AI 中文总结
本文研究离散惠更斯网络的相位振子动力学，推导映射z↦z+hK sin z的莫尔斯-斯梅尔准则，构造含非横截异宿交点的树，拓展了同步网络的动力学分析。
AI 中文摘要
受惠更斯同步离散模型的启发，我们研究具有正互权重和平滑奇周期相互作用的N+1个相同相位振子的离散网络。相互作用定律定义了一个内在势，消除公共相位后得到N维环面上的梯度映射。加权图拉普拉斯算子为严格李雅普诺夫递减和全局可逆性提供了显式步长。我们的主要结果是关于映射z↦z+hK sin z的莫尔斯-斯梅尔准则：若K对称、非奇异且弱对角占优，则当0<h<1/λ_max(K)时，该映射是莫尔斯-斯梅尔的，且此区间是最优的。证明结合了不变坐标片与伴随变分方程的最大值原理。该准则适用于任意长度的单位权重路径和三叶星，加权版本涵盖具有离散凹权重的路径。通过结构稳定性，它可扩展到每个包含生成路径且附加链接足够弱的图。反之，我们构造了无限多具有莫尔斯势和几乎全局汇的树，其映射却具有非横截异宿交点。
英文摘要
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 $0<h<1/λ_{\max}(K)$, 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.
Comments25 pages, 3 figures
