Remember Me
Or use your Academic/Social account:


Or use your Academic/Social account:


You have just completed your registration at OpenAire.

Before you can login to the site, you will need to activate your account. An e-mail will be sent to you with the proper instructions.


Please note that this site is currently undergoing Beta testing.
Any new content you create is not guaranteed to be present to the final version of the site upon release.

Thank you for your patience,
OpenAire Dev Team.

Close This Message


Verify Password:
Verify E-mail:
*All Fields Are Required.
Please Verify You Are Human:
fbtwitterlinkedinvimeoflicker grey 14rssslideshare1
Stewart, Ian; Golubitsky, Martin; Pivato, Marcus (2003)
Publisher: Society for Industrial and Applied Mathematics
Languages: English
Types: Article
Subjects: QA

Classified by OpenAIRE into

arxiv: Quantitative Biology::Cell Behavior
A coupled cell system is a network of dynamical systems, or “cells,” coupled together. Such systems\ud can be represented schematically by a directed graph whose nodes correspond to cells and whose\ud edges represent couplings. A symmetry of a coupled cell system is a permutation of the cells that\ud preserves all internal dynamics and all couplings. Symmetry can lead to patterns of synchronized\ud cells, rotating waves, multirhythms, and synchronized chaos. We ask whether symmetry is the only\ud mechanism that can create such states in a coupled cell system and show that it is not.\ud The key idea is to replace the symmetry group by the symmetry groupoid, which encodes information\ud about the input sets of cells. (The input set of a cell consists of that cell and all cells\ud connected to that cell.) The admissible vector fields for a given graph—the dynamical systems with\ud the corresponding internal dynamics and couplings—are precisely those that are equivariant under\ud the symmetry groupoid. A pattern of synchrony is “robust” if it arises for all admissible vector\ud fields. The first main result shows that robust patterns of synchrony (invariance of “polydiagonal”\ud subspaces under all admissible vector fields) are equivalent to the combinatorial condition that an\ud equivalence relation on cells is “balanced.” The second main result shows that admissible vector\ud fields restricted to polydiagonal subspaces are themselves admissible vector fields for a new coupled\ud cell network, the “quotient network.” The existence of quotient networks has surprising implications\ud for synchronous dynamics in coupled cell systems.
  • The results below are discovered through our pilot algorithms. Let us know how we are doing!

    • [1] H. Brandt, U¨ber eine Verallgemeinerung des Gruppenbegriffes, Math. Ann., 96 (1927), pp. 360-366.
    • [2] S. Boccaletti, L. M. Pecora, and A. Pelaez, Unifying framework for synchronization of coupled dynamical systems, Phys. Rev. E (3), 63 (2001), 066219.
    • [3] R. Brown, From groups to groupoids: A brief survey, Bull. London Math. Soc., 19 (1987), pp. 113-134.
    • [4] P. L. Buono and M. Golubitsky, Models of central pattern generators for quadruped locomotion I. Primary gaits, J. Math. Biol., 42 (2001), pp. 291-326.
    • [5] A. Dias and I. Stewart, Symmetry groupoids and admissible vector fields for coupled cell networks, submitted.
    • [6] M. Golubitsky, M. Nicol, and I. Stewart, Some curious phenomena in coupled cell networks, submitted.
    • [7] M. Golubitsky and I. Stewart, The Symmetry Perspective: From Equilibrium to Chaos in Phase Space and Physical Space, Progr. Math. 200, Birkh¨auser Verlag, Basel, 2002.
    • [8] M. Golubitsky and I. Stewart, Patterns of oscillation in coupled cell systems, in Geometry, Dynamics, and Mechanics: 60th Birthday Volume for J. E. Marsden, P. Holmes, P. Newton, and A. Weinstein, eds., Springer-Verlag, New York, 2002, pp. 243-286.
  • No related research data.
  • No similar publications.

Share - Bookmark

Funded by projects

Cite this article