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Nonlinear dynamics of networks: the groupoid formalism
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by Martin Golubitsky and Ian Stewart PDF
Bull. Amer. Math. Soc. 43 (2006), 305-364 Request permission


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.
  • Falih Aldosray and Ian Stewart, Enumeration of homogeneous coupled cell networks, Internat. J. Bifur. Chaos Appl. Sci. Engrg. 15 (2005), no. 8, 2361–2373. MR 2174556, DOI 10.1142/S0218127405013368
  • J. C. Alexander, James A. Yorke, Zhiping You, and I. Kan, Riddled basins, Internat. J. Bifur. Chaos Appl. Sci. Engrg. 2 (1992), no. 4, 795–813. MR 1206103, DOI 10.1142/S0218127492000446
  • Fernando Antoneli, Ana Paula Dias, Martin Golubitsky, and Yunjiao Wang, Patterns of synchrony in lattice dynamical systems, Nonlinearity 18 (2005), no. 5, 2193–2209. MR 2164738, DOI 10.1088/0951-7715/18/5/016
  • Peter Ashwin, Jorge Buescu, and Ian Stewart, Bubbling of attractors and synchronisation of chaotic oscillators, Phys. Lett. A 193 (1994), no. 2, 126–139. MR 1295394, DOI 10.1016/0375-9601(94)90947-4
  • Peter Ashwin, Jorge Buescu, and Ian Stewart, From attractor to chaotic saddle: a tale of transverse instability, Nonlinearity 9 (1996), no. 3, 703–737. MR 1393154, DOI 10.1088/0951-7715/9/3/006
  • Peter Ashwin and Peter Stork, Permissible symmetries of coupled cell networks, Math. Proc. Cambridge Philos. Soc. 116 (1994), no. 1, 27–36. MR 1274157, DOI 10.1017/S0305004100072364
  • P. Ashwin and J. W. Swift, The dynamics of $n$ weakly coupled identical oscillators, J. Nonlinear Sci. 2 (1992), no. 1, 69–108. MR 1158354, DOI 10.1007/BF02429852
  • Norman L. Biggs, Discrete mathematics, 2nd ed., Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, 1989. MR 1078626
  • BD89 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. BPP00 S. Boccaletti, L.M. Pecora, and A. Pelaez. A unifying framework for synchronization of coupled dynamical systems, Phys. Rev. E 63 (2001) 066219.
  • H. Brandt, Über eine Verallgemeinerung des Gruppenbegriffes, Math. Ann. 96 (1927), no. 1, 360–366 (German). MR 1512323, DOI 10.1007/BF01209171
  • B87 R. Brown. From groups to groupoids: a brief survey, Bull. London Math. Soc. 19 (1987) 113–134.
  • Pietro-Luciano Buono and Martin Golubitsky, Models of central pattern generators for quadruped locomotion. I. Primary gaits, J. Math. Biol. 42 (2001), no. 4, 291–326. MR 1834105, DOI 10.1007/s002850000058
  • CP83 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. C-MP-VV96 S. Chow, J. Mallet-Paret, and E. Van Vleck. Dynamics of lattice differential equations. Int. Jnl. Bifur. Chaos. 6 No. 9 (1996) 1605–1621. C-MP-VV96a S. Chow, J. Mallet-Paret, and E. Van Vleck. Pattern formation and spatial chaos in spatially discrete evolution equations. Random and Computational Dynamics 4 (1996) 109–178. CS00 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. CS93a J.J. Collins and I. Stewart. Hexapodal gaits and coupled nonlinear oscillator models, Biol. Cybern. 68 (1993) 287–298.
