### Abstract

Many systems in science and technology are networks: they consist of nodes with connections between them. Examples include electronic circuits, power grids, neuronal networks, and metabolic systems. Such networks are usually modeled by coupled nonlinear maps or differential equations, that is, as network dynamical systems. Network dynamical systems often behave very differently from regular dynamical systems that do not possess the structure of a network, and the interaction between the nodes of a network can spark surprising emergent behavior. An example is synchronization, the process by which neurons fire simultaneously and social consensus is reached. This paper is concerned with synchrony breaking, the phenomenon that less synchronous solutions emerge from more synchronous solutions as model parameters vary. It turns out that synchrony breaking often occurs via remarkable anomalous bifurcation scenarios. As an explanation for this it has been noted that homogeneous networks can be realized as quotient networks of so-called fundamental networks. The class of admissible dynamical systems for these fundamental networks is equal to the class of equivariant (symmetric) dynamical systems of the regular representation of a monoid (a monoid is an algebraic semigroup with unit). Using this geometric insight, we set up a framework for center manifold reduction in fundamental networks and their quotients. We then use this machinery to classify generic synchrony breaking bifurcations in three example networks with identical spectral properties and identical robust synchrony spaces.

Language | English |
---|---|

Pages | 121-155 |

Number of pages | 35 |

Journal | SIAM Review |

Volume | 61 |

Issue number | 1 |

Early online date | 7 Feb 2019 |

DOIs | |

Publication status | Published - Mar 2019 |

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### Keywords

- Bifurcation theory
- Center manifold reduction
- Coupled cell networks
- Dynamical systems

### Cite this

*SIAM Review*,

*61*(1), 121-155. https://doi.org/10.1137/18M1219977

}

*SIAM Review*, vol. 61, no. 1, pp. 121-155. https://doi.org/10.1137/18M1219977

**Center manifolds of coupled cell networks.** / Nijholt, Eddie; Rink, Bob; Sanders, Jan.

Research output: Contribution to Journal › Review article › Academic › peer-review

TY - JOUR

T1 - Center manifolds of coupled cell networks

AU - Nijholt, Eddie

AU - Rink, Bob

AU - Sanders, Jan

PY - 2019/3

Y1 - 2019/3

N2 - Many systems in science and technology are networks: they consist of nodes with connections between them. Examples include electronic circuits, power grids, neuronal networks, and metabolic systems. Such networks are usually modeled by coupled nonlinear maps or differential equations, that is, as network dynamical systems. Network dynamical systems often behave very differently from regular dynamical systems that do not possess the structure of a network, and the interaction between the nodes of a network can spark surprising emergent behavior. An example is synchronization, the process by which neurons fire simultaneously and social consensus is reached. This paper is concerned with synchrony breaking, the phenomenon that less synchronous solutions emerge from more synchronous solutions as model parameters vary. It turns out that synchrony breaking often occurs via remarkable anomalous bifurcation scenarios. As an explanation for this it has been noted that homogeneous networks can be realized as quotient networks of so-called fundamental networks. The class of admissible dynamical systems for these fundamental networks is equal to the class of equivariant (symmetric) dynamical systems of the regular representation of a monoid (a monoid is an algebraic semigroup with unit). Using this geometric insight, we set up a framework for center manifold reduction in fundamental networks and their quotients. We then use this machinery to classify generic synchrony breaking bifurcations in three example networks with identical spectral properties and identical robust synchrony spaces.

AB - Many systems in science and technology are networks: they consist of nodes with connections between them. Examples include electronic circuits, power grids, neuronal networks, and metabolic systems. Such networks are usually modeled by coupled nonlinear maps or differential equations, that is, as network dynamical systems. Network dynamical systems often behave very differently from regular dynamical systems that do not possess the structure of a network, and the interaction between the nodes of a network can spark surprising emergent behavior. An example is synchronization, the process by which neurons fire simultaneously and social consensus is reached. This paper is concerned with synchrony breaking, the phenomenon that less synchronous solutions emerge from more synchronous solutions as model parameters vary. It turns out that synchrony breaking often occurs via remarkable anomalous bifurcation scenarios. As an explanation for this it has been noted that homogeneous networks can be realized as quotient networks of so-called fundamental networks. The class of admissible dynamical systems for these fundamental networks is equal to the class of equivariant (symmetric) dynamical systems of the regular representation of a monoid (a monoid is an algebraic semigroup with unit). Using this geometric insight, we set up a framework for center manifold reduction in fundamental networks and their quotients. We then use this machinery to classify generic synchrony breaking bifurcations in three example networks with identical spectral properties and identical robust synchrony spaces.

KW - Bifurcation theory

KW - Center manifold reduction

KW - Coupled cell networks

KW - Dynamical systems

UR - http://www.scopus.com/inward/record.url?scp=85063811990&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85063811990&partnerID=8YFLogxK

U2 - 10.1137/18M1219977

DO - 10.1137/18M1219977

M3 - Review article

VL - 61

SP - 121

EP - 155

JO - SIAM Review

T2 - SIAM Review

JF - SIAM Review

SN - 0036-1445

IS - 1

ER -