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