Nonlinear dynamics of networks: the groupoid formalism

By Martin Golubitsky, Ian Stewart. 2006

423 citations recorded by the source corpus.

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.

Subfields: Astronomy and Astrophysics, Computer Networks and Communications, Molecular Biology

View Paper: https://www.semanticscholar.org/paper/8bc4ab1b4bc367c49813a9f69011863b39beadc2

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Astronomy and Astrophysics

In Computer Networks and Communications

In Molecular Biology

Citations in the corpus

Hierarchical Network Structure Promotes Dynamical Robustness (2014, background): “[52] Martin Golubitsky and Ian Stewart.”
Multilayer networks (2013, background): “A major thread has entailed the derivation of generic results that relate the qualitative behavior of a dynamical system to the topology of the corresponding coupled-cell network [139].”
Bifurcations from regular quotient networks: A first insight (2009, background): “A theory for coupled cell networks has been developed in [12,8,7].”