DIVERSE DYNAMICAL PATTERNS IN A COMPLEX DIGITAL NETWORK OF DELAYED ENGINEERING SYSTEM OF PHASE-LOCKED LOOPS

Abstract

Studies on complex networks with engineering systems have significant challenges for researchers. In this article, we demonstrate diverse intricate dynamical patterns in a one dimensional network of diffusivity coupled time-delayed digital phase-locked loops. Systems that utilize time-delayed feedback control are crucial in numerous engineering applications. In modern complex networks, time delayed systems increase the stability of the systems. Recently, many complex nonlinear networks are studied in different litterateurs, such as, complex computer networks, electrical grid systems, FPGA systems and many model neuronal networks. The dynamics of such networks exhibit extremely complex nonlinear phenomena. Therefore, it is very challenging to work with these in reality. To demonstrate such complex network we consider an one dimensional network of locally coupled time-delay feedback controlled digital-phase locked loop (TDFC-DPLL). TDFC-DPLL is more advanced version of DPLL compared to conventional DPLL. It has wider lock range, small steady state phase error and very less convergence time compared to ordinary single-sampler based positive zero-crossing DPLL (ZC1-DPLL). In a network of one dimensional array it shows several spatiotemporal patterns. Some of the important distinct patterns are synchronized fixed point (SFP), frozen random pattern (FRP), pattern selection (PS), travelling random pattern (TRP), spatiotemporal intermittency (STI), spatiotemporal chaos and chimera state. All those dynamical patterns are drawn on a phase diagram to grab the total dynamics. We derived the stability condition of the network for stable phase-locked operation. A comprehensive analysis of travelling random pattern is presented. This investigation explores the emergence of chimera states (in local coupled system) that occur predominantly in areas with very low coupling strength and higher nonlinearty parameter values. Additionally, the pathway of dynamic phase transition via the chimera state is demonstrated.