我们考虑具有固有频率的经典 Kuramoto 模型（KM）及其连续统极限（CL），并讨论同步解的存在性及其分岔与稳定性。我们特别假设在 CL 中频率函数是对称且线性的，从而在 KM 中固有频率是均匀间隔的。我们证明，在 KM 中，当节点数 $n$ 为奇数且趋于无穷大时，至少会产生 $O(2^n)$ 个单参数同步解族，并发生至少 $O(2^n)$ 个鞍结分岔和叉式分岔。此外，我们证明先前工作中获得的同步解族会经历鞍结分岔，其稳定性从渐近稳定变为不稳定，而其他同步解族在 KM 中是不稳定的。对于 CL，我们证明先前工作中获得的连续同步解单参数族是渐近稳定的，并且存在不可数个不连续同步解单参数族，它们与另一个连续同步解单参数族一样是不稳定的。
英文摘要：
We consider the classical Kuramoto model (KM) with natural frequencies and its continuum limit (CL), and discuss the existence of synchronized solutions and their bifurcations and stability. We specifically assume that the frequency function is symmetric and linear in the CL, so that the natural frequencies are evenly spaced in the KM. We show that in the KM, $O(2^n)$ one-parameter families of synchronized solutions are born and $O(2^n)$ saddle-node and pitchfork bifurcations occur at least, when the node number $n$ is odd and tends to infinity. Moreover, we prove that the family of synchronized solutions obtained in the previous work suffers a saddle-node bifurcation at which its stability changes from asymptotically stable to unstable and the other families of synchronized solutions are unstable in the KM. For the CL, we show that a one-parameter family of continuous synchronized solutions obtained in the previous work is asymptotically stable and that there exist uncountably many one-parameter families of discontinuous synchronized solutions and they are unstable along with another one-parameter family of continuous synchronized solutions.
Transient rebellions in the Kuramoto oscillator: Morse-Smale structural stability and connection graphs of finite 2-shift type
National Tsing Hua University（国立清华大学）；National Center for Theoretical Sciences National Taiwan University（国立台湾大学理论科学中心）；Institut für Mathematik Freie Universität Berlin（柏林自由大学数学研究所）；Department of Mathematics National Taiwan University（国立台湾大学数学系）