  • J. J. Collins and I. N. Stewart, Coupled nonlinear oscillators and the symmetries of animal gaits, J. Nonlinear Sci. 3 (1993), no. 3, 349–392. MR 1237096, DOI 10.1007/BF02429870
  • Ana Paula S. Dias and Ian Stewart, Symmetry groupoids and admissible vector fields for coupled cell networks, J. London Math. Soc. (2) 69 (2004), no. 3, 707–736. MR 2050042, DOI 10.1112/S0024610704005241
  • Ana Paula S. Dias and Ian Stewart, Linear equivalence and ODE-equivalence for coupled cell networks, Nonlinearity 18 (2005), no. 3, 1003–1020. MR 2134081, DOI 10.1088/0951-7715/18/3/004
  • Benoit Dionne, Martin Golubitsky, and Ian Stewart, Coupled cells with internal symmetry. I. Wreath products, Nonlinearity 9 (1996), no. 2, 559–574. MR 1384492, DOI 10.1088/0951-7715/9/2/016
  • Benoit Dionne, Martin Golubitsky, and Ian Stewart, Coupled cells with internal symmetry. II. Direct products, Nonlinearity 9 (1996), no. 2, 575–599. MR 1384493, DOI 10.1088/0951-7715/9/2/017
  • DBBO04 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. E02 T. Elmhirst. Symmetry and Emergence in Polymorphism and Sympatric Speciation, Ph.D. Thesis, Math. Inst., U. Warwick, 2002. EG05 T. Elmhirst and M. Golubitsky. Nilpotent Hopf bifurcations in coupled cell systems, SIAM J. Appl. Dynam. Sys. 5 (2006). To appear.
  • Irving R. Epstein and Martin Golubitsky, Symmetric patterns in linear arrays of coupled cells, Chaos 3 (1993), no. 1, 1–5. MR 1210158, DOI 10.1063/1.165974
  • Martin Feinberg, The existence and uniqueness of steady states for a class of chemical reaction networks, Arch. Rational Mech. Anal. 132 (1995), no. 4, 311–370. MR 1365832, DOI 10.1007/BF00375614
  • Michael Field, Lectures on bifurcations, dynamics and symmetry, Pitman Research Notes in Mathematics Series, vol. 356, Longman, Harlow, 1996. MR 1425388
  • F04 M. Field. Combinatorial dynamics. Dynamical Systems 19 (2004) 217–243. F61 R. FitzHugh. Impulses and physiological states in theoretical models of nerve membrane, Biophys. J. 1 (1961) 445–466. G74 P.P. Gambaryan. How Mammals Run: Anatomical Adaptations, Wiley, New York, 1974.
  • David Gillis and Martin Golubitsky, Patterns in square arrays of coupled cells, J. Math. Anal. Appl. 208 (1997), no. 2, 487–509. MR 1441450, DOI 10.1006/jmaa.1997.5347
  • GJS05 M. Golubitsky, K. Josić, and E. Shea-Brown. Rotation, oscillation and spike numbers in phase oscillator networks, J. Nonlinear Sci. To appear.
  • M. Golubitsky, M. Nicol, and I. Stewart, Some curious phenomena in coupled cell networks, J. Nonlinear Sci. 14 (2004), no. 2, 207–236. MR 2041431, DOI 10.1007/s00332-003-0593-6
  • M. Golubitsky, M. Pivato, and I. Stewart, Interior symmetry and local bifurcation in coupled cell networks, Dyn. Syst. 19 (2004), no. 4, 389–407. MR 2107649, DOI 10.1080/14689360512331318006
  • Martin Golubitsky and David G. Schaeffer, Singularities and groups in bifurcation theory. Vol. I, Applied Mathematical Sciences, vol. 51, Springer-Verlag, New York, 1985. MR 771477, DOI 10.1007/978-1-4612-5034-0
  • Martin Golubitsky and Ian Stewart, Hopf bifurcation with dihedral group symmetry: coupled nonlinear oscillators, Multiparameter bifurcation theory (Arcata, Calif., 1985) Contemp. Math., vol. 56, Amer. Math. Soc., Providence, RI, 1986, pp. 131–173. MR 855088, DOI 10.1090/conm/056/855088
  • Martin Golubitsky and Ian Stewart, Symmetry and pattern formation in coupled cell networks, Pattern formation in continuous and coupled systems (Minneapolis, MN, 1998) IMA Vol. Math. Appl., vol. 115, Springer, New York, 1999, pp. 65–82. MR 1708862, DOI 10.1007/978-1-4612-1558-5_{6}
  • Martin Golubitsky and Ian Stewart, The symmetry perspective, Progress in Mathematics, vol. 200, Birkhäuser Verlag, Basel, 2002. From equilibrium to chaos in phase space and physical space. MR 1891106, DOI 10.1007/978-3-0348-8167-8
  • Martin Golubitsky, Ian Stewart, Pietro-Luciano Buono, and J. J. Collins, A modular network for legged locomotion, Phys. D 115 (1998), no. 1-2, 56–72. MR 1616780, DOI 10.1016/S0167-2789(97)00222-4
  • GSBC99 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.
