆ E is a function of S , N and P as described in the paragraph above and so the system undergoes| ∩ a Hamiltonian Hopf bifurcation in ∆.

Example 4.5: We return to Example 4.3 and let E = (iω) be given by (13) with V1 and GM V2 as above. Suppose that U1, U2 are the eigenvector and generalized eigenvector of iω, again changing basis, one can verify that

span V Im(U ), V Re(U ), V Im(U ), V Re(U ) ∆ { 1 ⊗ 1 1 ⊗ 1 1 ⊗ 2 1 ⊗ 2 } ⊂ 1 and

span V Im(U ), V Re(U ), V Im(U ), V Re(U ) ∆ . { 2 ⊗ 1 2 ⊗ 1 2 ⊗ 2 2 ⊗ 2 } ⊂ 2 Therefore, Theorem 4.4 applies and in each synchrony subspace, guarantees the existence of families of periodic solutions depending on the sign of the coeﬃcient a as described in the paragraph preceding Theorem 4.4.

5. Discussion

As is shown in [6], each coupled cell system has a set of admissible vectors fields and this set is determined by the directed graph associated with the system. In this paper, we prove that only bidirectionally coupled systems may admit Hamiltonian vector fields. Thus, we have specified a connection between a class of dynamical systems and their topological representations in the coupled cell formalism. This discovery, as in the case of gradient coupled cell systems [21], suggests that other classes of dynamical systems and their topological representations may also be connected in ways that are yet to be discovered. Furthermore, our results on the topological criteria resolve issues presented in [14]. While the previous work provided mathematical proof that Hamiltonian unidirectionally coupled ring of

19 Hamiltonian oscillators cannot remain Hamiltonian, it did not provide particular insights into constructing larger systems. With our topological criteria, we have the means to systemically construct larger coupled Hamiltonian systems from existing Hamiltonian subsystems. This tech- nique is particularly useful towards applications that seek to investigate complex networks that are also Hamiltonian. We study also the sets of matrices leading to codimension one bifurcations and characterize the generalized eigenspaces at criticality in terms of the eigenspaces of the adjacency matrix of the network. Finally, we show two bifurcation results. The first one is the zero eigenvalue case which leads to the analysis of a one-degree of freedom Hamiltonian in each synchrony subspace intersecting the generalized kernel, for which the bifurcating branches can be easily determined. The second is the purely imaginary eigenvalue case which leads to Hamiltonian Hopf bifurcation in the synchrony subspaces intersecting the generalized eigenspace of the purely imaginary eigenvalue. One interesting direction we plan to pursue is to connect our results to the coupled cell network formalism developed by Rink and Sanders, see for instance [5]. Here, coupled cell systems are described as vector fields corresponding to a “fundamental network”, and each of these fundamental network vector field is equivariant with respect to the action of a monoid of mappings which encodes the coupling structure. Our goal would be to establish sufficient conditions on a Hamiltonian function to be invariant with respect to the monoid action so that each cell is also Hamiltonian. Therefore, from theory developed in [5] and Rink and Sanders (normal form), local equivariant bifurcation theory can be used systematically for Hamiltonian coupled cell systems.

Acknowledgements

PLB acknowledges the support of NSERC Canada in the form of a Discovery Grant. A.P. and B.S.C. were supported by the Complex Dynamics and Systems Program of the Army Research Oﬃce, supervised by Dr. Samuel Stanton, under grant W911NF-07-R-003-4. A.P. wishes to thank Prof. Melvin Leok for very fruitful discussions regarding Hamiltonian mechanics and its applications to optimal control of rigid bodies. We would like to thank the referees for very relevant comments and suggestions which helped improve the manuscript in signiﬁcant ways.

