Quasi-periodic states in coupled rings of cells

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 Z p × Z q in the unidirectional case and D p × 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.