P.L. Buono, M. Golubitsky and A. Palacios.
Heteroclinic Cycles in Rings of Coupled Cells
Physica D 143 (2000) 74-108
Symmetry is used to investigate the existence and stability of
heteroclinic cycles involving steady-state and periodic solutions in
coupled cell systems with D_n-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.