Institute for Research in Electronics and Applied Physics, University of Maryland, College Park, Maryland 20742, USA.
Chaos. 2013 Sep;23(3):033116. doi: 10.1063/1.4816361.
This paper addresses the behavior of large systems of heterogeneous, globally coupled oscillators each of which is described by the generic Landau-Stuart equation, which incorporates both phase and amplitude dynamics of individual oscillators. One goal of our paper is to investigate the effect of a spread in the amplitude growth parameter of the oscillators and of the effect of a homogeneous nonlinear frequency shift. Both of these effects are of potential relevance to recently reported experiments. Our second goal is to gain further understanding of the macroscopic system dynamics at large coupling strength, and its dependence on the nonlinear frequency shift parameter. It is proven that at large coupling strength, if the nonlinear frequency shift parameter is below a certain value, then there is a unique attractor for which the oscillators all clump at a single amplitude and uniformly rotating phase (we call this a single-cluster "locked state"). Using a combination of analytical and numerical methods, we show that at higher values of the nonlinear frequency shift parameter, the single-cluster locked state attractor continues to exist, but other types of coexisting attractors emerge. These include two-cluster locked states, periodic orbits, chaotic orbits, and quasiperiodic orbits.
本文研究了由广义 Landau-Stuart 方程描述的异类、全局耦合振子的大系统行为，该方程包含了单个振子的相位和幅度动力学。本文的目标之一是研究振子的幅度增长参数的分布和均匀非线性频率移动的影响。这两种效应都与最近报道的实验有关。我们的第二个目标是进一步了解在大耦合强度下的宏观系统动力学及其对非线性频率移动参数的依赖性。证明在大耦合强度下，如果非线性频率移动参数低于某个值，则存在唯一的吸引子，其中所有的振子都聚集在单一的幅度和均匀旋转的相位上（我们称之为单一簇“锁定状态”）。本文使用了分析和数值方法的结合，表明在更高的非线性频率移动参数值下，单一簇锁定状态的吸引子仍然存在，但也出现了其他类型的共存吸引子。这些共存吸引子包括双簇锁定状态、周期轨道、混沌轨道和准周期轨道。