  • Martin Golubitsky, Ian Stewart, and Benoit Dionne, Coupled cells: wreath products and direct products, Dynamics, bifurcation and symmetry (Cargèse, 1993) NATO Adv. Sci. Inst. Ser. C: Math. Phys. Sci., vol. 437, Kluwer Acad. Publ., Dordrecht, 1994, pp. 127–138. MR 1305373
  • Martin Golubitsky, Ian Stewart, and David G. Schaeffer, Singularities and groups in bifurcation theory. Vol. II, Applied Mathematical Sciences, vol. 69, Springer-Verlag, New York, 1988. MR 950168, DOI 10.1007/978-1-4612-4574-2
  • Martin Golubitsky, Ian Stewart, and Andrei Török, Patterns of synchrony in coupled cell networks with multiple arrows, SIAM J. Appl. Dyn. Syst. 4 (2005), no. 1, 78–100. MR 2136519, DOI 10.1137/040612634
  • M. Gabriela M. Gomes and Graham F. Medley, Dynamics of multiple strains of infectious agents coupled by cross-immunity: a comparison of models, Mathematical approaches for emerging and reemerging infectious diseases: models, methods, and theory (Minneapolis, MN, 1999) IMA Vol. Math. Appl., vol. 126, Springer, New York, 2002, pp. 171–191. MR 1938903, DOI 10.1007/978-1-4613-0065-6_{1}0
  • GPM98 S. Grillner, D. Parker, and A.J. El Manira. Vertebrate locomotion—a lamprey perspective, Ann. New York Acad. Sci. 860 (1998) 1–18. GW85 S. Grillner and P. Wallén. Central pattern generators for locomotion, with special reference to vertebrates, Ann. Rev. Neurosci 8 (1985) 233–261. G06 B. Gucciardi. Thesis, University of Houston, 2006. In preparation.
  • Philip J. Higgins, Notes on categories and groupoids, Van Nostrand Rienhold Mathematical Studies, No. 32, Van Nostrand Reinhold Co., London-New York-Melbourne, 1971. MR 0327946
  • M. W. Hirsch, C. C. Pugh, and M. Shub, Invariant manifolds, Lecture Notes in Mathematics, Vol. 583, Springer-Verlag, Berlin-New York, 1977. MR 0501173
  • Frank C. Hoppensteadt and Eugene M. Izhikevich, Weakly connected neural networks, Applied Mathematical Sciences, vol. 126, Springer-Verlag, New York, 1997. MR 1458890, DOI 10.1007/978-1-4612-1828-9
  • Anatole Katok and Boris Hasselblatt, Introduction to the modern theory of dynamical systems, Encyclopedia of Mathematics and its Applications, vol. 54, Cambridge University Press, Cambridge, 1995. With a supplementary chapter by Katok and Leonardo Mendoza. MR 1326374, DOI 10.1017/CBO9780511809187
  • N. Kopell and G. B. Ermentrout, Symmetry and phaselocking in chains of weakly coupled oscillators, Comm. Pure Appl. Math. 39 (1986), no. 5, 623–660. MR 849426, DOI 10.1002/cpa.3160390504
  • N. Kopell and G. B. Ermentrout, Coupled oscillators and the design of central pattern generators, Math. Biosci. 90 (1988), no. 1-2, 87–109. Nonlinearity in biology and medicine (Los Alamos, NM, 1987). MR 958133, DOI 10.1016/0025-5564(88)90059-4
  • KL94 N. Kopell and G. LeMasson. Rhythmogenesis, amplitude modulation, and multiplexing in a cortical architecture, Proc. Natl. Acad. Sci. USA 91 (1994) 10586–10590.
