Huygens synchronization of three clocks equidistant from each other

• Published in: Nonlinear dynamics Vol. 112; no. 5; pp. 3303 - 3317
• Main Authors: D’Aniello, Emma, Oliveira, Henrique M.
• Format: Journal Article
• Language: English
• Published: Dordrecht Springer Netherlands 01.03.2024; Springer Nature B.V
• Subjects: Amplitudes; Automotive Engineering; Classical Mechanics; Clocks; Clocks & watches; Control; Dynamical Systems; Energy; Engineering; Friction; Initial conditions; Mechanical Engineering; Original Paper; Oscillations; Oscillators; Synchronism; Vibration; Stability; Primary 34D06; Secondary 37E30; Andronov pendulum clocks; Mutual symmetric impact interaction; Amplitude increase in locked-state oscillations; Synchronization of oscillators
• ISSN: 0924-090X; 1573-269X
• DOI: 10.1007/s11071-023-09241-9

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 2 π 3 between successive clocks, either clockwise or counter-clockwise. Furthermore, these locked states exhibit an attracting set, the closure of which 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 change in the amplitude of the locked-state oscillations, as quantified in this paper.