Synchronization of oscillators with random nonlocal connectivity.
Synchronization of oscillators with random nonlocal connectivity.
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DOI：
10.1103/physreve.54.64
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发表时间：
1996-07
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影响因子：
--
通讯作者：
P. Gade
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文献类型：
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作者：
P. Gade
在本文中，我们研究了有关与随机非本地连通性@Chate和Manneville的大量耦合地图同步的现有观察结果，Chaos 2，307〜1992！和电路。从宏观限制中的特征值中的连通性矩阵中有有限的差距，我们对差距进行了定量解释。从低连通性限制中的这种高度集体行为，表明该行为几乎是非常低的连接性的统计学统计。在各种上下文中，耦合的地图〜CML。案例没有晶格几何 @2＃。 CML具有随机的非局部耦合 @6＃。其中局部连接没有任何空间意义。很容易实现。在这种随机的非本地连通性chate和Manneville的系统中，对不同的连通性及其效果有用。从各种角度来看，使用这种连接性在空间扩展系统（例如耦合的振荡器）中同步。通过实现同步，可以尝试构建较大但更具控制的系统，这些系统有时是较低的 @7＃ ，在过去的几年中，这个问题已经进行了几项研究，我们想证明，同步现象通常可以通过研究来理解连通性矩阵的特征范围可以帮助我们理解本文的现有观察结果。速率，在Ref中研究的均匀模式，其余的模型也将研究较低的连接性。同步状态，我们将研究具有随机非局部连通性的连通性矩阵的特征值。在参考文献中，Cook和Derrida与随机能量模型，广义随机能量模型以及随机介质中的定向聚合物有关。在材料稀疏，非零元素的分布函数的情况下，他们获得了此类物品的确切分析结果，但是在Chate和Manneville @6＃的模型中，所有连接都具有相同的功能。权重，即非零元素的分布函数是D函数。
In this paper we study the existing observation in literature about synchronization of a large number of coupled maps with random nonlocal connectivity @Chate and Manneville, Chaos 2, 307 ~1992!#. These connectivities which lack any spatial significance can be realized in neural nets and electrical circuits. It is quite interesting and of practical importance to note that a huge number of maps can be synchronized with this connectivity. We show that this synchronization stems from the fact that the connectivity matrix has a finite gap in the eigenvalue spectrum in the macroscopic limit. We give a quantitative explanation for the gap. We compare the analytic results with the ones quoted in the above reference. We also study the departures from this highly collective behavior in the low connectivity limit and show that the behavior is almost statistical for very low connectivity. Of late, there has been considerable attention paid to the study of coupled map lattices ~CML! in various contexts. They have been used as a computationally simple and analytically tractable model for spatiotemporal systems @1#. The studies on CML’s have been either in one and two dimensional lattices or with global coupling, in which case there is no notion of lattice geometry @2#. The higher dimensional connectivities @3# or hierarchial connectivities @4,5# are studied very little. One more system that has been studied is the CML with random nonlocal couplings @6#. The motivation is twofold. First, this is an effectively high dimensional system. The phenomenology in CML in higher dimensions has not been studied much, and needs further investigation. Second, there are systems like neural nets in which the local connections do not have any spatial significance. There also exist systems like electrical circuits @7# in which connectivity is at one’s will and such a coupling can be easily realized. Thus the studies of different connectivities and their effects will be useful in designing well controlled systems. In this system of random nonlocal connectivity Chate and Manneville have presented preliminary results @6# which show that synchronization of a large number of oscillators is easily achieved with this connectivity. Synchronization of oscillators in spatially extended systems such as coupled oscillators is important from various points of view. By achieving synchronization, one can try to build huge but more controllable and better behaved systems which are effectively low dimensional @7#. Sometimes, synchronization may serve other purposes, such as sending codes that are difficult to break @8#. In various contexts, this problem has been subjected to several studies in the past few years @9#. We would like to show that the phenomenon of synchronization can be generally understood by investigating the eigenvalue spectrum of the connectivity matrix and can help us to understand the existing observations. In this paper, we will explicitly illustrate how one can separate the mode leading to spatial homogeneity from the rest. We will show that there exists a finite gap between the growth rates, the spatially homogeneous mode, and the rest in the model studied in Ref. @6#. We will also study the departures from this behavior for lower connectivities. For the linear stability analysis of the synchronized state, we will study the eigenvalue spectrum of the connectivity matrix with random nonlocal connectivity. We will also study the eigenspectrum of the product of such matrices. We would note that a similar model of random connectivity matrix of size N3N with k nonzero elements in each row has been investigated by Cook and Derrida in Ref. @10# in connection with the random energy model, the generalized random energy model, and directed polymers in random media. They have obtained exact analytic results for products of such matrices in the case where the matrices are sparse and the distribution function of nonzero elements is not a d function. However, in the model studied by Chate and Manneville @6#, all connections have the same weight, i.e., the distribution function of nonzero elements is a d function.