具有紧支撑双峰频率分布的Kuramoto模型中的一阶相变
First-order phase transitions in the Kuramoto model with compact bimodal frequency distributions
AI总结：
本文通过结合Kuramoto自洽方法、Crawford对称性分析和Ott-Antonsen拟设，推导出紧支撑双峰频率分布Kuramoto模型的完整分岔图，证明其同步路径总是经过驻波区域，与无限支撑情形的双稳态相变形成对比。
摘要：
当自然频率分布在其最大值处具有有限平坦区域时，由耦合相位振子构成的网络的Kuramoto模型会表现出一阶相变。如果单个网络的频率分布是双峰的，则包括迟滞和双稳态在内的一阶相变也会出现。在本研究中，我们关注这两种构型的相互作用，并在连续极限下分析具有紧支撑双峰频率分布的Kuramoto模型。迄今为止，严格的解析处理一直难以实现。通过结合Kuramoto的自洽方法、Crawford的对称性考虑，并利用应用于一族收敛到紧支撑分布的有理分布函数的Ott-Antonsen拟设，我们推导出系统序参量动力学的完整分岔图。我们证明，当双峰分布由两个具有紧支撑的单峰分布复合而成时，通往同步的路径总是经过一个驻波区域。这与当两个复合单峰分布具有无限支撑时可能跨越双稳态区域的相变形成对比。
英文摘要：
The Kuramoto model of a network of coupled phase oscillators exhibits a first-order phase transition when the distribution of natural frequencies has a finite flat region at its maximum. First-order phase transitions including hysteresis and bistability are also present if the frequency distribution of a single network is bimodal. In this study we are interested in the interplay of these two configurations and analyze the Kuramoto model with compact bimodal frequency distributions in the continuum limit. As of yet, a rigorous analytic treatment has been elusive. By combining Kuramoto's self-consistency approach, Crawford's symmetry considerations, and exploiting the Ott-Antonsen ansatz applied to a family of rational distribution functions that converge towards the compact distribution, we derive a full bifurcation diagram for the system's order parameter dynamics. We show that the route to synchronization always passes through a standing wave regime when the bimodal distribution is compounded by two unimodal distributions with compact support. This is in contrast to a possible transition across a region of bistability when the two compounding unimodal distributions have infinite support.
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