Chaos and Complex Systems Research Laboratory, Department of Physics, University of Burdwan, Burdwan 713 104, West Bengal, India.
Chaos. 2019 Jan;29(1):013102. doi: 10.1063/1.5077052.
Digital phase-locked loops (DPLLs) are nonlinear feedback-controlled systems that are widely used in electronic communication and signal processing applications. In most of the applications, they work in coupled mode; however, a vast amount of the studies on DPLLs concentrate on the dynamics of a single isolated unit. In this paper, we consider both one- and two-dimensional networks of DPLLs connected through a practically realistic nonlocal coupling and explore their collective dynamics. For the one-dimensional network, we analytically derive the parametric zone of a stable phase-locked state in which DPLLs essentially work in their normal mode of operation. We demonstrate that apart from the stable phase-locked state, a variety of spatiotemporal structures including chimeras arise in a broad parameter zone. For the two-dimensional network under nonlocal coupling, we identify several variants of chimera patterns, such as strip and spot chimeras. We identify and characterize the chimera patterns through suitable measures like local curvature and correlation function. Our study reveals the existence of chimeras in a widely used engineering system; therefore, we believe that these chimera patterns can be observed in experiments as well.
数字锁相环（DPLL）是一种广泛应用于电子通信和信号处理应用的非线性反馈控制系统。在大多数应用中，它们以耦合模式工作；然而，大量关于 DPLL 的研究集中在单个隔离单元的动力学上。在本文中，我们考虑通过实际的非局部耦合连接的 DPLL 的一维和二维网络，并探索它们的集体动力学。对于一维网络，我们从理论上推导出 DPLL 基本在其正常工作模式下稳定锁定相位的参数区域。我们证明，除了稳定的锁定状态外，在一个广泛的参数区域内还会出现各种时空结构，包括嵌合体。对于非局部耦合下的二维网络，我们确定了几种嵌合体模式的变体，例如条纹和斑点嵌合体。我们通过局部曲率和相关函数等合适的措施来识别和描述嵌合体模式。我们的研究揭示了在一个广泛使用的工程系统中存在嵌合体，因此，我们相信这些嵌合体模式也可以在实验中观察到。
