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Nonlinear dynamics of networks: the groupoid formalism

Authors: Martin Golubitsky and Ian Stewart
Journal: Bull. Amer. Math. Soc. 43 (2006), 305-364
MSC (2000): Primary 37G40, 34C23, 34C25, 92B99, 37G35
Published electronically: May 3, 2006
MathSciNet review: 2223010
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Abstract: A formal theory of symmetries of networks of coupled dynamical systems, stated in terms of the group of permutations of the nodes that preserve the network topology, has existed for some time. Global network symmetries impose strong constraints on the corresponding dynamical systems, which affect equilibria, periodic states, heteroclinic cycles, and even chaotic states. In particular, the symmetries of the network can lead to synchrony, phase relations, resonances, and synchronous or cycling chaos.

Symmetry is a rather restrictive assumption, and a general theory of networks should be more flexible. A recent generalization of the group-theoretic notion of symmetry replaces global symmetries by bijections between certain subsets of the directed edges of the network, the `input sets'. Now the symmetry group becomes a groupoid, which is an algebraic structure that resembles a group, except that the product of two elements may not be defined. The groupoid formalism makes it possible to extend group-theoretic methods to more general networks, and in particular it leads to a complete classification of `robust' patterns of synchrony in terms of the combinatorial structure of the network.

Many phenomena that would be nongeneric in an arbitrary dynamical system can become generic when constrained by a particular network topology. A network of dynamical systems is not just a dynamical system with a high-dimensional phase space. It is also equipped with a canonical set of observables--the states of the individual nodes of the network. Moreover, the form of the underlying ODE is constrained by the network topology--which variables occur in which component equations, and how those equations relate to each other. The result is a rich and new range of phenomena, only a few of which are yet properly understood.

References [Enhancements On Off] (What's this?)

