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 syste ms with D-n-symmetry. Using the lattice of isotropy subgroups, we study the normal form equations restricted to invariant fixed-point subspaces and pr ove that it is possible fur the normal form equations to have robust, asymp totically stable, heteroclinic cycles connecting periodic solutions with st eady states and periodic solutions with periodic solutions. A center manifo ld reduction from the ring of cells to the normal form equations is then pe rformed. Using this reduction we find parameter values of the cell system w here asymptotically stable cycles exist. Simulations of the cycles show tra jectories visiting steady states and periodic solutions and reveal interest ing spatio-temporal patterns in the dynamics of individual cells. We discus s how these patterns are forced by normal form symmetries. (C) 2000 Elsevie r Science B.V. All rights reserved.