Physica D 143 (2000) 74–108 Heteroclinic cycles in rings of coupled cells Pietro-Luciano Buono a , Martin Golubitsky b,∗ , Antonio Palacios c a Mathematics Institute, University of Warwick, Coventry CV4 7AL, UK b Department of Mathematics, University of Houston, Houston, TX 77204-3476, USA c Department of Mathematics, San Diego State University, San Diego, CA 92182-7720, USA In the memory of John David Crawford Abstract 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. © 2000 Elsevier Science B.V. All rights reserved. Keywords: Heteroclinic cycles; Coupled cell systems; Spatio-temporal patterns; Equivariant bifurcation theory; D n -symmetry 1. Introduction Coupled systems of differential equations or cells are often used as models of physical systems. For example, they are used by Hadley et al. [15] and Aronson et al. [2] to model arrays of Josephson junctions and by Kopell and Ermentrout [16,17] and Rand et al. [23] to model central pattern generators (CPGs) in biological systems. Recently, Collins and Stewart [4–6] and Golubitsky et al. [12] have shown that many phase relations observed in animal gaits can be modeled by coupled cell systems. In these works the symmetry of the cell network is important in determining the patterns of oscillation that the system can support. See the works of Dionne et al. [7,8], Golubitsky and Stewart [11], and the related work of Lamb and Melbourne [20]. In this paper, we discuss the existence of heteroclinic cycles in coupled cell systems. Such cycles model inter- mittency and are known to occur robustly and asymptotically stably in systems with symmetry (see [9,10,14]). Armbruster et al. [1] and Melbourne et al. [22] show that heteroclinic cycles can occur stably in systems with O(2)-symmetry and, in previous numerical work [3], we show that these cycles are also found stably in systems ∗ Corresponding author. E-mail addresses: buono@maths.warwick.ac.uk (P.-L. Buono), mg@uh.edu (M. Golubitsky), palacios@euler.sdsu.edu (A. Palacios) 0167-2789/00/$ – see front matter © 2000 Elsevier Science B.V. All rights reserved. PII:S0167-2789(00)00097-X