A Lyapunov function for a Synchronisation diffeomorphism of three clocks

TL;DR

The paper constructs a discrete Lyapunov function for the Synchronisation diffeomorphism of three clocks arranged in a line with nearest-neighbour coupling, establishing that the phase-opposition state $(\pi,\pi)$ on $\mathbb{T}^2$ is the unique asymptotically stable fixed point with an open basin $S$. By carefully analyzing the signs of auxiliary functions $\xi_i$ and the orbital derivative $\dot{V}$ within a symmetry-adapted region $A_1$ and its four symmetric counterparts, the authors show $\dot{V}<0$ except at $(\pi,\pi)$, and extend this to the full invariant set via equivariance under the symmetry maps $\Phi_j$. The result demonstrates robust, almost-sure convergence (probability one with respect to initial conditions on $\mathbb{T}^2$) to phase opposition and provides a concrete method for discrete Lyapunov function construction in topological dynamics. This work complements prior dynamical-system analyses by providing a rigorous stability certificate grounded in a continuous Lyapunov potential for a discrete-time map.

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.

Figures (7)

• Figure 1: Three clocks on a line with nearest neighbour interaction.
• Figure 2: We show here the set $A_1$ in light yellow; the heteroclinics $\eta_j$ for $j=1,\dots,8$ connecting sources and saddles, and the set $\Theta$ in red; the sectors $S_i$ such that $S=\bigcup_{i=1}^{4} S_{i}$ are also depicted.
• Figure 3: The region where the Lyapunov function is definite positive and the orbital derivative of the Lyapunov function is non-positive is shaded in blue and light yellow, with the light yellow colour representing $A_1$. This region contains the invariant open set $S$, which is enclosed by $\Theta$ (in red). We can observe the intersection of the open domain $S$ with $A_1$, denoted by $S_1$. Our study focuses on this region. The results extend to $S$ by equivariance.
• Figure 4: Complete phase diagram showing heteroclinic connections between saddle points and stable/unstable foci. The diagram was generated numerically for $a = 0.1$. Our focus is on the yellow shaded region $A_1$ at the bottom of the figure, as the other regions are equivalent by symmetry.
• Figure 5: Detail of the region $A_1$ and its subdivisions, where the three possible combinations of signs of the absolute values of $\xi_{3}$ and $\xi_{4}$ in the orbital derivative can occur.

Theorems & Definitions (41)

• Definition 1
• Definition 2: Discrete orbital derivative
• Definition 3: Discrete Lyapunov function
• Theorem 1: Discrete Lyapunov Stability Theorem
• Definition 4
• Proposition 1
• Proposition 2
• proof
• Proposition 3
• proof