The book presents an extensive mathematical study of a group of one- and two-dimensional dynamical systems modeling phase-locked loops (PLLs), in continuous and discrete time.
PLLs are electronic devices designed to track changes in the input signal by synchronizing an implemented oscillator with that signal through use of a feedback loop. A PLL may work either in continuous, or in discrete time. The commercially produced PLLs are used mainly in radio- and telecommunications and can be quite complex, containing several feedback loops, integrators and high-order filters. PLLs can be also entirely digital (DPLLs). However, the authors are not much interested in the design and implementation aspects, and their text is definitely not on electronic engineering.
Even the simplest kinds of PLLs are known for their rich dynamics, and that is the subject of this nonlinear science book. It is truly amazing how so narrow range of ‘sample species’ allows the authors to reach in its concern for almost every known aspect of modern theory of dynamical systems. The book incorporates both classical and recent results as well as original research (though there is no stressed distinction between those), including a number of theorems on general theory of dynamical systems, which are not aimed specifically at PLLs. Some of the results are presented with complete proofs, while for the most difficult (and less relevant to PLLs) of them the proofs are omitted. The text is divided in five chapters, the first of which is introductory, and the others correspond to different types of PLLs. The input signal in most cases is assumed to be sinusoidal.
Chapter 2 deals with the first-order PLLs in continuous time. The dynamical system model is a smooth flow on a 2-d torus given by a single first-order differential equation with a nonlinear right-hand side that is periodic w.r.t.both the unknown function and time. The aim of the study is to find the asymptotically stable periodic solutions (if any), i.e.the steady states of a given PLL, and classify them accordingly to their rotation numbers. The tools used are the averaging method, Poincaré mapping and asymptotical analysis, supported by numeric simulations. The effects observed are the devil’s staircase and Arnold tongues, i.e.the mode-locking phenomena.
In Chapter 3, the second-order continuous-time PLLs are considered. An extra unknown function is added, so the model is now a 3-d flow determined by a system of two first-order differential equations. The methods of Chapter 2 work to some extent. The new concepts introduced are the 2-d phase-portrait for an autonomous system (saddle and node fixed points, invariant manifolds, basins of attraction, separatrices, heteroclinic trajectories), its periodic perturbations (perturbed equilibria, integral manifolds, Melnikov theorem), the chaos (homoclinic effects, Smale horseshoe, symbolic dynamics). For higher-order PLL systems, a theorem on their reducibility to the order two (in the sense of the existence of an asymptotically stable 2-d manifold) is given.
First-order discrete-time PLLs are considered in Chapter 4. The model is a 1-d continuous map of the form “the identity plus a periodic function” earlier looked at in Chapter 2 as the Poincaré mapping for the 2-d flow, but now it is not necessarily monotone. Being considered as a lifted non-invertible circle map, this object naturally implies the definition of so-called periodic points of type \(n/m\) (meaning that the \(m\)th iterate of the map winds this point \(n\) times around the circle, with an appropriate meaning in terms of the PLL’s mode). The concepts introduced in this chapter are the Sharkovsky’s order, rotation intervals, attractive sets of stable orbits, Schwartz derivative, saddle-node and period-doubling bifurcations, Feigenbaum cascade, Lyapunov exponents. The mode-locking phenomena are investigated to a new extent.
In Chapter 5, the authors look at second-order discrete-time PLLs, and the model is a 2-d continuous map. Here the study becomes much more specific with particular PLLs’ equations, though incorporating the ideas of Chapters 3 and 4. The periodic points are classified by their \(n/m\) type and the eigenvalues of the Jacobian matrix of the \(m\)th iterate of the map. The notions of attractor and strange attractor are also introduced in this chapter.
The book may be recommended to electronic engineers interested in nonlinear dynamics as a collection of mathematical tools and ideas relevant to the PLLs and similar systems. It should also be of interest for students in mathematics as a exposition of the dynamical systems theory through a robust series of examples derived from the modern technology. A minor flaw of the text worth mentioning is its sometimes unconventional terminology, both mathematical and engineering, which may potentially confuse an unprepared reader.