# Huygens synchronisation of three clocks equidistant from each other

## TL;DR

This paper analyzes the synchronization behavior of three identical clocks interacting through impacts, revealing a stable phase difference of 120 degrees and examining amplitude effects in coupled pendulum clocks.

## Contribution

It introduces a nonlinear discrete dynamical system model for three impact-coupled oscillators and characterizes their synchronized states and amplitude variations.

## Key findings

- Final synchronized state has a phase difference of 2π/3.
- The synchronized states form an attracting set covering all initial conditions.
- Amplitude of oscillations slightly increases at the locked state.

## 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 $\frac{2\pi}{3}$ 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.