Nonlinear dynamics of networks: the groupoid formalism

@article{Golubitsky2006NonlinearDO,
  title={Nonlinear dynamics of networks: the groupoid formalism},
  author={Martin Golubitsky and I. N. Stewart},
  journal={Bulletin of the American Mathematical Society},
  year={2006},
  volume={43},
  pages={305-364}
}
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… 

Recent advances in symmetric and network dynamics.

TLDR
The role of symmetry-breaking in the creation of patterns is emphasized, and synchrony and phase relations in network dynamics are interpreted as a type of pattern, in which space is discretized into finitely many nodes, while time remains continuous.

Periodic dynamics of coupled cell networks I: rigid patterns of synchrony and phase relations

It has recently been proved by Golubitsky and coworkers that in any network of coupled dynamical systems, the possible ‘rigid’ patterns of synchrony of hyperbolic equilibria are determined by purely

Symmetries of Quotient Networks for Doubly Periodic Patterns on the Square Lattice

Patterns of synchrony in networks of coupled dynamical systems can be represented as colorings of the nodes, in which nodes of the same color are synchronous. Balanced colorings, where nodes of the

Dynamics of Coupled Cell Networks: Synchrony, Heteroclinic Cycles and Inflation

TLDR
Focussing on transitive networks that have only one type of cell (identical cell networks), this work addresses three questions relating the network structure to dynamics, and investigates how the dynamics of coupled cell networks with different structures and numbers of cells can be related.

A Formal Setting for Network Dynamics

This chapter is an introduction to coupled cell networks, a formal setting in which to analyse general features of dynamical systems that are coupled together in a network. Such networks are common

Symmetry and Synchrony in Coupled Cell Networks 1: Fixed-Point Spaces

TLDR
It is proved the "folk theorem" that in any Γ-equivariant dynamical system on Rk the only flow-invariant subspaces are the fixed-point spaces of subgroups of Γ.

Nonlinear Network Dynamics with Consensus–Dissensus Bifurcation

TLDR
A nonlinear dynamical system on networks inspired by the pitchfork bifurcation normal form classifies the stability of stationary states in terms of the effective resistances of the underlying graph; this classification clearly discerns the influence of the specific topology in which the local pitchfork systems are interconnected.

Strange Dynamics in a Fractional Derivative of Complex-Order Network of Chaotic Oscillators

  • C. Pinto
  • Physics
    Int. J. Bifurc. Chaos
  • 2015
TLDR
The peculiar dynamical features of a fractional derivative of complex-order network of coupled cell system, composed of two unidirectional rings of cells, coupled through a "buffer" cell, are studied.

Nonlinear Laplacian Dynamics: Symmetries, Perturbations, and Consensus

In this paper, we study a class of dynamic networks called Absolute Laplacian Flows under small perturbations. Absolute Laplacian Flows are a type of nonlinear generalisation of classical linear
...

References

SHOWING 1-10 OF 101 REFERENCES

Interior symmetry and local bifurcation in coupled cell networks

A coupled cell system is a network of dynamical systems, or ‘cells’, coupled together. Such systems can be represented schematically by a directed graph whose nodes correspond to cells and whose

Periodic dynamics of coupled cell networks II: cyclic symmetry

A coupled cell network is a directed graph whose nodes represent dynamical systems and whose directed edges specify how those systems are coupled to each other. The typical dynamic behaviour of a

Symmetry Groupoids and Admissible Vector Fields for Coupled Cell Networks

The space of admissible vector fields, consistent with the structure of a network of coupled dynamical systems, can be specified in terms of the network's symmetry groupoid. The symmetry groupoid

Symmetry Groupoids and Patterns of Synchrony in Coupled Cell Networks

A coupled cell system is a network of dynamical systems, or "cells," coupled together. Such systems can be represented schematically by a directed graph whose nodes correspond to cells and whose

Periodic dynamics of coupled cell networks I: rigid patterns of synchrony and phase relations

It has recently been proved by Golubitsky and coworkers that in any network of coupled dynamical systems, the possible ‘rigid’ patterns of synchrony of hyperbolic equilibria are determined by purely

The dynamics ofn weakly coupled identical oscillators

SummaryWe present a framework for analysing arbitrary networks of identical dissipative oscillators assuming weak coupling. Using the symmetry of the network, we find dynamically invariant regions in

Rotation , Oscillation and Spike Numbers in Phase Oscillator Networks

We study networks of coupled phase oscillators and show that network architecture can force relations between average frequencies of the oscillators. The main tool of our analysis is coupled cell

Winding Numbers and Average Frequencies in Phase Oscillator Networks

TLDR
Coupled cell theory provides a direct way of testing how coevolving oscillators form collections with closely related dynamics and is given a generalization to synchronous clusters of phase oscillators using quotient networks.

Some Curious Phenomena in Coupled Cell Networks

Abstract We discuss several examples of synchronous dynamical phenomena in coupled cell networks that are unexpected from symmetry considerations, but are natural using a theory developed by

Permissible symmetries of coupled cell networks

  • P. AshwinP. Stork
  • Mathematics
    Mathematical Proceedings of the Cambridge Philosophical Society
  • 1994
Abstract We consider coupled sets of identical cells and address the problem of which symmetries are permissible in such networks. For example, n linearly coupled cells with one independent variable
...