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Lyapunov functions for Morse-Smale synchronisation diffeomorphisms

Jorge Buescu, Henrique M. Oliveira

TL;DR

The article addresses the problem of synchronisation for three identically coupled clocks by modelling the phase-difference dynamics with a two-dimensional diffeomorphism on the torus $T^2$. It constructs discrete Lyapunov functions tailored to a Morse–Smale structure, proving the existence of two asymptotically stable fixed points at $(\frac{2\pi}{3},\frac{4\pi}{3})$ and $(\frac{4\pi}{3},\frac{2\pi}{3})$ with open basins of attraction and full Lebesgue measure of convergence. The results remain robust under small $C^1$ perturbations (non-identical clocks) via topological conjugacy, confirming structural stability and generic synchronisation. The paper also details symmetry properties, provides a rigorous analysis of the orbital derivative, and discusses extensions to nearest-neighbour interaction and potential generalisations to larger clock networks, including a conjecture for a complete Lyapunov function on $\mathbb{T}^2$.

Abstract

This paper investigates the dynamical system governing the phase differences between three identical oscillators arranged symmetrically and coupled by burst interactions. By constructing a discrete Lyapunov function, we prove the existence of two asymptotically stable fixed points on the 2-torus T^2, which correspond to Huygens synchronisation of three clocks. The locked states have phase differences of (2 pi/3,4 pi/3) and (4 pi/3,2 pi/3). Each fixed point possesses an open basin of attraction. The closure of the union of the basins of attraction of the two asymptotically stable attractors is the torus T^2, implying that Huygens synchronisation occurs generically and with full Lebesgue measure with respect to initial conditions.

Lyapunov functions for Morse-Smale synchronisation diffeomorphisms

TL;DR

The article addresses the problem of synchronisation for three identically coupled clocks by modelling the phase-difference dynamics with a two-dimensional diffeomorphism on the torus . It constructs discrete Lyapunov functions tailored to a Morse–Smale structure, proving the existence of two asymptotically stable fixed points at and with open basins of attraction and full Lebesgue measure of convergence. The results remain robust under small perturbations (non-identical clocks) via topological conjugacy, confirming structural stability and generic synchronisation. The paper also details symmetry properties, provides a rigorous analysis of the orbital derivative, and discusses extensions to nearest-neighbour interaction and potential generalisations to larger clock networks, including a conjecture for a complete Lyapunov function on .

Abstract

This paper investigates the dynamical system governing the phase differences between three identical oscillators arranged symmetrically and coupled by burst interactions. By constructing a discrete Lyapunov function, we prove the existence of two asymptotically stable fixed points on the 2-torus T^2, which correspond to Huygens synchronisation of three clocks. The locked states have phase differences of (2 pi/3,4 pi/3) and (4 pi/3,2 pi/3). Each fixed point possesses an open basin of attraction. The closure of the union of the basins of attraction of the two asymptotically stable attractors is the torus T^2, implying that Huygens synchronisation occurs generically and with full Lebesgue measure with respect to initial conditions.

Paper Structure

This paper contains 18 sections, 24 theorems, 78 equations, 6 figures, 1 table.

Table of Contents

  1. Introduction and preparatory results
  2. Motivation and organisation of this article
  3. Discrete Lyapunov functions
  4. Structural stability and Morse-Smale systems
  5. The synchronisation diffeomorphism for three equidistant clocks
  6. Identical clocks
  7. Non-identical clocks
  8. Lyapunov function for $G$ and $\tilde{G}$
  9. Equivariance
  10. Technical details of the proof of Lyapunov Theorems
  11. Symmetry
  12. Orbital derivative
  13. Analysis of the signs of $\psi_{i}\left( x,y\right)$, $i=1,2,3,4$
  14. Sign of the orbital derivative
  15. Full picture
  16. ...and 3 more sections

Key Result

Theorem 1

If there exists a scalar function $V(x)$, called Lyapunov function, for the dynamical system and equilibrium point mentioned in Definition def:LyapDiff, satisfying the following conditions: then the equilibrium point $x_0$ is Lyapunov stable. If, additionally, $\frac{dV \left(x\left(t\right)\right)}{dt} < 0$ for all $x \neq x_0$, then $x_0$ is asymptotically stable.

Figures (6)

  • Figure 1: System of three symmetrically coupled clocks.
  • Figure 2: A planar representation of the torus $\mathbb{T}^2$ using coordinates $x$ and $y$. In light gray, we display the streamlines of the dynamical system. The opposite edges correspond to the same vertical and horizontal sections on the torus via the canonical identification map.
  • Figure 3: The $\left( \mathbb{T}^2, G \right)$ flow on the torus in two perspectives.
  • Figure 4: The $\left( \mathbb{T}^2, \tilde{G} \right)$ flow in two perspectives.
  • Figure 5: We show here the set $D$, its subdivisions into subsets and the phase portrait showing heteroclinic connections. Our focus is on the light shaded region $\overline{S}=T_1 \cup T_2$ at the upper part of the figure, as the darker region $\overline{R}=T_3 \cup T_4$ is related by symmetry to the former. The Lyapunov function $V$ is applied to the upper open triangle $S=\text{int} \overline{S}$ and its orbital derivative in that region is negative except at the fixed point $\left( \frac{2 \pi}{3}, \frac{4 \pi}{3} \right)$.
  • ...and 1 more figures

Theorems & Definitions (56)

  • Definition 1
  • Theorem 1: Lyapunov Stability Theorem
  • Definition 2
  • Definition 3: Discrete orbital derivative
  • Definition 4: Discrete Lyapunov function
  • Theorem 2: Discrete Lyapunov Stability Theorem
  • Definition 5
  • Theorem 3
  • Definition 6
  • Definition 7
  • ...and 46 more