Equations of Phase-Locked Loops: Dynamics on the Circle, Torus and Cylinder (World Scientific Series on Nonlinear Science Series a) (World Scientific Series on Nonlinear Science: Series a)

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......Page 8 Preface......Page 6 1.1 What is Phase-Locked Loop?......Page 12 1.2 PLL and differential or recurrence equations......Page 13 1.3 Averaging method......Page 16 1.4 Organization of the book......Page 18 2.1 Equations of the system......Page 20 2.2.1 Basic properties of solutions......Page 23 2.2.2 Application to Adler’s equation......Page 26 2.3.1 The Poincare mapping......Page 29 2.3.2 Periodic solutions......Page 31 2.3.3 Asymptotic formulae for periodic solutions......Page 32 2.3.4 Conclusions for the PLL equation......Page 34 2.4.1 Trajectories on the torus......Page 35 2.4.2 Periodic points......Page 37 2.4.3 Rotation number......Page 38 2.4.4 Rotation number as the function of a parameter......Page 39 2.5.1 Devil’s staircase......Page 41 2.5.2 Constructing of a devil’s staircase......Page 42 2.5.3 T-property......Page 45 2.5.4 A fundamental Theorem......Page 47 2.5.5 Consequences for forced oscillators......Page 49 2.5.6 Numerical and analytical approach......Page 50 2.6.1 The Poincare mapping......Page 54 2.6.2 The Arnold’s tongues......Page 57 2.6.3 Numerical results and consequences of a symmetry......Page 59 2.7.1 Small input signal......Page 61 2.7.2 Properties of the rotation number......Page 62 2.7.3 The number of periodic orbits......Page 64 3.1 The system with a low-pass filter......Page 66 3.2.1 The phase-plane trajectories......Page 68 3.2.2 The case > 1 . Phase-modulated output signals......Page 70 3.2.3 The case < 1. Hold-in region......Page 72 3.2.4 Boundary of pull-in region: S2 = S3......Page 76 3.2.5 The case = 1. Boundary of hold-in region......Page 77 3.2.6 The filter with high cut-off frequency......Page 78 3.2.7 The filter with low cut-off frequency......Page 79 3.3 Perturbation of the phase difference (wt)......Page 81 3.3.1 A basic theorem......Page 82 3.3.2 An approximate formula for periodic solutions......Page 83 3.3.3 Numerical experiments......Page 84 3.4.1 The basic notions and motivations......Page 86 3.4.2 An equation of the second order......Page 88 3.4.3 Proof of Theorem 3.3......Page 89 3.4.4 Uniqueness of the manifold......Page 94 3.5.1 Small values of parameter a = A T......Page 96 3.5.2 A neighborhood of the trajectory x = M ( )......Page 98 3.6.1 The Poincare mapping......Page 100 3.6.2 Invariant lines of hyperbolic fixed points......Page 102 3.6.3 Heteroclinic and homoclinic trajectories......Page 104 3.6.4 Melnikov’s theorem......Page 107 3.7 Boundaries of attractive domains......Page 111 3.7.1 Small values of the parameters:......Page 112 3.7.2 Large values of a......Page 114 3.7.3 A neighborhood of the line = H ( )......Page 116 3.7.4 Numerical experiments......Page 117 3.8 The Smale horseshoe. Transient chaos......Page 120 3.8.1 Invariant set of the Smale horseshoe......Page 121 3.8.2 Homeomorphism......Page 124 3.8.3 Comments......Page 126 3.9.1 The system with a filter of the higher order......Page 128 3.9.2 Two-dimensional integral manifold......Page 130 3.9.3 Proof of Theorem 3.10......Page 132 3.9.4 The local linearization......Page 134 4.1 Recurrence equations of the system......Page 138 4.2.1 Type of a periodic point......Page 140 4.2.2 Basic properties of periodic points......Page 142 4.2.3 Li and Yorke Theorem......Page 144 4.3.1 Definition and properties......Page 148 4.3.2 Selected frequency locking regions......Page 151 4.3.3 Application to the map (4.7)......Page 155 4.4.1 Stability of periodic points......Page 156 4.4.2 Stable periodic points of the type n/1 and n/2......Page 157 4.4.3 Attractive set of a fixed point......Page 161 4.4.4 Attractive set of a stable periodic orbit......Page 165 4.5 The number of stable orbits......Page 166 4.5.1 Schwarzian derivative......Page 167 4.5.2 Application to the map T(T) = T + 27rp + asin7......Page 169 4.6.1 Saddle-node bifurcation......Page 172 4.6.2 Period doubling bifurcation......Page 174 4.6.3 The Feigenbaum cascade......Page 176 4.6.4 Invariant measures......Page 179 4.6.5 The Liapunov exponent......Page 182 4.6.6 Skeleton of superstable orbits......Page 184 4.6.7 The Feigenbaum cascade (continuation)......Page 188 4.7 Bifurcation of the rotation interval......Page 191 4.7.1 A simplified mapping......Page 193 4.7.2 Superstable periodic orbits of the type 1/k......Page 195 4.7.3 Family of quadratic polynomials......Page 196 4.7.4 Dynamics restricted to the set I0......Page 198 4.7.5 Asymptotic properties for 0......Page 200 5.1 Description of the DPLL system by a two-dimensional map......Page 202 5.2 Stable periodic orbits......Page 206 5.2.1 Periodic points of the type n/1......Page 207 5.2.2 Stability of fixed points......Page 208 5.2.3 Hold-in regions......Page 211 5.2.4 Small values of......Page 212 5.3 Reduction to a one-dimensional system......Page 213 5.3.1 Existence of an invariant manifold......Page 214 5.3.2 Decay of the invariant manifold......Page 217 5.4.1 Maximal invariant set......Page 220 5.4.2 Attractors......Page 221 5.4.3 Attractive domains......Page 227 Bibliography......Page 232 Index......Page 236 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. "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."--Jacket