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

Dynamical Systems

Volumn 22, Issue 4, 2007, Pages 389-450

  Stewart, Ian ^a   Parker, Martyn ^a  

^a UNIVERSITY OF WARWICK   (United Kingdom)

Author keywords

[No Author keywords available]

Indexed keywords

EQUIVALENCE CLASSES; PERTURBATION TECHNIQUES; VECTORS;

HYPERBOLIC EQUILIBRIA; RIGID PHASE CONJECTURE; RIGID SYNCHRONY CONJECTURE; TIME-PERIODIC STATES;

DYNAMICAL SYSTEMS;

EID: 36949025701     PISSN: 14689367     EISSN: 14689375     Source Type: Journal    
DOI: 10.1080/14689360701450410     Document Type: Article

References (31)

1. Elmhirst, T. and Golubitsky, M., 2006, Nilpotent Hopf bifurcations in coupled cell systems. SIAM Journal of Applied Dynamical Systems, 5, 205-251.
2. Golubitsky, M., Josić, K. and Shea-Brown, E., 2006, Winding numbers and averaged frequencies in phase oscillator networks. Journal of Nonlinear Science, 16, 201-231.
3. Golubitsky, M. and Krupa, M., Stability computations for nilpotent Hopf bifurcations in coupled cell systems. International Journal of Bifurcation and Chaos, (To appear).
4. Golubitsky, M., Nicol, M., and Stewart, I., 2004, Some curious phenomena in coupled cell networks. Journal of Nonlinear Science, 14, 119-236.
5. Leite, M. and Golubitsky, M., 2006, Homogeneous three-cell networks. Nonlinearity, 19, 2313-2363.
6. Stewart, I., Golubitsky, M. and Pivato, M., 2003, Symmetry groupoids and patterns of synchrony in coupled cell networks. SIAM Journal of Applied Dynamical Systems, 2, 609-646.
7. Golubitsky, M., Stewart, I. and Török, A., 2005, Patterns of synchrony in coupled cell networks with multiple arrows. SIAM Journal of Applied Dynamical Systems, 4(1) 78-100.
8. Aldis, J., A polynomial time algorithm, to determine maximal balanced equivalence relations. International Journal of Bifurcation and Chaos (To appear).
9. Aldosray, F. and Stewart, I., 2005, Enumeration of homogeneous coupled cell networks. International Journal of Bifurcation and Chaos, 15, 2361-2373
10. Antoneli, F. and Stewart, I., 2006, Symmetry and synchrony in coupled cell networks 1: fixed-point spaces. International Journal of Bifurcation and Chaos, 16, 559-577.
11. Dias, A.P.S. and Stewart, I., 2004, Symmetry groupoids and admissible vector fields for coupled cell networks. Journal of London Mathematical Society, 69, 707-736.
12. Dias, A.P.S. and Stewart, I., 2005, Linear equivalence and ODE-equivalence for coupled cell networks. Nonlinearity, 18, 1003-1020.
13. Gucciardi, B., 2006, PhD thesis, University of Houston. (In preparation).
14. Leite, M., 2005, Homogeneous three-cell networks. PhD thesis. University of Houston, Texas.
15. Golubitsky, M. and Stewart, I., 2006, Nonlinear dynamics of networks: the groupoid formalism. Bulletin of the American Mathematical Society, 43(3), 305-364.
16. Field, M., 2004, Combinatorial dynamics. Dynamical Systems, 19, 217-243.
17. Golubitsky, M. and Stewart, I., 2002, The Symmetry Perspective: From Equilibrium to Chaos in Phase Space and Physical Space, Progress in Mathematics 200 (Basel: Birkhäuser).
18. Buono, L. and Golubitsky, M., 2001, Models of central pattern generators for quadruped locomotion: I. primary gaits. Journal of Mathematical Biology, 42, 291-326.
19. Katok, A. and Hasselblatt, B., 1995. Introduction to the Modern Theory of Dynamical Systems (Cambridge: Cambridge University Press).
20. Hirsch, M.W., Pugh, C.C. and Shub, M., 1977, Invariant Manifolds, Lect. Notes Math. 583 (New York: Springer).
21. Golubitsky, M., Stewart, I.N. and Schaeffer, D.G. 1988. Singularities and Groups in Bifurcation Theory: Vol. 2, Applied Mathematical Sciences 69 (New York: Springer).
22. K. Josić and Török, A., 2006. Network architecture and spatio-temporally symmetric dynamics. Physica D, 224, 52-68.
23. Stewart, I. and Parker, M., Periodic dynamics of coupled cell networks II: cyclic symmetry (To appear).
24. Brandt, H., 1927, Über eine Verallgemeinerung des Gruppenbegriftes. Mathematische Annalen, 96, 360-366.
25. Higgins, P.J., 1971, Notes on Categories and Groupoids, Van Nostrand Reinhold Mathematical Studies 32 (New York and London: Van Nostrand Reinhold).
26. Abraham, R., 1983, Marsden, J.E., and Ratiu, T.. Manifolds, Tensor Analysis, and Applications (Reading MA: Addison-Wesley).
27. Wu, F. and Desoer, C.A., 1972, Global inverse function theorem. Institute of Electrical and Electronics Engineers. Transactions on Circuits and Systems, 19, 199-201.
28. Gordon, W. B., 1972, On the diffeomorphisms of euclidean space. American Mathematical Monthly 79, 755-759.
29. Stewart, I. and Parker, M., Periodic dynamics of coupled cell networks III: rigid paterns of synchrony for homogeneous networks (In preparation).
30. Abraham, R. and Marsden, J.E., 1978, Foundations of Mechanics (New York: Benjamin/Cummings).
31. Adams, J.F., 1969. Lectures on Lie Groups (Chicago: University of Chicago Press) (Midway Reprint 1982.)