标题： 二维矩阵耦合Kuramoto模型上的独立维度相变 显示英文标题

标题： Independent dimensional phase transition on a two-dimensional Kuramoto model with matrix coupling

摘要： 高维版的单维Kuramoto范式是推动对现实世界系统动态更真实描述的重要步骤。 尽管这些广义模型中的振荡器具有多维特性，但到目前为止，任何一对振荡器之间的相互作用方案都由一个标量因子主导，最终导致所有维度上的同步。 作为适合于单维情况的标量耦合的自然扩展，我们尝试在数值和理论上研究$2\times2$实矩阵在二维Kuramoto振荡器上的耦合机制。 这种新机制的一个特点是，矩阵耦合使得振荡器的两个维度能够分别进入同步或去同步状态，这在其他高维广义模型中尚未见过。 在各种矩阵配置下，两个维度的同步和去同步组合成四种定性不同的系统位置和运动模式。 我们证明，当一个矩阵以特定方式转变为另一个矩阵时，系统模式也会相应地通过序参数的连续或爆炸性转变而切换，从而模仿信息科学和生物学中的一系列行为。 显示英文摘要

摘要： The high-dimensional generalization of the one-dimensional Kuramoto paradigm has been an essential step in bringing about a more faithful depiction of the dynamics of real-world systems. Despite the multi-dimensional nature of the oscillators in these generalized models, the interacting schemes so far have been dominated by a scalar factor unanimously between any pair of oscillators that leads eventually to synchronization on all dimensions. As a natural extension of the scalar coupling befitting for the one-dimensional case, we take a tentative step in studying numerically and theoretically the coupling mechanism of $2\times2$ real matrices on two-dimensional Kuramoto oscillators. One of the features stemmed from this new mechanism is that the matrix coupling enables the two dimensions of the oscillators to separate their transitions to either synchronization or desynchronization which has not been seen in other high-dimensional generalizations. Under various matrix configurations, the synchronization and desynchronization of the two dimensions combine into four qualitatively distinct modes of position and motion of the system. We demonstrate that as one matrix is morphed into another in a specific manner, the system mode also switches correspondingly either through continuous or explosive transitions of the order parameters, thus mimicking a range of behaviors in information science and biology.