Phase-Locked Loops (PLLs) are electronic systems that can be used as a synchronized oscillator, a driver or multiplier of frequency, a modulator or demodulator and as an amplifier of phase modulated signals. This book updates the methods used in the analysis of PLLs by drawing on the results obtained in the last 40 years. Many are published for the first time in book form. Nonlinear and deterministic mathematical models of continuous-time and discrete-time PLLs are considered and their basic properties are given in the form of theorems with rigorous proofs. The book exhibits very beautiful dynamics, and shows various physical phenomena observed in synchronized oscillators described by complete (not averaged) equations of PLLs. Specially selected mathematical tools are used — the theory of differential equations on a torus, the phase-plane portraits on a cyclinder, a perturbation theory (Melnikov's theorem on heteroclinic trajectories), integral manifolds, iterations of one-dimensional maps of a circle and two-dimensional maps of a cylinder. Using these tools, the properties of PLLs, in particular the regions of synchronization are described. Emphasis is on bifurcations of various types of periodic and chaotic oscillations. Strange attractors in the dynamics of PLLs are considered, such as those discovered by Rössler, Henon, Lorenz, May, Chua and others.Contents:Introduction:What Is Phase-Locked Loop?PLL and Differential or Recurrence EquationsAveraging MethodOrganization of the BookThe First Order Continuous-Time Phase-Locked Loops:Equations of the SystemThe Averaged EquationSolutions of the Basic FrequencyDifferential Equation on the TorusFractional SynchronizationThe System with Rectangular Waveform SignalsThe Mapping (f(p)=p+2πμ+a sin(p)The Second Order Continuous-Time Phase-Locked Loops:The System with a Low-Pass FilterPhase-Plane Portrait of the Averaged SystemPerturbation of the Phase Difference φ(wt)Stable Integral ManifoldThe PLL System Reducible to the First Order OneHomoclinic StructuresBoundaries of Attractive DomainsThe Smale Horseshoe, Transient ChaosHigher Order Systems Reducible to the Second Order OnesOne-Dimensional Discrete-Time Phase-Locked Loop:Recurrence Equations of the SystemPeriodic Output SignalsRotation Interval and Frequency Locking RegionsStable Orbits, Hold-In RegionsThe Number of Stable OrbitsBifurcations of Periodic OrbitsBifurcation of the Rotation IntervalTwo-Dimensional Discrete-Time Phase-Locked Loop:Description of the DPLL System by a Two-Dimensional MapStable Periodic OrbitsReduction to a One-Dimensional SystemStrange Attractors and Chaotic Steady-StatesReadership: Graduate students and researchers in nonlinear science and applied physics; mathematically inclined engineers and mathematicians.