时滞耦合锁相环振荡器同步态的对称分岔分析
Symmetric bifurcation analysis of synchronous states of time-delayed coupled Phase-Locked Loop oscillators
摘要：
近年来，人们对时滞耦合振荡器网络的研究越来越感兴趣，因为它们在许多实际应用中都存在。在许多情况下，对称模式可以出现在这些网络中，因此系统的一部分可能会重复自己，并且这个子系统的属性代表了整个相空间上的动力学。本文在Correa和Piqueira \cite{Correa2013}对二节点网络的工作基础上，对二阶N节点时滞全连通网络进行了分析。这项研究是使用对称群进行的。我们证明了对称性强迫的多重特征值的存在，以及存在的Hopf分支。三种不同的模型，即全相位，相位和相位差模型被用来分析网络的动力学。我们确定一有限集的频率$ω$，这可能对应于在每种情况下的延迟的临界值的Hopf分叉。该$S_n$地图是用来实际寻找霍普夫分叉连同数值计算使用朗伯W函数。为了证实分析结果，使用了数值模拟。虽然我们限制注意二阶节点，结果可以扩展到更高阶的网络提供的时间延迟在节点之间的连接保持相等。
英文摘要：
In recent years there has been an increasing interest in studying time-delayed coupled networks of oscillators since these occur in many real life applications. In many cases symmetry patterns can emerge in these networks, as a consequence a part of the system might repeat itself, and properties of this subsystem are representative of the dynamics on the whole phase space. In this paper an analysis of the second order N-node time-delay fully connected network is presented which is based on previous work by Correa and Piqueira \cite{Correa2013} for a 2-node network. This study is carried out using symmetry groups. We show the existence of multiple eigenvalues forced by symmetry, as well as the existence of Hopf bifurcations. Three different models are used to analyze the network dynamics, namely, the full-phase, the phase, and the phase-difference model. We determine a finite set of frequencies $ω$, that might correspond to Hopf bifurcations in each case for critical values of the delay. The $S_n$ map is used to actually find Hopf bifurcations along with numerical calculations using the Lambert W function. Numerical simulations are used in order to confirm the analytical results. Although we restrict attention to second order nodes, the results could be extended to higher order networks provided the time-delay in the connections between nodes remains equal.
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