  • 1. F. Aldosray and I. Stewart. Enumeration of homogeneous coupled cell networks. Internat. J. Bif. Chaos Appl. Sci. Engrg. MR 2174556
  • 2. J.C. Alexander, I. Kan, J.A. Yorke, and Zhiping You. Riddled basins, Internat. J. Bif. Chaos 2 (1992) 795-813.MR 1206103 (93k:58140)
  • 3. F. Antoneli, A.P.S. Dias, M. Golubitsky, and Y. Wang. Patterns of synchrony in lattice dynamical systems, Nonlinearity 18 (2005) 2193-2209. MR 2164738
  • 4. P. Ashwin, J. Buescu, and I. Stewart. Bubbling of attractors and synchronisation of oscillators, Phys. Lett. A 193 (1994) 126-139.MR 1295394 (95e:58114)
  • 5. P. Ashwin, J. Buescu, and I. Stewart. From attractor to chaotic saddle: a tale of transverse instability, Nonlinearity 9 (1996) 703-737. MR 1393154 (97k:58096)
  • 6. P. Ashwin and P. Stork. Permissible symmetries of coupled cell networks, Math. Proc. Camb. Phil. Soc. 116 (1994) 27-36. MR 1274157 (95j:92003)
  • 7. P. Ashwin and J.W. Swift. The dynamics of $ n$ identical oscillators with symmetric coupling. J. Nonlin. Sci. 2 (1992) 69-108. MR 1158354 (93g:58103)
  • 8. N.L. Biggs. Discrete Mathematics, Oxford University Press, Oxford, 1989. MR 1078626 (91h:00002)
  • 9. J. Blaszczyk and C. Dobrzecka. Alteration in the pattern of locomotion following a partial movement restraint in puppies, Acta. Neuro. Exp. 49 (1989) 39-46.
  • 10. S. Boccaletti, L.M. Pecora, and A. Pelaez. A unifying framework for synchronization of coupled dynamical systems, Phys. Rev. E 63 (2001) 066219.
  • 11. H. Brandt. Über eine Verallgemeinerung des Gruppenbegriffes, Math. Ann. 96 (1927) 360-366. MR 1512323
  • 12. R. Brown. From groups to groupoids: a brief survey, Bull. London Math. Soc. 19 (1987) 113-134.
  • 13. P.L. Buono and M. Golubitsky. Models of central pattern generators for quadruped locomotion: I. primary gaits. J. Math. Biol. 42, No. 4 (2001) 291-326. MR 1834105 (2002f:92003)
  • 14. R. Calabrese and E. Peterson. Neural control of heartbeat in the leech, Hirudo medicinalis, in Neural Origin of Rhythmic Movements (A. Roberts and B. Roberts, eds.), Symp. Soc. Exp. Biol. 37 (1983) 195-221.
  • 15. J. Cohen and I. Stewart. Polymorphism viewed as phenotypic symmetry-breaking, in: Nonlinear Phenomena in Physical and Biological Sciences (S.K. Malik, ed.), Indian National Science Academy, New Delhi, 1-67.
  • 16. J.J. Collins and I. Stewart. Hexapodal gaits and coupled nonlinear oscillator models, Biol. Cybern. 68 (1993) 287-298.
  • 17. J.J. Collins and I. Stewart. Coupled nonlinear oscillators and the symmetries of animal gaits, J. Nonlin. Sci. 3 (1993) 349-392. MR 1237096 (94g:92007)
  • 18. A.P.S. Dias and I. Stewart. Symmetry groupoids and admissible vector fields for coupled cell networks, J. London Math. Soc. 69 (2004) 707-736. MR 2050042 (2005j:37034)
  • 19. A.P.S. Dias and I. Stewart. Linear equivalence and ODE-equivalence for coupled cell networks, Nonlinearity 18 (2005) 1003-1020. MR 2134081 (2006e:37029)
  • 20. B. Dionne, M. Golubitsky, and I. Stewart. Coupled cells with internal symmetry, Part 1: wreath products, Nonlinearity 9 (1996) 559-574. MR 1384492 (97j:58110)
  • 21. B. Dionne, M. Golubitsky, and I. Stewart. Coupled cells with internal symmetry, Part 2: direct products, Nonlinearity 9 (1996) 575-599. MR 1384493 (97j:58111)
  • 22. R. Dobrin, Q.K. Beg, A.-L. Barabási, and Z.N. Oltvai. Aggregation of topological motifs in the Escherichia coli transcriptional regulatory network, BMC Bioinformatics 5 (2004) 1471-2105/5/10.
  • 23. T. Elmhirst. Symmetry and Emergence in Polymorphism and Sympatric Speciation, Ph.D. Thesis, Math. Inst., U. Warwick, 2002.
