Institute of Applied Physics of the Russian Academy of Sciences, 46 Ulyanov Street, 603950 Nizhny Novgorod, Russia.
Scientific Computing Laboratory, Center for the Study of Complex Systems, Institute of Physics Belgrade, University of Belgrade, Pregrevica 118, 11080 Belgrade, Serbia.
Phys Rev E. 2019 Dec;100(6-1):062211. doi: 10.1103/PhysRevE.100.062211.
We consider the macroscopic regimes and the scenarios for the onset and the suppression of collective oscillations in a heterogeneous population of active rotators composed of excitable or oscillatory elements. We analyze the system in the continuum limit within the framework of Ott-Antonsen reduction method, determining the states with a constant mean field and their stability boundaries in terms of the characteristics of the rotators' frequency distribution. The system is established to display three macroscopic regimes, namely the homogeneous stationary state, where all the units lie at the resting state, the global oscillatory state, characterized by the partially synchronized local oscillations, and the heterogeneous stationary state, which includes a mixture of resting and asynchronously oscillating units. The transitions between the characteristic domains are found to involve a complex bifurcation structure, organized around three codimension-two bifurcation points: a Bogdanov-Takens point, a cusp point, and a fold-homoclinic point. Apart from the monostable domains, our study also reveals two domains admitting bistable behavior, manifested as coexistence between the two stationary solutions or between a stationary and a periodic solution. It is shown that the collective mode may emerge via two generic scenarios, guided by a saddle-node of infinite period or the Hopf bifurcation, such that the transition from the homogeneous to the heterogeneous stationary state under increasing diversity may follow the classical paradigm, but may also be hysteretic. We demonstrate that the basic bifurcation structure holds qualitatively in the presence of small noise or small coupling delay, with the boundaries of the characteristic domains shifted compared to the noiseless and the delay-free case.
我们研究了由兴奋或振荡元件组成的活跃转子异质群体中集体振荡的出现和抑制的宏观状态和情景。我们在 Ott-Antonsen 约化方法的框架内分析了连续体极限下的系统，根据转子频率分布的特征确定具有恒定平均场的状态及其稳定性边界。该系统表现出三种宏观状态，即所有单元都处于静止状态的均匀稳定状态、局部振荡部分同步的全局振荡状态以及包含静止和异步振荡单元混合物的非均匀稳定状态。发现特征域之间的转换涉及复杂的分岔结构，围绕三个二维分歧点组织：Bogdanov-Takens 点、尖点和折叠同宿点。除了单稳域之外，我们的研究还揭示了两个允许双稳行为的域，表现为两个稳定解或一个稳定解和一个周期解共存。结果表明，集体模式可能通过两种通用情景出现，一种是由无限周期的鞍点或 Hopf 分岔引导，另一种是随着多样性的增加从均匀到非均匀稳定状态的转变可能遵循经典范例，但也可能滞后。我们证明了基本的分岔结构在存在小噪声或小耦合延迟的情况下定性成立，与无噪声和无延迟的情况相比，特征域的边界发生了偏移。
