Marvel Seth A, Strogatz Steven H
Center for Applied Mathematics, Cornell University, Ithaca, New York 14853, USA.
Chaos. 2009 Mar;19(1):013132. doi: 10.1063/1.3087132.
We study the nonlinear dynamics of series arrays of Josephson junctions in the large-N limit, where N is the number of junctions in the array. The junctions are assumed to be identical, overdamped, driven by a constant bias current, and globally coupled through a common load. Previous simulations of such arrays revealed that their dynamics are remarkably simple, hinting at the presence of some hidden symmetry or other structure. These observations were later explained by the discovery of N-3 constants of motion, the choice of which confines the resulting flow in phase space to a low-dimensional invariant manifold. Here we show that the dimensionality can be reduced further by restricting attention to a special family of states recently identified by Ott and Antonsen. In geometric terms, the Ott-Antonsen ansatz corresponds to an invariant submanifold of dimension one less than that found earlier. We derive and analyze the flow on this submanifold for two special cases: an array with purely resistive loading and another with resistive-inductive-capacitive loading. Our results recover (and in some instances improve) earlier findings based on linearization arguments.
我们研究了约瑟夫森结串联阵列在大N极限下的非线性动力学，其中N是阵列中结的数量。假设这些结是相同的、过阻尼的，由恒定偏置电流驱动，并通过公共负载进行全局耦合。此前对这类阵列的模拟显示，它们的动力学非常简单，这暗示着存在某种隐藏的对称性或其他结构。这些观察结果后来通过发现N - 3个运动常数得到了解释，这些常数的选择将相空间中产生的流限制在一个低维不变流形上。在这里，我们表明，通过将注意力限制在Ott和Antonsen最近确定的一类特殊状态上，可以进一步降低维度。从几何角度来看，Ott - Antonsen假设对应于一个维度比早期发现的少一维的不变子流形。我们针对两种特殊情况推导并分析了该子流形上的流：一种是纯电阻负载的阵列，另一种是电阻 - 电感 - 电容负载的阵列。我们的结果恢复了（在某些情况下还改进了）基于线性化论证的早期发现。
