Huygens synchronisation of three clocks equidistant from each other

TL;DR

The paper studies the synchronization of three identical pendulum clocks mounted on a rigid support, coupled by symmetric, instantaneous impacts. It develops a two-dimensional discrete phase map to capture phase-difference dynamics and proves that the asymptotically locked state features mutual phase offsets of $\frac{2\pi}{3}$, with two possible rotational orientations, and an attracting set that encompasses all initial conditions. It shows that the locked-state amplitude is slightly increased by coupling, with the fixed-point amplitude given by $v^* = \frac{(4\mu+\alpha)^2+h^2}{2(4\mu+\alpha)}$ (to leading order in $\alpha$), and that the two- and three-clock systems share this behavior under weak symmetric interactions. The work generalizes Huygens synchronization to a triad of weakly interacting oscillators and lays a foundation for extending to $n$ clocks and exploring Arnold tongues and experimental validation of the theoretical predictions.

Abstract

This paper investigates the synchronization of three identical oscillators, or clocks, suspended from a common rigid support. We consider scenarios where each clock interacts with the other two, achieving synchronization through small impacts exchanged between oscillator pairs. The fundamental outcome of our study reveals that the ultimate synchronized state maintains a phase difference of $\frac{2π}{3}$ between successive clocks, either clockwise or counter-clockwise. Furthermore, these locked states exhibit an attracting set, which closure encompasses the entire initial conditions space. Our analytical approach involves constructing a nonlinear discrete dynamical system in dimension two. These findings hold significance for sets of three weakly coupled periodic oscillators engaged in mutual symmetric impact periodic interaction, irrespective of the specific oscillator models employed. Lastly, we explore the amplitude of oscillations at the final locked state in the context of two and three interacting Andronov pendulum clocks. Our analysis reveals a precise small increase in the amplitude of the locked-state oscillations, as quantified in this paper.

Figures (7)

• Figure 1: Limit cycle of an isolated clock represented as a solid curve in the phase space. The horizontal axis represents the angular position, and the vertical axis represents velocity.
• Figure 2: The three clocks hang at the three vertices of a triangle.
• Figure 3: Graph of the map $g$, which is an homeomorphism in the interval $\left[ 0,2\pi\right]$.
• Figure 4: The homeomorphism $h_{1}$ with five fixed points.
• Figure 5: The homeomorphism $h_{1}$ with two fixed points, one repeller and the other attractor.

Theorems & Definitions (2)

• Remark 1
• Remark 2