Synchronization of beats in phase-locked loops

Dynamics of two phase-locked loops (PLL) with first order low-pass filters coupled via additional phase discriminator is studied. Mathematical models of the partial systems are pendulum-like type. Thus, mathematical model of the whole system consists of four ordinary differential equations. Phase space of the model is a cylindrical with two cyclic variables. In a case of low-inertial control loops the model transforms into dynamical system with toroidal phase space. The observed model has a great variety of dynamical modes both regular and chaotic. In the article, the main attention is paid to analysis of limit cycles and chaotic attractors, corresponding to beating modes of PLLs. In the beating mode oscillations with angular modulation exist at the PLL’s output. The characteristics of the modulation could be controlled by PLL’s parameters and, in case of ensemble, by coupling parameters. The actuality of the beating modes investigations is concerned with some possible applications in neurodynamics – oscillations in the beating modes are similar to the neuronal membrane potential oscillations. The goal of the paper is the analysis of beating modes of coupled PLLs to control the characteristics of modulated oscillations and to synchronize these oscillations in particular. The study is performed by numerical modelling with modern nonlinear dynamics methods and bifurcation theory applications. As a result, the regions of synchronization of beating modes are determined in parameter space of the model. The coexistence of beating modes synchronous and asynchronous dynamics are shown for PLLs with inertial control loop. Bifurcation mechanisms of beating mode synchronization loss are studied. The coupling of PLLs through additional phase discriminator is shown to synchronize beating mode oscillations.