  • Y. Kuramoto, Chemical oscillations, waves, and turbulence, Springer Series in Synergetics, vol. 19, Springer-Verlag, Berlin, 1984. MR 762432, DOI 10.1007/978-3-642-69689-3
  • Luiz A. Ferreira and Erica E. Leite, Integrable theories in any dimension and homogenous [homogeneous] spaces, Nuclear Phys. B 547 (1999), no. 3, 471–500. MR 1697351, DOI 10.1016/S0550-3213(99)00090-5
  • LG06 M. Leite and M. Golubitsky. Homogeneous three-cell networks. Nonlinearity. Submitted. MMZ04 S.C. Manrubia, A.S. Mikhailov, and D.H. Zanette, Emergence of Dynamical Order, World Scientific, Singapore, 2004.
  • John Milnor, On the concept of attractor, Comm. Math. Phys. 99 (1985), no. 2, 177–195. MR 790735
  • MSIKCA02 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.
  • Erik Mosekilde, Yuri Maistrenko, and Dmitry Postnov, Chaotic synchronization, World Scientific Series on Nonlinear Science. Series A: Monographs and Treatises, vol. 42, World Scientific Publishing Co., Inc., River Edge, NJ, 2002. Applications to living systems. MR 1939912, DOI 10.1142/9789812778260
  • NAY62 J.S. Nagumo, S. Arimoto, and S. Yoshizawa. An active pulse transmission line simulating nerve axon, Proc. IRE 50 (1962) 2061–2071. NHR02 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.
  • M. E. J. Newman, The structure and function of complex networks, SIAM Rev. 45 (2003), no. 2, 167–256. MR 2010377, DOI 10.1137/S003614450342480
  • OC96 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. OB02 Z.N. Oltvai and A.-L. Barabási. Life’s complexity pyramid, Science 298 (2002) 763–764. OS94 E. Ott and J.C. Sommerer. Blowout bifurcations: the occurrence of riddled basins and on-off intermittency, Phys. Lett. A 188 (1994) 39–47.
  • Heinz-Otto Peitgen, Hartmut Jürgens, and Dietmar Saupe, Chaos and fractals, Springer-Verlag, New York, 1992. New frontiers of science; With a foreword by Mitchell J. Feigenbaum; Appendix A by Yuval Fisher; Appendix B by Carl J. G. Evertsz and Benoit B. Mandelbrot. MR 1185709, DOI 10.1007/978-1-4757-4740-9
  • PST93 N. Platt, E.A. Spiegel, and C. Tresser. On-off intermittency: a mechanism for bursting, Phys. Rev. Lett. 70 (1993) 279–282. R76 O.E. Rössler. An equation for continuous chaos, Phys. Lett. 57A (1976) 397–398. S99 W. Singer. Neuronal synchrony: a versatile code for the definition of relations, Neuron 24 (1999) 49–65.
