To appear in Dynamical Systems: An International Journal Vol. 00, No. 00, Month 20XX, 1–22 Topology and Bifurcations in Hamiltonian Coupled Cell Systems 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 sys- tems 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 bifurca- tions 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 net- work 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, bi- furcations 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 en- ergy 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 gener- alization 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 func- tion 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 de- pending 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 neces- sary 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 e2 e3 e1 e4 v1 v2 v3 e2 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 .