AI 中文总结
研究图上相同耦合斯图尔特 - 兰道振子新集体运动，确定旋转扭曲态及同步态存在与稳定性条件，推广相互作用矩阵类别，发现非均匀振子系统也有非均匀旋转扭曲态，数值模拟展示多种系统行为。
AI 中文摘要
在本文中，我们研究了图上相同耦合的斯图尔特 - 兰道振子的一类新的集体运动。该模型之前已知对于某类初始数据会收敛到同步状态。我们表明，当相互作用矩阵是循环矩阵时，存在另一类吸引子，其中粒子均匀分布在一个圆上，类似于Kuramoto模型中的“扭曲态”，但绕原点旋转。与经典Kuramoto情况不同，这里的旋转是由图结构的不对称引起的，而非自然频率。我们确定了斯图尔特 - 兰道模型和Kuramoto模型中“旋转扭曲态”的存在和局部稳定性条件。还给出了同步态存在和稳定性的充分条件，其可在特定参数区域与旋转扭曲态共存。通过领导者 - 跟随者相互作用矩阵推广了能产生旋转扭曲态的相互作用矩阵类别。表明对于非均匀振子系统也能存在非均匀旋转扭曲态，吸引到不同的个体目标振幅。本研究伴有数值模拟，展示了系统的可能行为，包括亚稳动力学和奇异态。
英文摘要
In this article, we study a new class of collective motion for identical coupled Stuart-Landau oscillators on graphs. This model was previously known to converge to the synchronized state for a certain class of initial data. Here, we show that when the interaction matrix is circulant, there exists another class of attractors, in which the particles are uniformly distributed on a circle, reminiscent of the \textit{twisted states} known for the Kuramoto model, but which rotate around the origin. However, contrary to the classical Kuramoto case, here the rotation is caused by the asymmetry of the graph structure, and not by the natural frequencies. We identify conditions for the existence and local stability of the \textit{rotational twisted states}, both in the Stuart-Landau model and in the Kuramoto model. We also provide sufficient conditions for the existence and stability of the synchronized state, which can co-exist with the rotational twisted state in a certain parameter region. We provide a generalization of the class of interaction matrices able to generate rotational twisted states via leader-follower interaction matrices. We show that heterogeneous rotational twisted states can also exist for a system of heterogeneous oscillators, attracted to different individual target amplitudes. This study is accompanied by numerical simulations that illustrate the possible behaviors of the system, which also include metastable dynamics and a chimera state.
Comments39 pages, 8 figures
