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Synchrony in networks of Franklin bells

Sayli, Mustafa ; Thul, Rudiger

Authors

Mustafa Sayli

Rudiger Thul



Abstract

The Franklin bell is an electro-mechanical oscillator that can generate a repeating chime in the presence of an electric field. Benjamin Franklin famously used it as a lightning detector. The chime arises from the impact of a metal ball on a metal bell. Thus, a network of Franklin bells can be regarded as a network of impact oscillators. Although the number of techniques for analysing impacting systems has grown in recent years, this has typically focused on low dimensional systems and relatively little attention has been paid to networks. Here we redress this balance with a focus on synchronous oscillatory network states. We first study a single Franklin bell, showing how to construct periodic orbits and how to determine their linear stability and bifurcation. To cope with the non-smooth nature of the impacts we use saltation operators to develop the correct Floquet theory. We further introduce a new smoothing technique that circumvents the need for saltation and that recovers the saltation operators in some appropriate limit. We then consider the dynamics of a network of Franklin bells, showing how the master stability function approach can be adapted to treat the linear stability of the synchronous state for arbitrary network topolo-gies. We use this to determine conditions for network induced instabilities. Direct numerical simulations are shown to be in excellent agreement with theoretical results.

Journal Article Type Article
Publication Date Oct 1, 2019
Journal IMA Journal of Applied Mathematics
Print ISSN 0272-4960
Electronic ISSN 1464-3634
Publisher Oxford University Press
Peer Reviewed Peer Reviewed
Volume 84
Issue 5
Article Number hxz023
Pages 1001-1021
APA6 Citation Sayli, M., Lai, Y. M., Thul, R., & Coombes, S. (2019). Synchrony in networks of Franklin bells. IMA Journal of Applied Mathematics, 84(5), 1001-1021. https://doi.org/10.1093/imamat/hxz023
DOI https://doi.org/10.1093/imamat/hxz023
Keywords Franklin bell; impacting systems; smoothing; network dynamics; synchronisation; master stability function
Publisher URL https://academic.oup.com/imamat/article/84/5/1001/5574709