具有时滞相互作用的拓扑信号的全局同步
Global synchronization of topological signals with time-delayed interactions
- University of Namur（那慕尔大学）
- IMT School for Advanced Studies Lucca（卢卡高等研究学院）
- University of Leiden（莱顿大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文研究时滞相互作用对拓扑振子全局同步的影响，通过主稳定函数和Lambert W函数给出谱条件，证明单纯形耦合可促进或抑制同步，而Dirac耦合无法实现全局同步。
AI中文摘要：
拓扑信号是定义在高阶结构（如单纯复形或胞腔复形）上的动力学变量。在本工作中，我们研究了时滞相互作用对拓扑振子全局同步涌现的影响，并将其应用于Stuart-Landau系统。我们首先考察信号支撑在给定维度的单纯形或胞腔上并通过Hodge-Laplace矩阵耦合的情形。我们推导了主稳定函数，并证明对于合适的离散时滞值，同步解的稳定性在解析上变得可处理，并可通过使用Lambert W函数求解。该分析给出了全局拓扑同步涌现的显式谱条件，并表明允许的时滞可以促进或抑制同步。然后，我们研究了由Dirac算子耦合的不同维度单纯形或胞腔上信号之间的时滞相互作用。在类似的允许时滞下，变分问题（仍与主稳定函数相关）同样可以简化，然而在这种情况下，我们证明了全局拓扑Dirac同步不能涌现。在单纯复形和胞腔复形上的数值模拟证实并补充了这些发现。
英文摘要：
Topological signals are dynamical variables supported on higher-order structures such as simplicial or cell complexes. In this work, we investigate the impact of time-delayed interactions on the emergence of global synchronization of topological oscillators, with application to the Stuart-Landau system. We first examine the case where signals are supported on simplexes, or cells, of a given dimension and coupled through the Hodge-Laplace matrix. We derive the Master Stability Function and we prove that for suitable discrete delay values, the stability of the synchronous solution becomes analytically tractable and can be solved by using the Lambert W-function. This analysis yields explicit spectral conditions for Global Topological Synchronization to emerge and shows that admissible delays may either promote or suppress synchronization. We then study delayed interactions between signals supported on simplexes, or cells, of different dimensions coupled by the Dirac operator. Under analogous admissible time delays the variational problem, still related to the Master Stability Function, can again be simplified, however in this case we prove that Global Topological Dirac Synchronization cannot emerge. Numerical simulations on simplicial and cell complexes corroborate and complement these findings.
