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Clusters in nonsmooth oscillator networks

Nicks, Rachel; Chambon, Lucie; Coombes, Stephen

Authors

Lucie Chambon lucie.chambon@inria.fr



Abstract

For coupled oscillator networks with Laplacian coupling the master stability function (MSF) has proven a particularly powerful tool for assessing the stability of the synchronous state. Using tools from group theory this approach has recently been extended to treat more general cluster states. However, the MSF and its generalisations require the determination of a set of Floquet multipliers from variational equations obtained by linearisation around a periodic orbit. Since closed form solutions for periodic orbits are invariably hard to come by the framework is often explored using numerical techniques. Here we show that further insight into network dynamics can be obtained by focusing on piecewise linear (PWL) oscillator models. Not only do these allow for the explicit construction of periodic orbits, their variational analysis can also be explicitly performed. The price for adopting such nonsmooth systems is that many of the notions from smooth dynamical systems, and in particular linear stability, need to be modified to take into account possible jumps in the components of Jacobians. This is naturally accommodated with the use of \textit{saltation} matrices. By augmenting the variational approach for studying smooth dynamical systems with such matrices we show that, for a wide variety of networks that have been used as models of biological systems, cluster states can be explicitly investigated. By way of illustration we analyse an integrate-and-fire network model with event-driven synaptic coupling as well as a diffusively coupled network built from planar PWL nodes, including a reduction of the popular Morris--Lecar neuron model. We use these examples to emphasise that the stability of network cluster states can depend as much on the choice of single node dynamics as it does on the form of network structural connectivity. Importantly the procedure that we present here, for understanding cluster synchronisation in networks, is valid for a wide variety of systems in biology, physics, and engineering that can be described by PWL oscillators.

Citation

Nicks, R., Chambon, L., & Coombes, S. (2018). Clusters in nonsmooth oscillator networks. Physical Review E, 97(3), https://doi.org/10.1103/PhysRevE.97.032213

Journal Article Type Article
Acceptance Date Mar 6, 2018
Online Publication Date Mar 23, 2018
Publication Date Mar 30, 2018
Deposit Date Mar 12, 2018
Publicly Available Date Mar 23, 2018
Journal Physical Review E
Print ISSN 2470-0045
Electronic ISSN 2470-0053
Publisher American Physical Society
Peer Reviewed Peer Reviewed
Volume 97
Issue 3
DOI https://doi.org/10.1103/PhysRevE.97.032213
Keywords Master Stability Function, Oscillator networks, Nonsmooth dynamics, Group theory
Public URL http://eprints.nottingham.ac.uk/id/eprint/50363
Publisher URL https://journals.aps.org/pre/abstract/10.1103/PhysRevE.97.032213
Copyright Statement Copyright information regarding this work can be found at the following address: http://eprints.nottingh.../end_user_agreement.pdf
Additional Information Nicks, R.; Chambon L.; Coombes, S. Clusters in nonsmooth oscillator networks. Physical Review E. 2018 Mar 23; 97(3): 032213 doi:10.1103/PhysRevE.97.032213

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Copyright Statement
Copyright information regarding this work can be found at the following address: http://eprints.nottingham.ac.uk/end_user_agreement.pdf





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