Heteroclinic cycles in rings of coupled cells

Symmetry is used to investigate the existence and stability of heteroclinic cycles involving steady-state and periodic solutions in coupled cell systems with Dn-symmetry. Using the lattice of isotropy subgroups, we study the normal form equations restricted to invariant fixed-point subspaces and prove that it is possible for the normal form equations to have robust, asymptotically stable, heteroclinic cycles connecting periodic solutions with steady states and periodic solutions with periodic solutions. A center manifold reduction from the ring of cells to the normal form equations is then performed. Using this reduction we find parameter values of the cell system where asymptotically stable cycles exist. Simulations of the cycles show trajectories visiting steady states and periodic solutions and reveal interesting spatio-temporal patterns in the dynamics of individual cells. We discuss how these patterns are forced by normal form symmetries.