  • 24. T. Elmhirst and M. Golubitsky. Nilpotent Hopf bifurcations in coupled cell systems, SIAM J. Appl. Dynam. Sys. 5 (2006). To appear.
  • 25. I.R. Epstein and M. Golubitsky. Symmetric patterns in linear arrays of coupled cells, Chaos 3(1) (1993) 1-5. MR 1210158 (94a:80015)
  • 26. M. Feinberg. The existence and uniqueness of steady states for a class of chemical reaction networks, Arch. Rational Mech. Anal. 132 (1995) 311-370. MR 1365832 (97g:92028)
  • 27. M.J. Field. Lectures on Bifurcations, Dynamics and Symmetry, Research Notes in Mathematics 356, Pitman, San Francisco, 1996. MR 1425388 (97h:58115)
  • 28. R. FitzHugh. Impulses and physiological states in theoretical models of nerve membrane, Biophys. J. 1 (1961) 445-466.
  • 29. P.P. Gambaryan. How Mammals Run: Anatomical Adaptations, Wiley, New York, 1974.
  • 30. D. Gillis and M. Golubitsky. Patterns in square arrays of coupled cells, J. Math. Anal. App. 208 (1997) 487-509. MR 1441450 (98g:58158)
  • 31. M. Golubitsky, K. Josic, and E. Shea-Brown. Rotation, oscillation and spike numbers in phase oscillator networks, J. Nonlinear Sci. To appear.
  • 32. M. Golubitsky, M. Nicol, and I. Stewart. Some curious phenomena in coupled cell networks, J. Nonlinear Sci. 14 (2004) 119-236. MR 2041431 (2005a:37037)
  • 33. M. Golubitsky, M. Pivato, and I. Stewart. Interior symmetry and local bifurcation in coupled cell networks, Dyn. Sys. 19 (2004) 389-407 MR 2107649
  • 34. M. Golubitsky and D.G. Schaeffer. Singularities and Groups in Bifurcation Theory: Vol. I, Applied Mathematical Sciences 51, Springer-Verlag, New York, 1985. MR 0771477 (86e:58014)
  • 35. M. Golubitsky and I. Stewart. Hopf bifurcation with dihedral group symmetry: coupled nonlinear oscillators, in Multiparameter Bifurcation Theory (M. Golubitsky and J. Guckenheimer, eds.), Proceedings of the AMS-IMS-SIAM Joint Summer Research Conference, July 1985, Arcata; Contemporary Math. 56, Amer. Math. Soc., Providence, RI, 1986, 131-173. MR 0855088 (88c:58047)
  • 36. M. Golubitsky and I. Stewart. Symmetry and pattern formation in coupled cell networks, In: Pattern Formation in Continuous and Coupled Systems (M. Golubitsky, D. Luss, and S.H. Strogatz, eds.), IMA volumes in Math. and Appls. 115, Springer-Verlag, New York, 1999, 65-82. MR 1708862 (2001h:34060)
  • 37. M. Golubitsky and I. Stewart. The Symmetry Perspective: From Equilibrium to Chaos in Phase Space and Physical Space, Progress in Mathematics 200, Birkhäuser, Basel, 2002. MR 1891106 (2003e:37068)
  • 38. M. Golubitsky, I. Stewart, P.-L. Buono, and J.J. Collins. A modular network for legged locomotion, Physica D 115 (1998) 56-72. MR 1616780 (99d:92051)
  • 39. M. Golubitsky, I. Stewart, P.-L. Buono, and J.J. Collins. Symmetry in locomotor central pattern generators and animal gaits, Nature 401 (1999) 693-695.
  • 40. M. Golubitsky, I. Stewart, and B. Dionne. Coupled cells: wreath products and direct products, in Dynamics, Bifurcation and Symmetry, NATO ARW Series (P. Chossat and J.M. Gambaudo, eds.), Kluwer, Amsterdam, 1994, 127-138. MR 1305373 (96e:58115)
  • 41. M. Golubitsky, I. Stewart, and D.G. Schaeffer. Singularities and Groups in Bifurcation Theory II, Applied Mathematics Sciences, 69, Springer-Verlag, New York, 1988. MR 0950168 (89m:58038)
  • 42. M. Golubitsky, I. Stewart, and A. Török. Patterns of synchrony in coupled cell networks with multiple arrows, SIAM J. Appl. Dyn. Sys. 4(1) (2005) 78-100. MR 2136519 (2005k:34143)
  • 43. M.G.M. Gomes and G.F. Medley. Dynamics of multiple strains of infectious agents coupled by cross-immunity: a comparison of models, in Mathematical Approaches for Emerging and Reemerging Infectious Diseases (S. Blower, C. Castillo-Chavez, K.L. Cooke, D. Kirschner, and P. van der Driessche, eds.), Springer-Verlag, New York, 2002, 171-191. MR 1938903
