Synchrony in networks of Franklin bells

Sayli, Mustafa; Lai, Yi Ming; Thul, Rudiger; Coombes, Stephen

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.

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

• Journal Article Type: Article
• Acceptance Date: Aug 15, 2019
• Online Publication Date: Sep 27, 2019
• Publication Date: Oct 1, 2019
• Deposit Date: Aug 23, 2019
• Publicly Available Date: Apr 20, 2020
• 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
• DOI: https://doi.org/10.1093/imamat/hxz023
• Keywords: Franklin bell; impacting systems; smoothing; network dynamics; synchronisation; master stability function
• Public URL: https://nottingham-repository.worktribe.com/output/2438752
• Publisher URL: https://academic.oup.com/imamat/article/84/5/1001/5574709
• Contract Date: Aug 23, 2019