Phase Synchronization in the Globally Coupled Non linear chaotic Maps and the Scaling Theory

我们对全球耦合映射格（GCML）的湍流状态进行了广泛的统计调查，GCML被认为是一个具有异常统计性质的系统；图的平均场甚至在热力学极限处波动，违反了大数定律（一种隐藏的相干性）。我们的研究结果可以总结如下。尽管在湍流状态下映射之间的耦合很小，但我们发现存在显著的簇吸引子，其中映射分裂成几个簇，并且簇之间以一定的周期性相互振荡。最显著的周期性表现是第三周期运动中最大对称的三簇吸引子（p3c3 MSCA），由于总体对称性，平均场波动几乎可以忽略不计。如果耦合设置略高于MSCA，则会出现相关的吸引态，其中星团的数量减少，但星团轨道大致相同。这里的平均场波动异常地增强了。我们可以成功地推导出调谐条件，该条件决定了在给定非线性映射下形成各种簇态所需的耦合。GCML湍流状态的控制动力学是元素逻辑映射周期窗口的叶化。隐相干可以看作是最温和的周期性表现。我们的调整条件预测了模型参数空间中的曲线，这些曲线将具有不同非线性和耦合但具有相同周期簇态的GCML联系在一起。GCML是情报活动的基本模型。我们认为我们的发现对于该研究领域的未来发展至关重要，因为它们简洁地告诉我们，即使在非常弱的耦合下，大型复杂系统也可以形成同步状态。

We performed an extensive statistical survey of Globally Coupled Map Lattice (GCML) in its turbulent regime, which had been regarded as a system with an anomalous statistical property; the mean field of the map fluctuates even at the thermo dynamical limit violating the law of large numbers (a hidden coherence) .Our findings may be summarized as follows.1. Even though the coupling between the maps is set extremely small in the turbulent regime, we found that there emerge remarkable cluster attractors, in which the maps split into a few clusters and the clusters mutually oscillate in a certain periodicity. The most remarkable periodicity manifestation is the maximally symmetric three-clustered attractor in period-three motion (p3c3 MSCA), where the mean field fluctuation is almost negligible due to the population symmetry.2. If the coupling is set slightly higher than that for the MSCA, there emerge associated attractor-states, where the number of the clusters has decreased but the cluster orbits are approximately the same. Here the mean field fluctuation is anomalously enhanced.3. We could successfully derive the tuning condition, which determines the necessary coupling at a given non-linearity of maps for the formation of various cluster states. The controlling dynamics in the turbulent regime of GCML is the foliation of the periodic windows of the element logistic map. The hidden-coherence may be regarded most modest periodicity manifestation. Our tuning condition predicts curves in the model parameter space, which link together those GCML with distinct non-linearity and coupling but exhibiting the same periodic cluster state. The GCML is a basic model for the intelligence activity. We consider that our findings as listed above are crucial for the future progress in this research field because they succinctly tell that a large complex system can form synchronized states even at the very weak coupling.