  • Ian Stewart, Toby Elmhirst, and Jack Cohen, Symmetry-breaking as an origin of species, Bifurcation, symmetry and patterns (Porto, 2000) Trends Math., Birkhäuser, Basel, 2003, pp. 3–54. MR 2014354
  • Ian Stewart, Martin Golubitsky, and Marcus Pivato, Symmetry groupoids and patterns of synchrony in coupled cell networks, SIAM J. Appl. Dyn. Syst. 2 (2003), no. 4, 609–646. MR 2050244, DOI 10.1137/S1111111103419896
  • SP05a I. Stewart and M. Parker. Periodic dynamics of coupled cell networks I: rigid patterns of synchrony. Preprint. SP05b I. Stewart and M. Parker. Periodic dynamics of coupled cell networks II: cyclic symmetry. Preprint. SP05c I. Stewart and M. Parker. Periodic dynamics of coupled cell networks III: rigid phase patterns. Preprint.
  • W. T. Tutte, Graph theory, Encyclopedia of Mathematics and its Applications, vol. 21, Addison-Wesley Publishing Company, Advanced Book Program, Reading, MA, 1984. With a foreword by C. St. J. A. Nash-Williams. MR 746795
  • TCN03 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. TCN02 J.J. Tyson, A. Csikasz-Nagy, and B. Novak. The dynamics of cell cycle regulation, BioEssays 24 (2002) 1095–1109. VV00 T.L. Vincent and T.L.S. Vincent. Evolution and control system design, IEEE Control Systems Magazine (October 2000) 20–35.
  • Xiao Fan Wang, Complex networks: topology, dynamics and synchronization, Internat. J. Bifur. Chaos Appl. Sci. Engrg. 12 (2002), no. 5, 885–916. Chaos control and synchronization (Shanghai, 2001). MR 1913980, DOI 10.1142/S0218127402004802
  • Yunjiao Wang and Martin Golubitsky, Two-colour patterns of synchrony in lattice dynamical systems, Nonlinearity 18 (2005), no. 2, 631–657. MR 2122678, DOI 10.1088/0951-7715/18/2/010
  • WS98 D.J. Watts and S.H. Strogatz. Collective dynamics of small-world networks, Nature 393 440–442.
  • Alan Weinstein, Groupoids: unifying internal and external symmetry. A tour through some examples, Notices Amer. Math. Soc. 43 (1996), no. 7, 744–752. MR 1394388
  • Robin J. Wilson, Introduction to graph theory, 3rd ed., Longman, New York, 1985. MR 826772
  • WA03 D.M. Wolf and A.P. Arkin. Motifs, modules, and games in bacteria, Current Opinion in Microbiol. 6 (2003) 125–134. Wo95 D. Wood. Coupled Oscillators with Internal Symmetries, Ph.D. Thesis, Univ. Warwick, 1995.
  • David Wood, Hopf bifurcations in three coupled oscillators with internal $Z_2$ symmetries, Dynam. Stability Systems 13 (1998), no. 1, 55–93. MR 1624208, DOI 10.1080/02681119808806254
  • David Wood, A cautionary tale of coupling cells with internal symmetries, Internat. J. Bifur. Chaos Appl. Sci. Engrg. 11 (2001), no. 1, 123–132. MR 1815530, DOI 10.1142/S0218127401001980
  • Chai Wah Wu, Synchronization in coupled chaotic circuits and systems, World Scientific Series on Nonlinear Science. Series A: Monographs and Treatises, vol. 41, World Scientific Publishing Co., Inc., River Edge, NJ, 2002. MR 1891843, DOI 10.1142/9789812778420
  • Chai Wah Wu, Synchronization in networks of nonlinear dynamical systems coupled via a directed graph, Nonlinearity 18 (2005), no. 3, 1057–1064. MR 2134084, DOI 10.1088/0951-7715/18/3/007
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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.
  • © Copyright 2006 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Bull. Amer. Math. Soc. 43 (2006), 305-364
  • MSC (2000): Primary 37G40, 34C23, 34C25, 92B99, 37G35
  • DOI:
  • MathSciNet review: 2223010