发表机构
Narva College, University of Tartu（塔尔图大学纳尔瓦学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文利用内在Kobayashi度量研究全纯动力系统的增量稳定性，建立了相关收缩判据与不变性结果，通过耦合全纯振荡器网络实验验证了方法的有效性。
AI 中文摘要
本文通过无穷小Kobayashi度量研究全纯动力系统的增量稳定性，Kobayashi度量是复流形上的内在伪度量，在全纯变换下不变，且无辅助黎曼或埃尔米特公式中固有的坐标依赖性。收缩被形式化为沿轨迹的Kobayashi度量的上Dini导数不等式；从该微分条件到指数距离收缩的过渡遵循Forni和Sepulchre的经典芬斯勒度量收缩机制，此处针对芬斯勒结构为Kobayashi度量本身的情况实例化。由于内在条件难以直接验证，通过光滑埃尔米特度量开发了实用判据，该度量是经典实矩阵收缩不等式的复埃尔米特类似物：对此类度量的收缩意味着前向不变紧子集上的内在收缩，两个概念通过显式局部等价常数相关联。在此基础上，针对拉普拉斯耦合全纯网络建立了Nagumo型不变性结果，为一类此前未以该方式处理的系统提供了前向不变性的可验证条件，且该框架可扩展至反馈控制全纯系统，其平衡态和周期轨道的结论直接源于内在收缩。耦合全纯振荡器网络的数值实验在已证明的不变集上解析验证了埃尔米特条件，且揭示观测到的同步速率显著超过该保证速率；该差距与逐节点速率和网络图拉普拉斯谱间隙的闭式组合匹配至三位小数，此处将其识别为面向网络的扩展的目标，而非完全解决。
英文摘要
This paper studies incremental stability of holomorphic dynamical systems through the infinitesimal Kobayashi metric, an intrinsic pseudometric on complex manifolds invariant under holomorphic transformations and free of the coordinate dependence inherent in auxiliary Riemannian or Hermitian formulations. Contraction is formalized as an upper Dini-derivative inequality on the Kobayashi metric along trajectories; the passage from this differential condition to exponential distance contraction follows the classical Finsler-metric contraction mechanism of Forni and Sepulchre, instantiated here for the case in which the Finsler structure is the Kobayashi metric itself. Since the intrinsic condition is difficult to verify directly, a practical criterion is developed through a smooth Hermitian metric, the complex-Hermitian analogue of the classical real matrix contraction inequality: contraction with respect to such a metric implies intrinsic contraction on forward-invariant compact subsets, the two notions related through explicit local equivalence constants. Building on this, a Nagumo-type invariance result is established for Laplacian-coupled holomorphic networks, giving verifiable conditions for forward invariance in a class of systems not previously treated this way, and the framework extends to feedback-controlled holomorphic systems, with consequences for equilibria and periodic orbits following directly from intrinsic contraction. Numerical experiments on a network of coupled holomorphic oscillators verify the Hermitian condition analytically on a proven invariant set, and reveal that the observed synchronization rate substantially exceeds this guaranteed rate; the gap matches, to three decimal places, a closed-form combination of the node-wise rate and the network graph-Laplacian spectral gap, identified here as a target for a network-aware extension rather than resolved in full.
