发表机构
Indian Statistical Institute（印度统计研究所）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究针对单纯形Kuramoto模型，通过Hodge分解解析推导了临界耦合强度以上的定点状态，发现系统脆弱性随拓扑子空间维度超线性增长。
AI 中文摘要
同步通常被理解为相互作用的动力学单元之间出现锁相的集体状态。然而，将这一概念扩展到动力学变量定义在高维单纯形上的系统时，会引入来自底层单纯复形拓扑的全新约束。在单纯形Kuramoto模型中，非平凡拓扑循环会产生高维调和子空间，该子空间不受耦合影响，因此可以无限漂移，从而阻止全局同步状态的出现。为解决这一问题，我们对单纯形Kuramoto动力学应用Hodge分解并研究其各分量。尽管存在这种拓扑阻碍，我们证明当耦合强度超过临界值时，单纯形Kuramoto模型在Hodge分解的恰当和余恰当扇区中存在定点状态，我们对这两个扇区都进行了解析推导。这种分解提供了同步的广义概念，其中非调和分量收敛到固定状态。我们进一步研究这些定点状态对外部扰动的鲁棒性。排除漂移且无相互作用的调和分量后，扰动响应由加权拉普拉斯谱控制，并恢复出与标准Kuramoto情况类似的基尔霍夫指数依赖性。相比之下，调和扇区是非耗散的，因此系统的脆弱性随该拓扑子空间的维度增加而增大。我们使用具有不同第一贝蒂数的三角剖分环面对该效应进行数值表征，发现系统脆弱性与调和子空间维度之间存在超线性标度关系。
英文摘要
Synchronization is conventionally understood as the emergence of a phase-locked collective state among interacting dynamical units. However, extending this notion to systems whose dynamical variables are defined on higher-dimensional simplices introduces fundamentally new constraints arising from the topology of the underlying simplicial complex. In the simplicial Kuramoto model, nontrivial topological cycles give rise to a higher dimensional harmonic subspace that is unaffected by the coupling and can therefore drift indefinitely, preventing a globally synchronized state. To resolve this we employ Hodge decomposition on simplicial Kuramoto dynamics and investigated the components. Despite this topological obstruction, we show that the simplicial Kuramoto model admits fixed-point states in the exact and coexact sectors of the Hodge decomposition above critical coupling strengths, which we derive analytically for both sectors. This decomposition provides a generalized notion of synchronization in which the non-harmonic components converge to fixed states. We further investigate the robustness of these fixed-point states to external perturbations. Excluding the drifting and non-interacting harmonic component, the perturbation response is governed by the spectrum of weighted Laplacian and recovers a Kirchhoff index dependence analogous to standard Kuramoto case. In contrast, the harmonic sector is non-dissipative, and consequently the fragility of the system increases with the dimension of this topological subspace. We characterize this effect numerically using triangulated tori with varying first Betti number and find a superlinear scaling relation between system fragility and the dimension of the harmonic sector.
Comments14 pages, 11 figures
