Ku Wai Lim, Girvan Michelle, Ott Edward
Institute for Research in Electronics and Applied Physics, University of Maryland, College Park, Maryland 20742, USA.
Chaos. 2015 Dec;25(12):123122. doi: 10.1063/1.4938534.
In this paper, we study dynamical systems in which a large number N of identical Landau-Stuart oscillators are globally coupled via a mean-field. Previously, it has been observed that this type of system can exhibit a variety of different dynamical behaviors. These behaviors include time periodic cluster states in which each oscillator is in one of a small number of groups for which all oscillators in each group have the same state which is different from group to group, as well as a behavior in which all oscillators have different states and the macroscopic dynamics of the mean field is chaotic. We argue that this second type of behavior is "extensive" in the sense that the chaotic attractor in the full phase space of the system has a fractal dimension that scales linearly with N and that the number of positive Lyapunov exponents of the attractor also scales linearly with N. An important focus of this paper is the transition between cluster states and extensive chaos as the system is subjected to slow adiabatic parameter change. We observe discontinuous transitions between the cluster states (which correspond to low dimensional dynamics) and the extensively chaotic states. Furthermore, examining the cluster state, as the system approaches the discontinuous transition to extensive chaos, we find that the oscillator population distribution between the clusters continually evolves so that the cluster state is always marginally stable. This behavior is used to reveal the mechanism of the discontinuous transition. We also apply the Kaplan-Yorke formula to study the fractal structure of the extensively chaotic attractors.
在本文中，我们研究了这样一类动力系统：其中大量（数量为N）相同的朗道 - 斯图尔特振荡器通过平均场进行全局耦合。此前已经观察到，这类系统能够展现出多种不同的动力学行为。这些行为包括时间周期簇态，即每个振荡器处于少数几个组中的一个，每个组内的所有振荡器具有相同的状态，且不同组的状态不同；以及另一种行为，即所有振荡器具有不同的状态，且平均场的宏观动力学是混沌的。我们认为，后一种行为是“广泛的”，因为系统全相空间中的混沌吸引子具有分形维数，该维数随N线性缩放，并且吸引子的正李雅普诺夫指数的数量也随N线性缩放。本文的一个重要关注点是当系统经历缓慢绝热参数变化时，簇态与广泛混沌之间的转变。我们观察到簇态（对应于低维动力学）与广泛混沌态之间的不连续转变。此外，在研究簇态时，当系统接近向广泛混沌的不连续转变时，我们发现簇之间的振荡器种群分布不断演化，使得簇态始终处于边缘稳定状态。这种行为被用于揭示不连续转变的机制。我们还应用卡普兰 - 约克公式来研究广泛混沌吸引子的分形结构。
