## Nonlinear dynamics of networks: the groupoid formalism (2006)

Venue: | Bull. Amer. Math. Soc |

Citations: | 33 - 6 self |

### BibTeX

@ARTICLE{Golubitsky06nonlineardynamics,

author = {Martin Golubitsky and Ian Stewart},

title = {Nonlinear dynamics of networks: the groupoid formalism},

journal = {Bull. Amer. Math. Soc},

year = {2006},

volume = {43},

pages = {305--364}

}

### Years of Citing Articles

### OpenURL

### Abstract

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. Contents 1.

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11 |
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10 |
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8 |
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8 |
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Citation Context ...ecise definition of a quotient network. Proofs of these results are found in [42, 72]. We apply this characterization to some patterns that arise in lattice dynamics—with a few surprises (Wang et al. =-=[3, 81]-=-). Robust synchrony is a very strong requirement: it occurs because of the presence of subspaces that are flow-invariant for all admissible ODEs. Section 8 shifts the emphasis to synchronous states ra... |

6 |
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6 |
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Citation Context ...ks, and there is a huge literature, most of which we shall not cite here. In models of speciation insNONLINEAR DYNAMICS OF NETWORKS: THE GROUPOID FORMALISM 309 evolutionary biology (Cohen and Stewart =-=[15]-=-, Elmhirst [23], Stewart et al. [71], Vincent and Vincent [79]) the nodes of the network represent (coarse-grained sets of) organisms, the edges represent interactions (breeding, competition for resou... |

6 | Symmetry groupoids and admissible vector fields for coupled cell networks
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Citation Context ...l coupling and valence 1 (see Figure 21) and 34 such networks with valence 2 (see Leite [55, 56] and Figure 22). It is possible for two networks to generate the same systems of differential equations =-=[18, 19, 42]-=-. Such redundancies have been eliminated in this enumeration. The number of identical coupling networks grows exponentially with the number of cells. For example, there are precisely 13,505,066,262,00... |

6 |
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Citation Context ...l coupling and valence 1 (see Figure 21) and 34 such networks with valence 2 (see Leite [55, 56] and Figure 22). It is possible for two networks to generate the same systems of differential equations =-=[18, 19, 42]-=-. Such redundancies have been eliminated in this enumeration. The number of identical coupling networks grows exponentially with the number of cells. For example, there are precisely 13,505,066,262,00... |

6 | Interior symmetry and local bifurcation in coupled cell networks, Dyn Syst
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Citation Context ...global; groupoid symmetries are local. It turns out that there is a useful intermediate concept, a strengthening of input-equivalence ‘symmetries’ that we call interior symmetry. It was introduced in =-=[33]-=-, and in particular it leads to some systematic results on local bifurcation. Like all local bifurcation theories, a key role is played by the eigenvalue structure of linear (admissible) vector fields... |

5 |
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Citation Context ... from that of groups; see Brandt [11], Higgins [47], and Brown [12]. The symmetry group of a network (that is, the automorphism group of the graph) is known to have a strong influence on its dynamics =-=[6, 20, 21, 25, 30, 36, 37, 40, 41, 86, 87, 88]-=-. Evidence is accumulating that many generic features of the dynamics of asymmetric networks can be understood from the groupoid viewpoint (where ‘generic’ is relative to the network structure). As fo... |

5 |
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Citation Context ...ISM 307 uses a Liapunov function to derive conditions for stable synchronization of systems with identical internal dynamics, assuming linear (time-dependent) coupling. Feinberg [26] and Tyson et al. =-=[77, 78]-=- study cell systems where the dynamical equations are chemical reaction equations. Feinberg has beautiful results about when the network can support steady states and periodic solutions; Tyson et al. ... |

4 | Patterns of synchrony in lattice dynamical systems. Nonlinearity 18
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Citation Context ...ecise definition of a quotient network. Proofs of these results are found in [42, 72]. We apply this characterization to some patterns that arise in lattice dynamics—with a few surprises (Wang et al. =-=[3, 81]-=-). Robust synchrony is a very strong requirement: it occurs because of the presence of subspaces that are flow-invariant for all admissible ODEs. Section 8 shifts the emphasis to synchronous states ra... |