Abstract
We have investigated a fractional-order phase-locked loop characterised by a third-order differential equation. The integer-order mathematical model of the phase-locked loop (PLL) is first converted to fractional order using the Caputo-Fabrizio method. The stability of the equilibrium points is discussed in detail in both parameter and fractional-order domain. The proposed fractional-order phase-locked loop (FOPLL) model shows multiple coexisting attractors which was not discussed in the earlier literature of PLL. The significance of these infinite coexisting attractors is that they exist in the operation region of the PLL between [−π,π] which increases the complexity of operation of the PLLs. Mainly when such FOPLLs are used in large-scale networks, the synchronisation of the FOPLLs becomes complicated and will result in unstable control conditions. Hence, studying the network dynamics of such FOPLLs is significant which motivates us to investigate the synchronisation phenomenon of the FOPLLs constructed in a square network. We could show that, because of the multiple coexisting attractors, the FOPLLs show various synchronisation phenomena, and more importantly in the chaotic region for lower fractional-order values, we could show that the FOPLLs are synchronised and this finding is very useful to completely analyse the FOPLL networks in high-frequency operations.
Highlights
It was in [3, 4] that the first known mathematical model of the phase-locked loop (PLL) is proposed, and since the interest is to investigate the dynamics of the PLLs. e simplest form of a PLL consists of a voltagecontrolled oscillator (VCO) and a phase detector in the feedback which is to generate the phase error between the input and the oscillator signals. e PLL gets locked to the frequency when the loop error becomes constant [5]
A chaotic jerk system holding both self-excited and hidden attractors is investigated for the effect of time delay [38, 39] on secure communication, and the results revealed the existence of special properties such as multistability [40]
We have investigated a fractional-order PLL with three differential equations
Summary
When we consider the third-order model of the PLL, there have been several pieces of literature discussing the chaotic oscillations and the parameter dependence of such complex oscillations [6, 14, 15]. There have been no many investigations on the synchronisation of real-time circuits like PLL when coupled together in networks.
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