发表机构
School of Professional Studies, Northwestern University（西北大学专业学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本研究将Kaye的黎明合唱迟滞模型推广到网络，证明弱反馈下唯一平衡，且公平划分下弱耦合可产生组合多稳态，并推导尖点分岔的标度律。
AI 中文摘要
在黎明合唱的机制模型中，Kaye 证明了异质激活阈值和由群体活跃比例决定的共享反馈信号能够产生突发的集体激活和迟滞现象。我们将这一机制扩展到相互作用的智能体网络。每个节点具有连续的激活水平，一个非负行随机矩阵决定节点活动如何贡献于个体反馈。我们证明，足够弱的反馈会产生唯一的全局吸引平衡态。对于所有反馈强度，齐次动力学精确复现 Kaye 的标量方程；因此，网络拓扑不会改变齐次分支的折叠或尖点，且没有异质模式在齐次模式之前变得不稳定。对于公平划分，网络允许一个精确的商系统，其中块内节点从每个块接收相同的聚合输入。当块之间解耦时，商系统简化为 Kaye 标量方程的独立副本。我们证明，在足够弱的块间耦合下，每个稳定的标量平衡态分配都能持续存在，产生一个组合族的稳定商平衡态，这些平衡态可提升为稳定的全网络平衡态，并在破坏精确公平性的小扰动下保持不变。对于两个对称耦合的块，具有不等块活动的分支终止于一对对称相关的尖点分岔。在双稳态起始附近，我们推导了这些分岔发生时的块间耦合、分岔时块间活动差异以及外部刺激相对于标量尖点的相应偏移的标度律。数值延拓证实了 gamma、逻辑斯蒂和正态阈值分布的标度律。
英文摘要
In a mechanistic model of the dawn chorus, Kaye showed that heterogeneous activation thresholds and a shared feedback signal determined by the population's active fraction can produce abrupt collective activation and hysteresis. We extend this mechanism to a network of interacting agents. Each node has a continuous activation level, and a nonnegative row-stochastic matrix determines how node activities contribute to individual feedback. We prove that sufficiently weak feedback yields a unique globally attracting equilibrium. For all feedback strengths, the homogeneous dynamics exactly reproduce Kaye's scalar equation; consequently, network topology does not alter the folds or cusp of the homogeneous branch, and no heterogeneous mode becomes unstable before the homogeneous mode. For equitable partitions, the network admits an exact quotient system in which nodes within a block receive the same aggregate input from every block. When blocks are uncoupled, the quotient reduces to independent copies of Kaye's scalar equation. We show that every assignment of stable scalar equilibria to blocks persists under sufficiently weak interblock coupling, producing a combinatorial family of stable quotient equilibria that lift to stable full-network equilibria and remain under small perturbations that break exact equitability. For two symmetrically coupled blocks, branches with unequal block activities terminate at a pair of symmetry-related cusp bifurcations. Near the onset of bistability, we derive scaling laws for the interblock coupling at which these bifurcations occur, the activity difference between the blocks at bifurcation, and the corresponding shift of the external stimulus from the scalar cusp. Numerical continuation confirms the scaling laws for gamma, logistic, and normal threshold distributions.