References

[1] Golubitsky M, Stewart I. Nonlinear dynamics of networks: the groupoid formalism. Bull Amer Math Soc. 2006;43:305–364. [2] Belykh I, Hasler M, Lauret M, Nijmeijer H. Synchronization and graph topology. Int Journal of Bifurcation and Chaos. 2005;15:34233433. [3] Aguiar M, Ashwin P, Dias A, Field M. Dynamics of coupled cell systems: synchrony, heteroclinic cycles and inflation. J Nonlinear Science. 2011;21:271–323. [4] DeVille L, Lerman E. Modular dynamical systems on networks. accepted in J European Math Soc. 2013;. [5] Rink B, Sanders J. Coupled cell networks and their hidden symmetries. SIAM J Math Anal. 2013; 46:1577–1609. [6] Golubitsky M, Stewart I, Török A. Patterns of synchrony in coupled cell networks with multiple arrows. SIAM Journal on Applied Dynamical Systems. 2005;4(1):78–100. [7] Elmhirst T, M G. Nilpotent hopf bifurcations in coupled cell systems. SIAM J Appl Dynam Sys. 2006;5:205–251. [8] Golubitsky M, Lauterbach R. Bifurcations from synchrony in homogeneous networks: linear theory. SIAM Journal on Applied Dynamical Systems. 2009;8(1):40–75.

20 [9] Antoneli F, Dias R APSand Paiva. Hopf bifurcation in coupled cell networks with interior symmetries. SIAM J on Applied Dynamical Systems. 2008;7:220–248. [10] Aguiar M, Dias A. Regular synchrony lattices for product coupled cell networks. Chaos. 2015; 25:013108. [11] Rink B, Sanders J. Amplified hopf bifurcations in feed-forward networks. SIAM J Appl Dyn Syst. 2013;12:1135–1157. [12] Meiss JD. Differential dynamical systems. Vol. 14. SIAM; 2007. [13] Golubitsky M, Stewart I. Generic bifurcation of hamiltonian systems with symmetry. Physica D. 1987;24:391–405. [14] Buono PL, Chan BS, Palacios A, In V. Dn symmetric Hamiltonian system: A network of coupled gyroscopes as a case study. Physica D. 2015;290:8–23. [15] Matus-Vargas A, Gonzàlez-HernandezHG, Chan BS, PalaciosA, BuonoPL, In V,Naik S, PhippsA, LonghiniP. Dynamics, bifurcations and normal forms in arrays of magnetostrictive energy harvesters with all-to-all coupling. Intl J Bif and Chaos. 2014;25:1550026. [16] Meyer K, Hall G, Offin D. Introduction to hamiltonian dynamical systems and the n-body problem. Springer-Verlag New York; 2009. [17] Montaldi J, Roberts M. Relative equilibria of molecules. J Nonlinear Sc. 1999;9:53–88. [18] Fermi E, Pasta J, Ulam S. Studies of nonlinear problems i. Los Alamos report LA-1940. 1955;. [19] Rink B. Proof of nishida’s conjecture on anharmonic lattices. Commun Math Phys. 2006;261:613– 627. [20] Sreenath N, Oh Y, Krishnaprasad P, Marsden J. The dynamics of coupled planar rigid bodies. part i: Reduction, equilibria and stability. Dyn Stab Syst. 1988;3:25–49. [21] Manoel M, Roberts M. Gradient systems on coupled cell networks. Nonlinearity. 2015;28:3487 – 3509. [22] Leite MC, Golubitsky M. Homogeneous three-cell networks. Nonlinearity. 2006;19:2313–2363. [23] Brouwer AE, Haemers WH. Spectra of graphs. Universitext; Springer, New York; 2012. [24] Dellnitz M, Melbourne I, Marsden JE. Generic bifurcation of hamiltonian vector fields with symmetry. Nonlinearity. 1992;5(4):979. [25] Hanßmann. Local and semi-local bifurcations in hamiltonian dynamical systems: Results and examples (lecture notes in mathematics vol 1893). Springer; 2007. [26] Stewart I, Golubitsky M, Pivato M. Symmetry groupoids and patterns of synchrony in coupled cell networks. SIAM J Appl Dynam Sys. 2003;2:609–646. [27] Buono PL, Laurent-Polz F, Montaldi J. Symmetric Hamiltonian bifurcations. In: Geometric mechanics and symmetry. Vol. 306 of London Math. Soc. Lecture Note Ser.; Cambridge Univ. Press, Cambridge; 2005. p. 357–402. [28] Van Der Meer JC. The hamiltonian hopf bifurcation. Springer; 1985.