Skip to main content
Article thumbnail
Location of Repository

Patterns of synchrony in coupled cell networks with multiple arrows

By Martin Golubitsky, Ian Stewart and Andrei Torok

Abstract

A coupled cell system is a network of dynamical systems, or “cells,” coupled together. The architecture\ud of a coupled cell network is a graph that indicates how cells are coupled and which cells are\ud equivalent. Stewart, Golubitsky, and Pivato presented a framework for coupled cell systems that\ud permits a classification of robust synchrony in terms of network architecture. They also studied\ud the existence of other robust dynamical patterns using a concept of quotient network. There are\ud two difficulties with their approach. First, there are examples of networks with robust patterns of\ud synchrony that are not included in their class of networks; and second, vector fields on the quotient\ud do not in general lift to vector fields on the original network, thus complicating genericity arguments.\ud We enlarge the class of coupled systems under consideration by allowing two cells to be coupled in\ud more than one way, and we show that this approach resolves both difficulties. The theory that we\ud develop, the “multiarrow formalism,” parallels that of Stewart, Golubitsky, and Pivato. In addition,\ud we prove that the pattern of synchrony generated by a hyperbolic equilibrium is rigid (the pattern\ud does not change under small admissible perturbations) if and only if the pattern corresponds to\ud a balanced equivalence relation. Finally, we use quotient networks to discuss Hopf bifurcation in\ud homogeneous cell systems with two-color balanced equivalence relations

Topics: QA
Publisher: Society for Industrial and Applied Mathematics
Year: 2005
OAI identifier: oai:wrap.warwick.ac.uk:183

Suggested articles

Citations

  1. (1992). A Long memory Property of Stock Market Returns and a New Model. Discussion paper 92-21, doi
  2. (1980). An Introduction to Long-Memory Time Series Models and Fractional Differencing. doi
  3. (1994). Asymptotic Theory for Long-Memory Time Series. doi
  4. (1993). Bias in an Estimator of the Fractional Difference Parameter. doi
  5. (1990). DF i DF i DF DF k DF / k s t k DF / k t doi
  6. (1989). Efficient Parameter Estimation for Self-Similar Processes. doi
  7. (1994). Estimation of the Memory Parameter for Nonstationary or Noninvertible Fractionally Integrated Processes. doi
  8. (1968). Fractional Brownian Motions, Fractional Noises and Applications. doi
  9. (1981). Fractional Differencing. doi
  10. (1986). Large-Sample Properties of Parameter Estimates for Strongly Dependent Stationary Gaussian Time Series. doi
  11. (1995). Log-periodogram Regression of Time Series with Long Range Dependence. doi
  12. (1989). Long Memory and Persistence in Aggregate Output. doi
  13. (1993). Long Memory in Foreign-Exchange Rates. doi
  14. (1992). Maximum Likelihood Estimation of Stationary Univariate Fractionally Integrated Time Series Models. doi
  15. (1994). On Maximum Likelihood Estimation of the Differencing Parameter of Fractionally-Integrated Noise with Unknown Mean. doi
  16. (1994). Semiparametric Analysis of Long-memory Time Series. doi
  17. (1996). Semiparametric Estimation of the Long-range Parameter. doi
  18. (1983). The Estimation and Application of Long Memory Time Series Models. doi
  19. (1996). The Estimation of Continuous Parameter Long-memory Time Series Models. doi
  20. (1990). The Fractional Unit Root Distribution. doi
  21. (1994). Therefore, we can use equation (4) to derive DSF s ( , ) by simply replacing s with s.

To submit an update or takedown request for this paper, please submit an Update/Correction/Removal Request.