  • 44. S. Grillner, D. Parker, and A.J. El Manira. Vertebrate locomotion--a lamprey perspective, Ann. New York Acad. Sci. 860 (1998) 1-18.
  • 45. S. Grillner and P. Wallén. Central pattern generators for locomotion, with special reference to vertebrates, Ann. Rev. Neurosci 8 (1985) 233-261.
  • 46. B. Gucciardi. Thesis, University of Houston, 2006. In preparation.
  • 47. P.J. Higgins. Notes on Categories and Groupoids, Van Nostrand Reinhold Mathematical Studies 32, Van Nostrand Reinhold, London, 1971. MR 0327946 (48:6288)
  • 48. M.W. Hirsch, C.C. Pugh, and M. Shub. Invariant Manifolds, Lect. Notes Math. 583, Springer, New York, 1977. MR 0501173 (58:18595)
  • 49. F.C. Hoppensteadt and I. Izhekevich. Weakly Connected Neural Nets, Applied Mathematical Sciences 126, Springer, New York, 1997. MR 1458890 (98k:92004)
  • 50. A. Katok and B. Hasselblatt. Introduction to the Modern Theory of Dynamical Systems, Cambridge U. Press, Cambridge, 1995. MR 1326374 (96c:58055)
  • 51. N. Kopell and G.B. Ermentrout. Symmetry and phaselocking in chains of weakly coupled oscillators, Comm. Pure Appl. Math. 39 (1986) 623-660. MR 0849426 (87m:34054)
  • 52. N. Kopell and G.B. Ermentrout. Coupled oscillators and the design of central pattern generators, Math. Biosci. 90 (1988) 87-109. MR 0958133 (89m:92024)
  • 53. N. Kopell and G. LeMasson. Rhythmogenesis, amplitude modulation, and multiplexing in a cortical architecture, Proc. Natl. Acad. Sci. USA 91 (1994) 10586-10590.
  • 54. Y. Kuramoto. Chemical Oscillations, Waves, and Turbulence, Springer, Berlin, 1984. MR 0762432 (87e:92054)
  • 55. M. Leite. Homogeneous three-cell networks. Ph.D. Thesis, University of Houston, August 2005. MR 1697351 (2000f:37111)
  • 56. M. Leite and M. Golubitsky. Homogeneous three-cell networks. Nonlinearity. Submitted.
  • 57. S.C. Manrubia, A.S. Mikhailov, and D.H. Zanette, Emergence of Dynamical Order, World Scientific, Singapore, 2004.
  • 58. J. Milnor. On the concept of attractor, Commun. Math. Phys. 99 (1985) 177-195. MR 0790735 (87i:58109a)
  • 59. R. Milo, S. Shen-Orr, S. Itkovitz, N. Kashtan, D. Chklovskii, and U. Alon. Network motifs: simple building blocks of complex networks, Science 298 (2002) 824.
  • 60. E. Mosekilde, Y. Maistrenko, and D. Postonov, Chaotic Synchronization, World Scientific, Singapore, 2002. MR 1939912 (2004h:37045)
  • 61. J.S. Nagumo, S. Arimoto, and S. Yoshizawa. An active pulse transmission line simulating nerve axon, Proc. IRE 50 (1962) 2061-2071.
  • 62. A.-M. Neutel, J.A.P. Heesterbeek, and P.C. de Ruiter. Stability in real food webs: weak links in long loops, Science 96 (2002) 1120-1123.
  • 63. M.E.J. Newman. The structure and function of complex networks, SIAM Review 45 (2003) 167-256. MR 2010377 (2005a:05206)
  • 64. O.H. Olsen and R.L. Calabrese. Activation of intrinsic and synaptic currents in leech heart interneurons by realistic waveforms, J. Neuroscience 16 4958-4970.
  • 65. Z.N. Oltvai and A.-L. Barabási. Life's complexity pyramid, Science 298 (2002) 763-764.
  • 66. E. Ott and J.C. Sommerer. Blowout bifurcations: the occurrence of riddled basins and on-off intermittency, Phys. Lett. A 188 (1994) 39-47.
  • 67. H.-O. Peitgen, H. Jürgens, and D. Saupe. Chaos and Fractals, Springer-Verlag, New York, 1992. MR 1185709 (93k:58157)
  • 68. N. Platt, E.A. Spiegel, and C. Tresser. On-off intermittency: a mechanism for bursting, Phys. Rev. Lett. 70 (1993) 279-282.
  • 69. O.E. Rössler. An equation for continuous chaos, Phys. Lett. 57A (1976) 397-398.
