Abstract
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 topologies. We use this to determine conditions for network induced instabilities. Direct numerical simulations are shown to be in excellent agreement with theoretical results.
Highlights
The history of the Franklin bell is long and well established
Γ2|a| both fixed points will be between the two plates, and otherwise they will be virtual. This latter case will guarantee the existence of impacts, and is the one we focus on for the rest of the paper since it is a necessary condition for the existence of periodic orbits, and chiming in a Franklin bell
Franklin bells have provided the blueprint for numerous electro-mechanical impact oscillators (Asano, 1975; Isacsson et al, 1998; Disna Jayampathi Karunanayake & Hoshino, 2010; Fig. 13. (A) master stability function (MSF) together with the values of ηl for the directed network shown in Fig. 11 for μ−1 = 1.9. (B) Time evolution of v1, v2 and v3
Summary
The history of the Franklin bell is long and well established. named after the American scientist Benjamin Franklin it was invented by the Scottish Benedictine monk Andrew Gordon in Erfurt, Germany, around 1742. This means that the stability of the synchronous orbit can be assessed in terms of a set of lower dimensional Floquet problems parameterized by the (possibly complex) eigenvalues of the network connectivity matrix This method has been extended to treat diffusively coupled networks of non-smooth Filippov type (Filippov, 1988) and integrate-and-fire piecewise linear (PWL) oscillator models (Coombes & Thul, 2016; Nicks et al, 2018; Lai et al, 2018), making use of saltation operators. As well as developing the mathematical techniques for handling a truly non-smooth Franklin bell network, we further introduce a new form of smoothing that circumvents the need for constructing saltation operators At heart, this technique introduces a virtual linear dynamical system that smoothly connects the orbits before and after impact.
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