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Huygens Synchronization of Three Aligned Clocks

Jorge Buescu, Emma D'Aniello, Henrique M. Oliveira

TL;DR

This work analyzes synchronization in a linear array of three identical oscillators with nearest-neighbor coupling by deriving a planar discrete diffeomorphism for the phase differences $(x,y)$. The authors establish a diffeomorphic, perturbative map $F$, study its invariants and symmetries, and reduce the global dynamics to a fundamental square $S_{10}$ via equivariance; within this domain they prove the existence of heteroclinic separatrices that partition the space into two basins, each converging to an attractor at $(\pi,\pi)$, corresponding to a final state of alternating phase opposition. Numerical evidence at $a=0.1$ supports the theoretical picture, and symmetry arguments extend the local dynamics to the whole plane, yielding a 2π-periodic tiling of invariant regions. The results provide a tractable framework for weakly interacting oscillator chains and suggest extensions to oscillator strings and to multi-agent swarms or neural networks where nearest-neighbor coupling dominates.

Abstract

This study examines the synchronization of three identical oscillators arranged in an array and coupled by small impacts, wherein each oscillator interacts solely with its nearest neighbor. The synchronized state, which is asymptotically stable, is characterized by phase opposition among alternating oscillators. We analyze the system using a non-linear discrete dynamical system based on a difference equation derived from the iteration of a plane diffeomorphism. We illustrate these results with the application to a system of three aligned Andronov clocks, showcasing their applicability to a broad range of oscillator systems.

Huygens Synchronization of Three Aligned Clocks

TL;DR

This work analyzes synchronization in a linear array of three identical oscillators with nearest-neighbor coupling by deriving a planar discrete diffeomorphism for the phase differences . The authors establish a diffeomorphic, perturbative map , study its invariants and symmetries, and reduce the global dynamics to a fundamental square via equivariance; within this domain they prove the existence of heteroclinic separatrices that partition the space into two basins, each converging to an attractor at , corresponding to a final state of alternating phase opposition. Numerical evidence at supports the theoretical picture, and symmetry arguments extend the local dynamics to the whole plane, yielding a 2π-periodic tiling of invariant regions. The results provide a tractable framework for weakly interacting oscillator chains and suggest extensions to oscillator strings and to multi-agent swarms or neural networks where nearest-neighbor coupling dominates.

Abstract

This study examines the synchronization of three identical oscillators arranged in an array and coupled by small impacts, wherein each oscillator interacts solely with its nearest neighbor. The synchronized state, which is asymptotically stable, is characterized by phase opposition among alternating oscillators. We analyze the system using a non-linear discrete dynamical system based on a difference equation derived from the iteration of a plane diffeomorphism. We illustrate these results with the application to a system of three aligned Andronov clocks, showcasing their applicability to a broad range of oscillator systems.
Paper Structure (12 sections, 11 theorems, 41 equations, 8 figures)

This paper contains 12 sections, 11 theorems, 41 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Model for Three Aligned Clocks
  3. Basic Analysis of the Model
  4. Jacobian Matrix
  5. Nullclines
  6. Invariance and symmetry
  7. Invariant lines
  8. Equivariance
  9. Symmetry
  10. Simplification of the dynamics
  11. The study of the dynamics restricted to $S_{10}$
  12. Conclusion

Key Result

Lemma 1

Consider the one-parameter families of straight lines $\Gamma_{k}^{+} = \{ (x,y): y=x+2k\pi \}_{k\in\mathbb{Z}}$ and $\Gamma_{l}^{-} = \{ (x,y): y=-x+2l\pi \}_{l\in\mathbb{Z}}$. Then:

Figures (8)

  • Figure 1: Nullclines of the field $F$ for $a=0.1$.
  • Figure 2: Invariant lines near $\left( \pi,\pi\right) .$
  • Figure 3: The four squares.
  • Figure 4: The square $S_{10}$.
  • Figure 5: Heteroclinic curves connecting $\left( 0,0\right)$ with $\left( \pi,0\right)$ and $\left( 2\pi,0\right)$ with $\left( \pi,0\right)$. These heteroclinics are precisely the stable manifold of the saddle $\left( \pi,0\right)$. The heteroclinics above were obtained numerically with parameter $a=0.1$.
  • ...and 3 more figures

Theorems & Definitions (23)

  • Lemma 1
  • proof
  • Proposition 2
  • Proposition 3
  • proof
  • Proposition 4
  • proof
  • Proposition 5
  • proof
  • Proposition 6
  • ...and 13 more