Symmetry and Synchrony in Coupled Cell Networks 3 Exotic Patterns

This paper continues the study of patterns of synchrony (equivalently, balanced colorings or flow-invariant subspaces) in symmetric coupled cell networks, and their relation to fixed-point spaces of subgroups of the symmetry group. Our aim is to provide a group-theoretic explanation of the "exotic" balanced coloring previously discussed in Part 2. Here we show that the pattern can be obtained as a projection into two dimensions of a fixed-point pattern in a three-dimensional lattice. We prove a general theorem giving sufficient conditions for such a construction to lead to a balanced coloring, for an arbitrary direct product of group networks.

[1]  W. T. Tutte.  Graph Theory , 1984 .

[2]  T. Hahn.  International tables for crystallography , 2002 .

[3]  Brian A. Davey,et al.  An Introduction to Lattices and Order , 1989 .

[4]  Robin J. Wilson.  Introduction to Graph Theory , 1974 .

[5]  Ian Stewart,et al.  The lattice of balanced equivalence relations of a coupled cell network , 2007, Mathematical Proceedings of the Cambridge Philosophical Society.

[6]  D. West.  Introduction to Graph Theory , 1995 .

[7]  M. Golubitsky,et al.  The Symmetry Perspective , 2002 .

[8]  Fernando Antoneli,et al.  Patterns of synchrony in lattice dynamical systems , 2005 .

[9]  M. Golubitsky,et al.  The Symmetry Perspective: From Equilibrium to Chaos in Phase Space and Physical Space , 2002 .

[10]  Marcus Pivato,et al.  Symmetry Groupoids and Patterns of Synchrony in Coupled Cell Networks , 2003, SIAM J. Appl. Dyn. Syst..

[11]  Fernando Antoneli,et al.  Symmetry and Synchrony in Coupled Cell Networks 1: Fixed-Point Spaces , 2006, Int. J. Bifurc. Chaos.

[12]  Martin Golubitsky,et al.  Two-colour patterns of synchrony in lattice dynamical systems , 2005 .

[13]  István Hargittai.  Quasicrystals, networks, and molecules of fivefold symmetry , 1990 .

[14]  M. Hall.  The Theory Of Groups , 1959 .

[15]  P. M. Neumann,et al.  Groups and Geometry , 1994 .

[16]  Peter Kramer,et al.  PLANAR PATTERNS WITH FIVEFOLD SYMMETRY AS SECTIONS OF PERIODIC STRUCTURES IN 4-SPACE , 1990 .

[17]  Fernando Antoneli,et al.  Symmetry and Synchrony in Coupled Cell Networks 2: Group Networks , 2007, Int. J. Bifurc. Chaos.