多维 Kuramoto 模型中的广义阻挫
Generalized frustration in the multidimensional Kuramoto model
AI总结：
本文将含矩阵耦合的广义阻挫 Kuramoto 模型推广至任意维度，分析零频与非零频情形下同步状态及其稳定性，并揭示维数奇偶性导致的连续或不连续相变差异。
摘要：
Kuramoto 模型最近被推广到任意维度，其方式是把振子重新解释为在 D 维空间中沿单位球面运动的粒子。于是每个粒子由一个 D 维单位向量表示。当 D=2 时，粒子在单位圆上运动，向量可以由单一相位描述，从而恢复原始 Kuramoto 模型。这种多维描述还可以进一步推广：把粒子之间的耦合常数提升为作用于单位向量的矩阵，用以表示一类广义阻挫。在最近的一篇论文中，我们详细分析了 D=2 时耦合矩阵的作用。本文把这一分析扩展到任意维度，研究同步状态及其稳定性。我们表明，当粒子的固有频率被设为零时，系统要么收敛到具有明确相位的稳态同步状态，要么收敛到一种有效的二维动力学，其中同步粒子在球面上旋转。这些状态的稳定性取决于耦合矩阵的特征值和特征向量。当固有频率不为零时，同步取决于 D 是偶数还是奇数。在偶数维中，向同步的转变是连续的，旋转状态被主动状态取代；在主动状态中，序参量旋转，同时其模发生振荡。如果 D 为奇数，相变是不连续的，主动状态受到抑制，只对一类受限的耦合矩阵出现。
英文摘要：
The Kuramoto model was recently extended to arbitrary dimensions by reinterpreting the oscillators as particles moving on the surface of unit spheres in a D-dimensional space. Each particle is then represented by a D-dimensional unit vector. For $D=2$ the particles move on the unit circle and the vectors can be described by a single phase, recovering the original Kuramoto model. This multidimensional description can be further extended by promoting the coupling constant between the particles to a matrix that acts on the unit vectors, representing a type of generalized frustration. In a recent paper we have analyzed in detail the role of the coupling matrix for $D=2$. Here we extend this analysis to arbitrary dimensions, presenting a study of synchronous states and their stability. We show that when the natural frequencies of the particles are set to zero, the system converges either to a stationary synchronized state with well defined phase, or to an effective two-dimensional dynamics, where the synchronized particles rotate on the sphere. The stability of these states depend on the eigenvalues and eigenvectors of the coupling matrix. When the natural frequencies are not zero, synchronization depends on whether $D$ is even or odd. In even dimensions the transition to synchronization is continuous and rotating states are replaced by active states, where the order parameter rotates while its module oscillates. If $D$ is odd the phase transition is discontinuous and active states are suppressed, occurring only for a restricted class of coupling matrices.
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