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Synchrony in networks of coupled nonsmooth dynamical systems: extending the master stability function

Coombes, Stephen; Thul, Rudiger



The master stability function is a powerful tool for determining synchrony in high-dimensional networks of coupled limit cycle oscillators. In part, this approach relies on the analysis of a low-dimensional variational equation around a periodic orbit. For smooth dynamical systems, this orbit is not generically available in closed form. However, many models in physics, engineering and biology admit to non-smooth piece-wise linear caricatures, for which it is possible to construct periodic orbits without recourse to numerical evolution of trajectories. A classic example is the McKean model of an excitable system that has been extensively studied in the mathematical neuroscience community. Understandably, the master stability function cannot be immediately applied to networks of such non-smooth elements. Here, we show how to extend the master stability function to non-smooth planar piece-wise linear systems, and in the process demonstrate that considerable insight into network dynamics can be obtained. In illustration, we highlight an inverse period-doubling route to synchrony, under variation in coupling strength, in globally linearly coupled networks for which the node dynamics is poised near a homoclinic bifurcation. Moreover, for a star graph, we establish a mechanism for achieving so-called remote synchronisation (where the hub oscillator does not synchronise with the rest of the network), even when all the oscillators are identical. We contrast this with node dynamics close to a non-smooth Andronov–Hopf bifurcation and also a saddle node bifurcation of limit cycles, for which no such bifurcation of synchrony occurs.


Coombes, S., & Thul, R. (2016). Synchrony in networks of coupled nonsmooth dynamical systems: extending the master stability function. European Journal of Applied Mathematics, 27(6), 904-922.

Journal Article Type Article
Acceptance Date Feb 25, 2016
Online Publication Date Mar 28, 2016
Publication Date Dec 1, 2016
Deposit Date Aug 13, 2018
Publicly Available Date Dec 17, 2018
Print ISSN 0956-7925
Electronic ISSN 1469-4425
Publisher Cambridge University Press (CUP)
Peer Reviewed Peer Reviewed
Volume 27
Issue 6
Pages 904-922
Keywords General applied mathematics, Synchronisation, Non-smooth equations, Complex networks, Neural networks
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