Quasi-periodic states in coupled rings of cells

Abstract

We study some dynamical features of certain coupled cell networks that consist of two (unidirectional or bidirectional) rings of cells coupled through a ‘buffer’ cell. Depending on how the rings and the buffer cell are coupled, the full network may have a non-trivial group of symmetries or a non-trivial group of ‘interior’ symmetries. This group is ${\mathbf{Z}}_p\times {\mathbf{Z}}_q$ in the unidirectional case and ${\mathbf{D}}_p\times {\mathbf{D}}_q$ in the bidirectional case. We are interested in finding quasi-periodic motion in these networks, motivated by an example presented by Golubitsky, Nicol and Stewart (Some curious phenomena in coupled cell systems, J Nonlinear Sci 2004;14(2):207–36). In the examples considered here, we obtain quasi-periodic states through a sequence of Hopf bifurcations. Interestingly, we observe relaxation oscillation phenomena appearing further away from the last Hopf bifurcation point. We use XPPAUT and MATLAB to compute numerically the relevant states.

Introduction

Networks of dynamical systems arise in many areas of science such as biology, economics, physics and ecology. See for example Strogatz [25], Wang [27], Stewart [23], Lieberman et al. [21], Boccaletti et al. [9], Alon [3], Albert et al. [2], Watts et al. [28], Milo et al. [22], and references therein. Several general formal theories have emerged and aiming to relate the dynamics of the networks and the network structure. We follow here the theory of coupled cell networks developed in Stewart et al. [24] and Golubitsky et al. [19]. For a survey see Golubitsky and Stewart [16]. In this theory, networks of dynamical systems are idealized through coupled cell networks – directed graphs where the nodes represent the individual systems and the edges the couplings between cells. The idea is that the graph abstracts important properties of the systems. Certain dynamical phenomena such as synchronization, phase-locking, quasi-periodic states, synchronized chaos, recurrent behavior, etc., are common in networks of dynamical systems and the architecture of the network (graph) plays an important role in the appearance of such phenomena. See for example Aguiar et al. [1], Ashwin et al. [7], [8] and Golubitsky et al. [13].

In this paper we consider networks consisting of two (unidirectional or bidirectional) (see Fig. 1) rings coupled through a ‘buffer’ cell, exhibiting two types of architecture: ‘exact symmetry’ and ‘interior symmetry’ (see Fig. 2). Specifically, we study periodic and quasi-periodic solutions arising through Hopf bifurcations in the coupled cell systems associated to the networks with the above architecture.

Recall that a symmetry of a network represented by a directed graph is a permutation of the nodes (cells) that preserves the edges (arrows). The notion of interior symmetry, recently introduced by Golubitsky et al. [14], generalizes the usual definition of symmetry. The schematic diagram on the right of Fig. 2 is an example of a network with interior symmetry. Observe that if we delete the arrows directed from the rings to the ‘buffer’ cell then we obtain the schematic diagram on the left of Fig. 2 which is an example of a network with exact symmetry (assuming that the arrows directed from the buffer cell to the rings respect the symmetry of the rings).

We consider four networks that represent the abstract framework presented above. The first two examples are networks of two unidirectional rings, in which the first ring consists of three cells and the second ring of five cells, see Fig. 3. The network in Fig. 3(a) has ${\mathbf{Z}}_3\times {\mathbf{Z}}_5$ exact symmetry and the network in Fig. 3(b) has ${\mathbf{Z}}_3\times {\mathbf{Z}}_5$ interior symmetry. The remaining networks are similar to these where now the cells in the two rings are coupled bidirectionally, see Fig. 4. These latter networks have ${\mathbf{D}}_3\times {\mathbf{D}}_5$ exact symmetry (Fig. 4(a)) and ${\mathbf{D}}_3\times {\mathbf{D}}_5$ interior symmetry (Fig. 4(b)), respectively. Observe that network in Fig. 3(a) is the result of ignoring the couplings from the $x_i$ and the $y_j$ cells to the buffer cell in the network in Fig. 3(b), similarly for the networks in Figs. 4(a) and (b).

It is known that symmetric ODE’s exhibit robust patterns of oscillations which possess spatio-temporal symmetries. In Golubitsky et al. [17] it is shown how to determine and classify all permitted types of spatio-temporal symmetry that a periodic solution of a system of ODE’s, with symmetry group $\Gamma$, can robustly support in terms of pairs of subgroups of $\Gamma$. This is called $H/K$-Theorem.

The first question that comes to mind is: Which of these periodic solutions can be obtained by Hopf bifurcation? The answer to this question is that not all types of spatio-temporal symmetry will arise in a Hopf bifurcation, however if one considers secondary (tertiary, etc.) Hopf bifurcations then it may be possible that the answer is ‘all solutions’. For example, in [17] there are several examples of networks displaying periodic solutions exhibiting spatio-temporal symmetries obtained through a secondary Hopf bifurcation, but would not appear in a primary Hopf bifurcation. Recently, Filipsky [12] has addressed this question in the context of equivariant dynamical systems with finite abelian symmetry and gives necessary and sufficient conditions in order to obtain a periodic solution with prescribed spatio-temporal symmetry by a primary Hopf bifurcation.

Motivated by this difficult question, we have looked at some simple examples of networks with symmetry which could indicate a possible approach. In the course of our investigation we came across several other questions which are interesting by themselves and maybe relevant to the above problem. In this paper we further explore some of our findings that were already reported in Antoneli, Dias and Pinto [5], [6].

In part, our choice of examples was motivated by some of the phenomena presented by Golubitsky et al. [13], where a quasi-periodic motion was observed in a numerical simulation of a coupled cell network of the same type as considered here with interior symmetry. By further exploring their example, numerically, we propose a bifurcation scenario where this quasi-periodic behavior is obtained through a sequence of Hopf bifurcations. The next step was to consider other network examples with similar structure in order to inspect if the same behavior could occur and through a similar mechanism.

We conclude that the kind of network architecture studied here favors the appearance of quasi-periodic states. Moreover, several interesting questions and conjectures have arisen:

• When does a secondary Hopf bifurcation produces a quasi-periodic motion and when does it produces a periodic solution in the network?

• The presence of symmetry (exact or interior) constrains the dynamical behavior of the cells in each of the rings, but the structure of the vector field seems to select the periodicity or quasi-periodicity of the global solution of the network. More specifically, we believe that resonance is strongly dependant on the choice of the vector field.

• Another interesting phenomena observed is the appearance of relaxation oscillations, after a sequence of Hopf bifurcations, that seems to be explained by the network structure. This structure imposes a symmetry group that is a direct product of two (interior) symmetry groups, each of which is a symmetry group of a distinguished sub-network. Moreover, it is surprising to observe this type of behavior in these coupled systems since they are not a priori multiple time scales systems, where these solutions frequently occur (Krupa and Szmolyan [20]). Can this relaxation oscillation phenomena be explained in the context of fast–slow systems through a canard explosion?

• In these questions, the type of symmetry, either exact or interior, does not seem to affect in an essential way the answer to these questions.