Stability of coupled Huygens oscillators

The Slow Clock is the Master

“…We refer to this limit cycle as a limit circle. The theory presented in [11,12] can be applied to this case, as in those works, the limit cycles were considered to be sectionally circular.…”

“…In [12], we explored the stability of a system representing the pendulum clock, which shares similar characteristics with a pure circular limit cycle, including Arnold tongues for near multiples of frequencies.…”

“…In the book by Andronov et al [2], the authors establish the existence and stability of the limit cycle for the motion in this dynamical system. Additionally, a comprehensive review of this theory along with an analysis of Huygens synchronization involving two Andronov clocks with similar frequencies can be found in [11,12].…”

“…The authors of this present article have previously investigated the phenomenon of synchronization and its stability. Our paper [12] was developed based on the research conducted in [11], which, in turn, was founded on the ideas proposed by Vassalo Pereira [13] and Ralph Abraams [1], who explored Andronov's model [2] of the Pendulum Clock. One of the most fruitful theories describing the working mechanism of a pendulum clock is presented in the book [2] by A. Andronov and his co-authors.…”

“…When the mutual influence of symmetric oscillators is reciprocal on each other, the type of synchrony that is generally obtained is called mastermaster. This is the typical situation of identical clocks perturbing each other as seen in [11,12]. A master-slave relation means that the pace, i.e., the frequency of one of the oscillators dictates the frequency of both oscillators in the final coupled, phase-locked state of the system.…”