Local bifurcation in symmetric coupled cell networks: Linear theory

Physica D: Nonlinear Phenomena

Volumn 223, Issue 1, 2006, Pages 93-108

  Dias, Ana Paula S ^a   Lamb, Jeroen S W ^b  

^a UNIVERSITY OF PORTO   (Portugal)

^b IMPERIAL COLLEGE LONDON   (United Kingdom)

Author keywords

Coupled cell network; Local bifurcation; Symmetry

Indexed keywords

BIFURCATION (MATHEMATICS); DIFFERENTIAL EQUATIONS; FINITE ELEMENT METHOD; GAMMA RAYS; GRAPH THEORY; LINEAR EQUATIONS;

COUPLED CELL NETWORKS; FINITE SYMMETRY GROUP; LOCAL BIFURCATION; SYMMETRIC EQUILIBRIUM;

ELECTRIC BATTERIES;

EID: 33750077831     PISSN: 01672789     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.physd.2006.08.014     Document Type: Article

References (25)

1. Abbott L.F., and van Vreeswijk C. Asynchronous states in neural networks of pulse-coupled oscillators. Phys. Rev. E 48 2 (1993) 1483-1490
2. Adams J.F. Lectures on Lie Groups (1969), W.A. Benjamin, Inc., New York, Amsterdam
3. Braiman Y., Lindner J.F., and Ditto W.L. Taming spatiotemporal chaos with disorder. Nature 378 (1995) 465-467
4. Bressloff P.C., Coombes S., and De Souza B. Dynamics of a ring of pulse-coupled oscillators: A group theoretic approach. Phys. Rev. Lett. 79 (1997) 2791-2794
5. Collins J.J., Chow C.C., and Imhoff T.T. Stochastic resonance without tuning. Nature 376 (1995) 236-238
6. Dionne B., Golubitsky M., and Stewart I. Coupled cells with internal symmetry Part 1: Wreath products. Nonlinearity 9 (1996) 559-574
7. Dionne B., Golubitsky M., and Stewart I. Coupled cells with internal symmetry Part 2: Direct products. Nonlinearity 9 (1996) 575-599
8. Field M. Combinatorial dynamics. Dyn. Syst. 19 3 (2004) 217-243
9. Gerhardt M., Schuster H., and Tyson J.J. Acellular automaton model of excitable media including curvature and dispersion. Science 247 (1990) 1563-1566
10. Golubitsky M., and Stewart I. The Symmetry Perspective: From Equilibrium to Chaos in Phase Space and Physical Space. Progress in Mathematics vol. 200 (2002), Birkhäuser, Basel
11. Golubitsky M., and Stewart I. Patterns of oscillation in coupled cell systems. In: Holmes P., Newton P., and Weinstein A. (Eds). Geometry, Dynamics, and Mechanics: 60th Birthday Volume for J.E. Marsden (2002), Springer-Verlag, New York 243-286
12. Golubitsky M., Stewart I., Buono P.-L., and Collins J.J. A modular network for legged locomotion. Physica D 115 (1998) 56-72
13. Golubitsky M., Stewart I.N., and Schaeffer D.G. Singularities and Groups in Bifurcation Theory: vol. II. Applied Mathematical Sciences vol. 69 (1988), Springer-Verlag, New York
14. Golubitsky M., Stewart I., and Török A. Patterns of synchrony in coupled cell networks with multiple arrows. SIAM J. Appl. Dyn. Syst. 4 1 (2005) 78-100
15. Hopfield J.J., and Herz A.V.M. Rapid local synchronization of action potentials: Toward computation with coupled integrate-and-fire neurons. Proc. Natl Acad. Sci. USA 92 (1995) 6655-6662
16. James G., and Liebeck M. Representations and Characters of Groups. 2nd edition (2001), Cambridge University Press, New York
17. Kauffman S.A. Metabolic stability and epigenesis in randomly constructed genetic nets. J. Theoret. Biol. 22 (1969) 437-467
18. Kuramoto Y. Chemical Oscillations, Waves, and Turbulence (1984), Springer, Berlin
19. Nowak M.A., and May R.M. Evolutionary games and spatial chaos. Nature 359 (1992) 826-829
20. Stewart I. Networking opportunity. Nature 427 (2004) 601-604
21. Stewart I., Golubitsky M., and Pivato M. Symmetry groupoids and patterns of synchrony in coupled cell networks. SIAM J. Appl. Dyn. Syst. 2 4 (2003) 609-646
22. Strogatz S.H., and Stewart I. Coupled oscillators and biological synchronization. Sci. Am. 269 6 (1993) 102-109
23. Wang X.F. Complex networks: Topology, dynamics and synchronization. Internat. J. Bifur. Chaos 12 (2002) 885-916
24. Wiesenfeld K. New results on frequency-locking dynamics of disordered Josephson arrays. Physica B 222 (1996) 315-319
25. Winfree A.T. The Geometry of Biological Time (1980), Springer, New York