标题： 双图依赖的两模相互作用连续极限下的 Kuramoto 模型中同步解的分岔 显示英文标题

标题： Bifurcations of synchronized solutions in a continuum limit of the Kuramoto model with two-mode interaction depending on two graphs

摘要： 我们研究了在连续极限（CL）下，具有依赖于两个图的双模相互作用的相同振荡器的Kuramoto模型（KM）中完全同步状态的分岔。 这里其中一个图是均匀的，但可能是确定性的稠密图、随机的稠密图或随机的稀疏图，另一个是确定性的有限最近邻图。 我们使用中心流形约简技术，这是动力系统中的标准方法，并证明CL在这些情况下会发生分岔，使得一参数完全同步状态变得不稳定，并出现一个稳定的两参数的$\ell$-峰正弦形状的定态解（$\ell\ge 2$），其中$n$表示节点数。 这与作者最近在经典KM上的结果形成对比，在经典KM中，其CL中的分岔行为与KM中的分岔行为非常不同，难以用动力系统中的标准技术如中心流形约简来解释。 此外，基于先前的基本结果，显示类似的分岔行为也发生在KM中。 在之前的一项研究中，通过确定性图的数值模拟提出了这种分岔的发生。 我们还通过零自然频率的KM的数值模拟验证了我们的理论结果。 显示英文摘要

摘要： We study bifurcations of the completely synchronized state in a continuum limit (CL) for the Kuramoto model (KM) of identical oscillators with two-mode interaction depending on two graphs. Here one of the graphs is uniform but may be deterministic dense, random dense or random sparse, and the other is a deterministic finite nearest neighbor. We use the center manifold reduction technique, which is a standard one in dynamical systems, and prove that the CL suffers bifurcations at which the one-parameter family of completely synchronized state becomes unstable and a stable two-parameter family of $\ell$-humped sinusoidal shape stationary solutions ($\ell\ge 2$) appears, where $n$ represents the node number. This contrasts the author's recent result on the classical KM in which bifurcation behavior in its CL is very different from ones in the KM and difficult to explain by standard techniques in dynamical systems such as the center manifold reduction. Moreover, similar bifurcation behavior is shown to occur in the KM, based on the previous fundamental results. The occurrence of such bifurcations were suggested by numerical simulations for the deterministic graphs in a previous study. We also demonstrate our theoretical results by numerical simulations for the KM with the zero natural frequency.