Nonlinear dynamics of networks: the groupoid formalism. (English) Zbl 1119.37036
Coupled cell networks arise naturally in the context of dynamical systems modeled on systems of ordinary differential equations. It is well known by now that the presence of symmetries of the network can result in interesting properties of single solutions, the phase portrait and possible bifurcations.
The present paper gives an account of a generalization of the symmetry group of a network, namely the symmetry groupoid of a network, which has been developed by the authors over the years. We are guided through their countless results and a wealth of striking applications thereof. The most important proofs can be found in [M. Golubitsky, M. Pivato and I. Stewart, Dyn. Syst. 19, No. 4, 389–407 (2004; Zbl 1067.37066)], [M. Golubitsky, I. Stewart and A. Török, SIAM J. Appl. Dyn. Syst. 4, No. 1, 78–100 (2005; Zbl 1090.34030)], and [(*) I. Stewart, M. Golubitsky and M. Pivato, SIAM J. Appl. Dyn. Syst. 2, No. 4, 609–646 (2003; Zbl 1089.34032)] of the list of references. (Unfortunately, the title of reference [(*)] is incomplete.) For an excellent summary, the reader is referred to the abstract.
We indicate where groupoids come in. For any cell (of the network), the set of cells having an arrow pointing to that cell is the input set of that cell. The elements of the symmetry groupoid of a network are the arrow type preserving bijections between the various input cells, called input isomorphisms. Thus, the groupoid consists of certain local symmetries of the network and depends only on the network architecture. Associativity (if the compositions are defined) and existence of identities and inverses, the groupoid axioms, are fulfilled. Of course, the vector fields considered must be admissible, i.e., compatible with the groupoid of the network. The results display that interesting phenomena in coupled cell networks so far attributed to global symmetry to occur in the absence of the latter and can often be traced to local symmetries.
The present paper gives an account of a generalization of the symmetry group of a network, namely the symmetry groupoid of a network, which has been developed by the authors over the years. We are guided through their countless results and a wealth of striking applications thereof. The most important proofs can be found in [M. Golubitsky, M. Pivato and I. Stewart, Dyn. Syst. 19, No. 4, 389–407 (2004; Zbl 1067.37066)], [M. Golubitsky, I. Stewart and A. Török, SIAM J. Appl. Dyn. Syst. 4, No. 1, 78–100 (2005; Zbl 1090.34030)], and [(*) I. Stewart, M. Golubitsky and M. Pivato, SIAM J. Appl. Dyn. Syst. 2, No. 4, 609–646 (2003; Zbl 1089.34032)] of the list of references. (Unfortunately, the title of reference [(*)] is incomplete.) For an excellent summary, the reader is referred to the abstract.
We indicate where groupoids come in. For any cell (of the network), the set of cells having an arrow pointing to that cell is the input set of that cell. The elements of the symmetry groupoid of a network are the arrow type preserving bijections between the various input cells, called input isomorphisms. Thus, the groupoid consists of certain local symmetries of the network and depends only on the network architecture. Associativity (if the compositions are defined) and existence of identities and inverses, the groupoid axioms, are fulfilled. Of course, the vector fields considered must be admissible, i.e., compatible with the groupoid of the network. The results display that interesting phenomena in coupled cell networks so far attributed to global symmetry to occur in the absence of the latter and can often be traced to local symmetries.
Reviewer: Dieter Erle (Dortmund)
MSC:
| 37G40 | Dynamical aspects of symmetries, equivariant bifurcation theory |
| 34C25 | Periodic solutions to ordinary differential equations |
| 34C14 | Symmetries, invariants of ordinary differential equations |
| 20L05 | Groupoids (i.e. small categories in which all morphisms are isomorphisms) |
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