Nonlinear dynamics of networks : the groupoid formalism

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Golubitsky, Martin ; Stewart, Ian (2006)

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.
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