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Nonlinear dynamics of networks : the groupoid formalism

By Martin Golubitsky and Ian Stewart


A formal theory of symmetries of networks of coupled dynamical\ud systems, stated in terms of the group of permutations of the nodes that preserve\ud the network topology, has existed for some time. Global network symmetries\ud impose strong constraints on the corresponding dynamical systems,\ud which affect equilibria, periodic states, heteroclinic cycles, and even chaotic\ud states. In particular, the symmetries of the network can lead to synchrony,\ud phase relations, resonances, and synchronous or cycling chaos.\ud Symmetry is a rather restrictive assumption, and a general theory of networks\ud should be more flexible. A recent generalization of the group-theoretic\ud notion of symmetry replaces global symmetries by bijections between certain\ud subsets of the directed edges of the network, the ‘input sets’. Now the symmetry\ud group becomes a groupoid, which is an algebraic structure that resembles\ud a group, except that the product of two elements may not be defined. The\ud groupoid formalism makes it possible to extend group-theoretic methods to\ud more general networks, and in particular it leads to a complete classification\ud of ‘robust’ patterns of synchrony in terms of the combinatorial structure of the\ud network.\ud Many phenomena that would be nongeneric in an arbitrary dynamical\ud system can become generic when constrained by a particular network topology.\ud A network of dynamical systems is not just a dynamical system with\ud a high-dimensional phase space. It is also equipped with a canonical set of\ud observables—the states of the individual nodes of the network. Moreover, the\ud form of the underlying ODE is constrained by the network topology—which\ud variables occur in which component equations, and how those equations relate\ud to each other. The result is a rich and new range of phenomena, only a few of\ud which are yet properly understood

Topics: QA
Publisher: American Mathematical Society
Year: 2006
OAI identifier: oai:wrap.warwick.ac.uk:181

