Periodic dynamics of coupled cell networks I: rigid patterns of synchrony and phase relations

It has recently been proved by Golubitsky and coworkers that in any network of coupled dynamical systems, the possible ‘rigid’ patterns of synchrony of hyperbolic equilibria are determined by purely combinatorial properties of the network, known as ‘balanced equivalence relations’. A pattern is ‘rigid’ if it persists under small ‘admissible’ perturbations of the differential equation — ones that respect the network structure. We discuss a natural generalisation of these ideas to time-periodic states, and motivate two basic conjectures, the Rigid Synchrony Conjecture and the Rigid Phase Conjecture. These conjectures state that for rigid hyperbolic time-periodic patterns, cells with synchronous dynamics must have synchronous input cells, and cells with phase-related dynamics must have input cells that have the same phase relations. We provide evidence supporting the two conjectures, by proving them for a special class of periodic orbits, which we call ‘tame’, under strong assumptions on the network architecture and the symmetries of the periodic state. The discussion takes place in the formal setting of coupled cell networks. We prove that rigid patterns of synchrony are balanced, together with the analogous result for rigid patterns of phase relations. The assumption on the network architecture simplifies the geometry of admissible vector fields, while tameness rules out patterns with non-trivial local or multilocal symmetry. The main idea is to perturb an admissible vector field in a way that retains sufficient control over the associated perturbed periodic orbit. We present two techniques for constructing these perturbations, both using a general theorem on groupoid-symmetrisation of vector fields, which has independent interest. In particular we introduce a method of ‘patching’ that makes local changes to an admissible vector field. Having established these results for all-to-all coupled networks and tame periodic orbits we prove more general versions that require these assumptions only on a suitable quotient network. These conditions are weaker and encompass a larger class of networks and periodic orbits. We give an example to show that rigidity cannot be relaxed to hyperbolicity. We also prove, without any technical assumptions, that rigidly synchronous or phase-related cells must be input-isomorphic, a necessary precondition for the two conjectures to hold.

[1]  J. Zukas Introduction to the Modern Theory of Dynamical Systems , 1998 .

[2]  John W. Aldis A Polynomial Time Algorithm to Determine Maximal Balanced Equivalence Relations , 2008, Int. J. Bifurc. Chaos.

[3]  Ian Stewart,et al.  Patterns of Synchrony in Coupled Cell Networks with Multiple Arrows , 2005, SIAM J. Appl. Dyn. Syst..

[4]  M. Golubitsky,et al.  Models of central pattern generators for quadruped locomotion II. Secondary gaits , 2001, Journal of mathematical biology.

[5]  Lambros Lambrou,et al.  Combinatorial Dynamics , 2004 .

[6]  C. Desoer,et al.  Global inverse function theorem , 1972 .

[7]  Martin Golubitsky,et al.  Stability Computations for Nilpotent Hopf bifurcations in Coupled Cell Systems , 2007, Int. J. Bifurc. Chaos.

[8]  Ian Stewart,et al.  Enumeration of Homogeneous Coupled Cell Networks , 2005, Int. J. Bifurc. Chaos.

[9]  Martin Golubitsky,et al.  Homogeneous three-cell networks , 2006 .

[10]  Martin Golubitsky,et al.  Nilpotent Hopf Bifurcations in Coupled Cell Systems , 2006, SIAM J. Appl. Dyn. Syst..

[11]  Ian Stewart,et al.  Periodic dynamics of coupled cell networks II: cyclic symmetry , 2008 .

[12]  H. Brandt Über eine Verallgemeinerung des Gruppenbegriffes , 1927 .

[13]  Ian Stewart,et al.  Some Curious Phenomena in Coupled Cell Networks , 2004, J. Nonlinear Sci..

[14]  W. Gordon On the Diffeomorphisms of Euclidean Space , 1972 .

[15]  Peter Ashwin,et al.  THE SYMMETRY PERSPECTIVE: FROM EQUILIBRIUM TO CHAOS IN PHASE SPACE AND PHYSICAL SPACE (Progress in Mathematics 200) , 2003 .

[16]  Ian Stewart,et al.  Symmetry Groupoids and Admissible Vector Fields for Coupled Cell Networks , 2004 .

[17]  Eric Shea-Brown,et al.  Winding Numbers and Average Frequencies in Phase Oscillator Networks , 2006, J. Nonlinear Sci..

[18]  Ian Stewart,et al.  Linear equivalence and ODE-equivalence for coupled cell networks , 2005 .

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

[20]  M. Golubitsky,et al.  Singularities and groups in bifurcation theory , 1985 .

[21]  Ralph Abraham,et al.  Foundations Of Mechanics , 2019 .

[22]  K. Josić,et al.  Network architecture and spatio-temporally symmetric dynamics , 2006 .

[23]  M. Golubitsky,et al.  Models of central pattern generators for quadruped locomotion I. Primary gaits , 2001, Journal of mathematical biology.

[24]  M. Golubitsky,et al.  Nonlinear dynamics of networks: the groupoid formalism , 2006 .

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

[26]  P. J. Higgins Notes on categories and groupoids , 1971 .

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

[28]  J. Frank Adams,et al.  Lectures on Lie groups , 1969 .