Networks of coupled time-delay digital tanlock loops: chimeras and other emergent spatiotemporal dynamics
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ORIGINAL PAPER
Networks of coupled time-delay digital tanlock loops: chimeras and other emergent spatiotemporal dynamics Bishwajit Paul · Tanmoy Banerjee
Received: 15 June 2020 / Accepted: 25 August 2020 © Springer Nature B.V. 2020
Abstract Phase-locked loops are important engineering systems that are widely used in electronic communication systems. Although, in almost all the engineering applications they operate in coupled conditions; however, most of the studies explored their dynamics in the isolated condition. In this paper, we explore in detail the collective dynamics of a network of coupled time-delay digital tanlock loops (TDTLs). Both oneand two-dimensional networks of TDTLs with nonlocal coupling are studied in this paper. We carry out detailed stability analysis for the one-dimensional network and determine the stable zone of operation. We explore the appearance of exotic spatiotemporal patterns like chimeras and solitary states. For the twodimensional network, a variety of chimera patterns, e.g., spot, linear stripe, wavy stripe, and inverted spot chimeras, are reported. We also identify the occurrence of breathing spot chimeras in two-dimensional network. We corroborate our results by characterizing the chimeras by proper measures in both one- and twodimensional cases.
B. Paul Department of Physics, Krishnagar Government College, Nadia, West Bengal 741 101, India e-mail: [email protected] T. Banerjee (B) Chaos and Complex Systems Research Laboratory, Department of Physics, University of Burdwan, Burdwan, West Bengal 713 104, India e-mail: [email protected]
Keywords Time-delay digital tanlock loop · Circulant matrix · Stability analysis · Spatiotemporal patterns · Chimeras
1 Introduction In most of the electronics and communication engineering systems, phase-locking technique plays a crucial role. To achieve phase-locking, phase-locked loops (PLLs) are widely used as the elementary building block of those systems [1,2]. The inherent nonlinearity of PLLs makes their dynamical behaviors extremely complex and therefore difficult to predict in practical situation [3]. In the era of modern communication systems, digital version of PLL, namely the digital phaselocked loop (DPLL), has rapidly replaced its analogue version [4,5]. There exist a variety of DPLLs; the most popular are the non-uniform sampling zero-crossing DPLL (ZC-DPLL), digital tanlock loop [4,6–8], and the recently proposed superior version of all DPLLs, namely the time-delay digital tanlock loop (TDTL) [9]. The TDTLs are better alternative of all the existing DPLLs owing to their insensitivity to the incoming signal strength, wider lock-range, and less steady-state phase error [10–15]. This makes TDTLs more useful in digital electronic systems and signal processing applications. Like other PLLs [2,16,17], an individual TDTL also exhibits complex dynamical behaviors like bifurcation and chaos due to its rich nonlinear characteristics [18].
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B. Paul, T. Banerjee
In modern communication systems, particularly in ele
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