发表机构
School of Mathematical Sciences, Dalian University of Technology（大连理工大学数学科学学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究探讨非对称Kuramoto网络中舞蹈平衡点的存在性与稳定性，引入两类公平划分，针对前向m-邻居模型给出存在性判据，结合离散傅里叶变换等明确其稳定与不稳定条件。
AI 中文摘要
我们研究非对称耦合Kuramoto网络中的非零频率锁相运动，这类运动是具有固定相位差和非零公共角速度的相对平衡点，我们称之为舞蹈平衡点。其存在性要求所有耦合和具有相同的非零值。我们证明对称耦合或非循环关联有向图均无法支撑舞蹈平衡点。我们引入结构公平划分和q-扭曲状态公平划分，针对所得的类常数轮廓证明了基于划分的判据，其中标准带标记q-扭曲轮廓可从单元素划分中恢复。对于前向m-邻居模型，我们通过精确的不可整除性判据刻画其存在性。在公共相移方向模下研究稳定性：对于一般有向网络，强连通性和边相位差位于(-π/2, π/2)内意味着局部轨道指数稳定性，并给出包含在局部吸引域内的显式正不变集；对于任意扭曲指数，该收缩论证给出具有显式正不变邻域的低缠绕稳定 regime。对于前向模型的每个现有q-扭曲分支，离散傅里叶变换判据表明，当所有非零傅里叶模式因子为正时，该分支具有局部轨道指数稳定性；当至少一个因子为负时，具有非线性不稳定性，不稳定情形下的证明构造了显式逃逸实傅里叶扰动。我们进一步针对任意扭曲指数，结合常数N、m和q推导了额外的显式稳定与不稳定范围；对于前两个扭曲分支，更精细的论证给出q=1时的第一模式转变判据，以及q=2时的完整有限大小分类，每个分支中的退化情形均单独处理。
英文摘要
We study nonzero-frequency phase-locked motions in asymmetrically coupled Kuramoto networks. Such motions are relative equilibria with fixed phase differences and a nonzero common angular velocity, and we call them dancing equilibria. Their existence requires all coupling sums to have the same nonzero value. We show that neither symmetric coupling nor an acyclic associated digraph can support a dancing equilibrium. We introduce structurally equitable and $q$-twisted state equitable partitions and prove a partition-based criterion for the resulting class-constant profiles, with standard labeled $q$-twisted profiles recovered from singleton partitions. For the forward $m$-neighbor model, we characterize existence by an exact indivisibility criterion. Stability is studied modulo the common phase-shift direction. For general directed networks, strong connectivity and edgewise phase differences in $\left(-π/2,π/2\right)$ imply local orbital exponential stability and yield an explicit positively invariant set contained in the local basin of attraction. For arbitrary twisted indices, this contraction argument gives a low-winding stability regime with explicit positively invariant neighborhoods. For each existing $q$-twisted branch of the forward model, a discrete Fourier transform criterion yields local orbital exponential stability when all nonzero Fourier-mode factors are positive and nonlinear instability when at least one is negative. In the unstable case, the proof constructs explicit escaping real Fourier perturbations. We further derive additional explicit stability and instability ranges for arbitrary twisted indices in terms of constants $N$, $m$, and $q$. For the first two twisted branches, sharper arguments yield a first-mode transition criterion for $q=1$ and a complete finite-size classification for $q=2$, with the degenerate case in each branch handled separately.
