Clusters in nonsmooth oscillator networks

Nicks, Rachel and Chambon, Lucie and Coombes, Stephen (2018) Clusters in nonsmooth oscillator networks. Physical Review E . (In Press)

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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.

Item Type: Article
Keywords: Master Stability Function, Oscillator networks, Nonsmooth dynamics, Group theory
Schools/Departments: University of Nottingham, UK > Faculty of Science > School of Mathematical Sciences
Depositing User: Coombes, Prof Stephen
Date Deposited: 12 Mar 2018 09:15
Last Modified: 12 Mar 2018 11:56

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