A Lyapunov function for a Synchronisation diffeomorphism of three clocks

TL;DR

This paper constructs a Lyapunov function to analyze the stability of a synchronization diffeomorphism of three clocks, demonstrating robust phase opposition synchronization with a well-defined basin of attraction.

Contribution

It introduces a systematic method for constructing a Lyapunov function for the three-clock synchronization system, confirming stability and robustness of the synchronized state.

Findings

The system has a unique asymptotically stable fixed point on the torus T2.

The basin of attraction of the stable fixed point is the entire torus T2.

Synchronization occurs with probability one from random initial conditions.

Abstract

Lyapunov functions are essential tools in dynamical systems, as they allow the stability analysis of equilibrium points without the need to explicitly solve the system's equations. Despite their importance, no systematic method exists for constructing Lyapunov functions. In a previous paper, we examined a diffeomorphism arising from the problem of Huygens Synchronisation for three identical limit cycle clocks arranged in a line, proving that the system possesses a unique asymptotically stable fixed point on the torus T2, corresponding to synchronisation in phase opposition. In this paper, we re-derive this result by constructing a discrete Lyapunov function for the system. The closure of the basin of attraction of the asymptotically stable attractor is the torus T2, showing that Huygens Synchronisation exhibits generic and robust behaviour, occurring with probability one with respect to initial conditions.