### Abstract

We define a graph network to be a coupled cell network where there are only one type of cell and one type of symmetric coupling between the cells. For a difference-coupled vector field on a graph network system, all the cells have the same internal dynamics, and the coupling between cells is identical, is symmetric, and depends only on the difference of the states of the interacting cells. We define four nested sets of difference-coupled vector fields by adding further restrictions on the internal dynamics and the coupling functions. These restrictions require that these functions preserve zero or are odd or linear. We characterize the synchrony and antisynchrony subspaces with respect to these four subsets of admissible vector fields. Synchrony and antisynchrony subspaces are determined by partitions and matched partitions of the cells that satisfy certain balance conditions. We compute the lattice of synchrony and antisynchrony subspaces for some examples of graph networks. We also apply our theory to systems of coupled van der Pol oscillators.

Original language | English (US) |
---|---|

Pages (from-to) | 904-938 |

Number of pages | 35 |

Journal | SIAM Journal on Applied Dynamical Systems |

Volume | 18 |

Issue number | 2 |

DOIs | |

State | Published - Jan 1 2019 |

### Fingerprint

### Keywords

- Bifurcation
- Coupled systems
- Synchrony

### ASJC Scopus subject areas

- Analysis
- Modeling and Simulation

### Cite this

**Synchrony and antisynchrony for difference-coupled vector fields on graph network systems.** / Neuberger, John M.; Sieben, Nándor; Swift, James W.

Research output: Contribution to journal › Article

}

TY - JOUR

T1 - Synchrony and antisynchrony for difference-coupled vector fields on graph network systems

AU - Neuberger, John M.

AU - Sieben, Nándor

AU - Swift, James W.

PY - 2019/1/1

Y1 - 2019/1/1

N2 - We define a graph network to be a coupled cell network where there are only one type of cell and one type of symmetric coupling between the cells. For a difference-coupled vector field on a graph network system, all the cells have the same internal dynamics, and the coupling between cells is identical, is symmetric, and depends only on the difference of the states of the interacting cells. We define four nested sets of difference-coupled vector fields by adding further restrictions on the internal dynamics and the coupling functions. These restrictions require that these functions preserve zero or are odd or linear. We characterize the synchrony and antisynchrony subspaces with respect to these four subsets of admissible vector fields. Synchrony and antisynchrony subspaces are determined by partitions and matched partitions of the cells that satisfy certain balance conditions. We compute the lattice of synchrony and antisynchrony subspaces for some examples of graph networks. We also apply our theory to systems of coupled van der Pol oscillators.

AB - We define a graph network to be a coupled cell network where there are only one type of cell and one type of symmetric coupling between the cells. For a difference-coupled vector field on a graph network system, all the cells have the same internal dynamics, and the coupling between cells is identical, is symmetric, and depends only on the difference of the states of the interacting cells. We define four nested sets of difference-coupled vector fields by adding further restrictions on the internal dynamics and the coupling functions. These restrictions require that these functions preserve zero or are odd or linear. We characterize the synchrony and antisynchrony subspaces with respect to these four subsets of admissible vector fields. Synchrony and antisynchrony subspaces are determined by partitions and matched partitions of the cells that satisfy certain balance conditions. We compute the lattice of synchrony and antisynchrony subspaces for some examples of graph networks. We also apply our theory to systems of coupled van der Pol oscillators.

KW - Bifurcation

KW - Coupled systems

KW - Synchrony

UR - http://www.scopus.com/inward/record.url?scp=85073641658&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85073641658&partnerID=8YFLogxK

U2 - 10.1137/18M1186769

DO - 10.1137/18M1186769

M3 - Article

AN - SCOPUS:85073641658

VL - 18

SP - 904

EP - 938

JO - SIAM Journal on Applied Dynamical Systems

JF - SIAM Journal on Applied Dynamical Systems

SN - 1536-0040

IS - 2

ER -