Morse-Smale dynamics in discrete Huygens networks

Abstract

We study discrete networks of $N+1$ identical phase oscillators with positive reciprocal weights and smooth odd periodic interactions, motivated by discrete models of Huygens synchronisation. The interaction laws define an intrinsic potential, and removing the common phase gives a gradient map on the $N$-torus. A weighted graph Laplacian yields explicit step sizes for strict Lyapunov decrease and global invertibility. Our main result is a Morse--Smale criterion for maps $z\mapsto z+hK\sin z$: if $K$ is symmetric, nonsingular and weakly diagonally dominant, the map is Morse--Smale for $0<h<1/\lambda_{\max}(K)$, and this interval is optimal. The proof combines invariant coordinate slabs with a maximum principle for the adjoint variational equation. The criterion applies to unit-weight paths of any length and to the three-leaf star, and a weighted version covers paths with discretely concave weights. By structural stability, it extends to every graph containing a spanning path with sufficiently weak additional links. Conversely, we construct infinitely many trees with Morse potentials and an almost-global sink, whose maps nevertheless have non-transverse heteroclinic intersections.