  • 70. W. Singer. Neuronal synchrony: a versatile code for the definition of relations, Neuron 24 (1999) 49-65.
  • 71. I. Stewart, T. Elmhirst, and J. Cohen. Symmetry-breaking as an origin of species. In: Bifurcation, Symmetry and Patterns (J. Buescu, S.B.S.D. Castro, A.P.S. Dias, and I.S. Labouriau, eds.), Birkhäuser, Basel, 2003, 3-54. MR 2014354 (2005a:92032)
  • 72. I. Stewart, M. Golubitsky, and M. Pivato. Patterns of synchrony in coupled cell networks, SIAM J. Appl. Dynam. Sys. 2 (2003) 609-646. [DOI: 10.1137/S1111111103419896] MR 2050244 (2005i:37030)
  • 73. I. Stewart and M. Parker. Periodic dynamics of coupled cell networks I: rigid patterns of synchrony. Preprint.
  • 74. I. Stewart and M. Parker. Periodic dynamics of coupled cell networks II: cyclic symmetry. Preprint.
  • 75. I. Stewart and M. Parker. Periodic dynamics of coupled cell networks III: rigid phase patterns. Preprint.
  • 76. W.T. Tutte. Graph Theory, Encyclopedia of Mathematics and Its Applications (G.-C. Rota, ed.), 21, Addison-Wesley, Reading, MA, 1984. MR 0746795 (87c:05001)
  • 77. J.J. Tyson, K.C. Chen, and B. Novak. Sniffers, buzzers, toggles and blinkers: dynamics of regulatory and signaling pathways in the cell, Curr. Opin. Cell Biol. 15 (2003) 221-231.
  • 78. J.J. Tyson, A. Csikasz-Nagy, and B. Novak. The dynamics of cell cycle regulation, BioEssays 24 (2002) 1095-1109.
  • 79. T.L. Vincent and T.L.S. Vincent. Evolution and control system design, IEEE Control Systems Magazine (October 2000) 20-35.
  • 80. X.-F. Wang. Complex networks: Topology, dynamics and synchronization, International Journal of Bifurcation and Chaos 12 (2002) 885-916. MR 1913980
  • 81. Y. Wang and M. Golubitsky. Two-colour patterns of synchrony in lattice dynamical systems, Nonlinearity 18 (2005) 631-657. MR 2122678 (2005i:37032)
  • 82. D.J. Watts and S.H. Strogatz. Collective dynamics of small-world networks, Nature 393 440-442.
  • 83. A. Weinstein. Groupoids: unifying internal and external symmetry, Notices Amer. Math. Soc. 43 (1996) 744-752. MR 1394388 (97f:20072)
  • 84. R.J. Wilson. Introduction to Graph Theory (3rd ed.), Longman, Harlow, 1985. MR 0826772 (87a:05051)
  • 85. D.M. Wolf and A.P. Arkin. Motifs, modules, and games in bacteria, Current Opinion in Microbiol. 6 (2003) 125-134.
  • 86. D. Wood. Coupled Oscillators with Internal Symmetries, Ph.D. Thesis, Univ. Warwick, 1995.
  • 87. D. Wood. Hopf bifurcations in three coupled oscillators with internal $ \mathbf{Z}_2$ symmetries, Dyn. Stab. Sys. 13 (1998) 55-93. MR 1624208 (99e:58136)
  • 88. D. Wood. A cautionary tale of coupling cells with internal symmetries, Internat. J. Bif. Chaos 11 (2001) 123-132. MR 1815530 (2001m:37106)
  • 89. C.W. Wu. Synchronization in Coupled Chaotic Circuits and Systems, World Scientific, Singapore, 2002. MR 1891843 (2003g:34106)
  • 90. C.W. Wu. Synchronization in networks of nonlinear dynamical systems coupled via a directed graph, Nonlinearity 18 (2005) 1057-1064. MR 2134084 (2005m:37059)

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Additional Information

Martin Golubitsky
Affiliation: Department of Mathematics, University of Houston, Houston, Texas 77204-3008

Ian Stewart
Affiliation: Mathematics Institute, University of Warwick, Coventry CV4 7AL, UK

Received by editor(s): May 2, 2005
Published electronically: May 3, 2006
Additional Notes: Part of this material was presented by M. Golubitsky in the SIAM plenary lecture “Coupled cell systems: A potpourri of theory and examples", given at the Joint Mathematics Meetings in Phoenix, AZ, January 2004.
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