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  1. (2001). A cautionary tale of coupling cells with internal symmetries, doi
  2. (1998). A modular network for legged locomotion, doi
  3. (2001). A unifying framework for synchronization of coupled dynamical systems, doi
  4. Activation of intrinsic and synaptic currents in leech heart interneurons by realistic waveforms,
  5. (2004). Aggregation of topological motifs in the Escherichia coli transcriptional regulatory network,
  6. (1989). Alteration in the pattern of locomotion following a partial movement restraint in puppies, A c t a .N e u r o .E x p .49
  7. (1962). An active pulse transmission line simulating nerve axon, doi
  8. (1976). An equation for continuous chaos,
  9. (1994). Blowout bifurcations: the occurrence of riddled basins and on-off intermittency, doi
  10. (1994). Bubbling of attractors and synchronisation of oscillators, doi
  11. (1985). Central pattern generators for locomotion, with special reference to vertebrates, doi
  12. (1992). Chaos and Fractals, doi
  13. (2002). Chaotic Synchronization, World Scientific, doi
  14. (1984). Chemical Oscillations, Waves, and Turbulence, doi
  15. Collective dynamics of small-world networks,
  16. (2002). Complex networks: Topology, dynamics and synchronization, doi
  17. (1996). Coupled cells with internal symmetry, Part 1: wreath products, doi
  18. (1996). Coupled cells with internal symmetry, Part 2: direct products, doi
  19. (1994). Coupled cells: wreath products and direct products, doi
  20. (1993). Coupled nonlinear oscillators and the symmetries of animal gaits, doi
  21. (1988). Coupled oscillators and the design of central pattern generators, doi
  22. Coupled Oscillators with Internal Symmetries,P h . D .T h e s i s ,U n i v .W a r w i c k doi
  23. (1989). Discrete Mathematics, doi
  24. (2002). Dynamics of multiple strains of infectious agents coupled by cross-immunity: a comparison of models, doi
  25. (2004). Emergence of Dynamical Order,W o r l d Scientific, doi
  26. Enumeration of homogeneous coupled cell networks. doi
  27. (2000). Evolution and control system design, doi
  28. (1996). From attractor to chaotic saddle: a tale of transverse instability, doi
  29. (1987). From groups to groupoids: a brief survey, doi
  30. (1984). Graph Theory, Encyclopedia of Mathematics and Its Applications doi
  31. (1996). Groupoids: unifying internal and external symmetry, doi
  32. (1993). Hexapodal gaits and coupled nonlinear oscillator models, doi
  33. Homogeneous three-cell networks. doi
  34. (1985). Hopf bifurcation with dihedral group symmetry: coupled nonlinear oscillators, in Multiparameter Bifurcation Theory doi
  35. (1998). Hopf bifurcations in three coupled oscillators with internal Z2 symmetries, doi
  36. How Mammals Run: Anatomical Adaptations,W i l e y ,N e wY o r k doi
  37. (1961). Impulses and physiological states in theoretical models of nerve membrane, doi
  38. (2004). Interior symmetry and local bifurcation in coupled cell networks, doi
  39. (1985). Introduction to Graph Theory doi
  40. (1995). Introduction to the Modern Theory of Dynamical Systems, doi
  41. (1977). Invariant Manifolds,L e c t .N o t e sM a t h .583, doi
  42. (2002). Life’s complexity pyramid, doi
  43. (2005). Linear equivalence and ODE-equivalence for coupled cell networks, doi
  44. (2001). Models of central pattern generators for quadruped locomotion: I. primary gaits. doi
  45. (2003). Motifs, modules, and games in bacteria, doi
  46. (2002). Network motifs: simple building blocks of complex networks, doi
  47. (1983). Neural control of heartbeat in the leech, Hirudo medicinalis, doi
  48. (1999). Neuronal synchrony: a versatile code for the definition of relations, doi
  49. (2006). Nilpotent Hopf bifurcations in coupled cell systems, doi
  50. (1971). Notes on Categories and Groupoids,
  51. (1985). On the concept of attractor, doi
  52. (1993). On-off intermittency: a mechanism for bursting, doi
  53. (1997). Patterns in square arrays of coupled cells, doi
  54. (2005). Patterns of synchrony in coupled cell networks with multiple arrows, doi
  55. (2003). Patterns of synchrony in coupled cell networks, doi
  56. (2005). Patterns of synchrony in lattice dynamical systems, doi
  57. Periodic dynamics of coupled cell networks I: rigid patterns of synchrony. doi
  58. Periodic dynamics of coupled cell networks II: cyclic symmetry. doi
  59. Periodic dynamics of coupled cell networks III: rigid phase patterns. doi
  60. (1994). Permissible symmetries of coupled cell networks, doi
  61. Polymorphism viewed as phenotypic symmetry-breaking, in:
  62. (1994). Rhythmogenesis, amplitude modulation, and multiplexing in a cortical architecture, doi
  63. (1992). Riddled basins, doi
  64. Rotation, oscillation and spike numbers in phase oscillator networks,
  65. (1985). Singularities and Groups doi
  66. (1988). Singularities and Groups in Bifurcation Theory II, doi
  67. (2003). Sniffers, buzzers, toggles and blinkers: dynamics of regulatory and signaling pathways in the cell, doi
  68. (2004). Some curious phenomena in coupled cell networks, doi
  69. (2002). Stability in real food webs: weak links in long loops, doi
  70. (1993). Symmetric patterns in linear arrays of coupled cells, doi
  71. (2002). Symmetry and Emergence in Polymorphism and Sympatric Speciation,P h
  72. (1999). Symmetry and pattern formation in coupled cell networks, In: doi
  73. (1986). Symmetry and phaselocking in chains of weakly coupled oscillators, doi
  74. (2004). Symmetry groupoids and admissible vector fields for coupled cell networks, doi
  75. (1999). Symmetry in locomotor central pattern generators and animal gaits, doi
  76. (2003). Symmetry-breaking as an origin of species. In: Bifurcation, Symmetry and Patterns doi
  77. (2002). Synchronization in Coupled Chaotic Circuits and Systems, doi
  78. (2005). Synchronization in networks of nonlinear dynamical systems coupled via a directed graph, doi
  79. (2002). The dynamics of cell cycle regulation, doi
  80. (1992). The dynamics of n identical oscillators with symmetric coupling. doi
  81. (1995). The existence and uniqueness of steady states for a class of chemical reaction networks, doi
  82. (2003). The structure and function of complex networks, doi
  83. (2002). The Symmetry Perspective: From Equilibrium to Chaos in Phase Space and Physical Space, doi
  84. (2006). Thesis,
  85. (2005). Two-colour patterns of synchrony in lattice dynamical systems, doi
  86. (1927). Uber eine Verallgemeinerung des Gruppenbegriffes, doi
  87. (1998). Vertebrate locomotion—a lamprey perspective, doi

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