# 1.1 Synchronization in historical perspective 1

1.1 Synchronization in historical perspective 1
1.2 Synchronization: just a description 7
1.2.1 What is synchronization? 8
1.2.2 What is NOT synchronization? 14
1.3 Synchronization: an overview of different cases 18
1.3.1 Terminological remarks 22
1.4 Main bibliography 23
Part I: Synchronization without formulae
Chapter 2 Basic notions: the self-sustained oscillator and its phase 27
2.1 Self-sustained oscillators: mathematical models of natural systems 28
2.1.1 Self-sustained oscillations are typical in nature 28
2.1.2 Geometrical image of periodic self-sustained oscillations: limit cycle 29
2.2 Phase: definition and properties 31
2.2.1 Phase and amplitude of a quasilinear oscillator 31
2.2.2 Amplitude is stable, phase is free 32
2.2.3 General case: limit cycle of arbitrary shape 33
2.3 Self-sustained oscillators: main features 35
2.3.1 Dissipation, stability and nonlinearity 35
2.3.2 Autonomous and forced systems: phase of a forced system is not free! 38
2.4 Self-sustained oscillators: further examples and discussion 40
2.4.1 Typical self-sustained system: internal feedback loop 40
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2.4.2 Relaxation oscillators 41
Chapter 3 Synchronization of a periodic oscillator by external force 45
3.1 Weakly forced quasilinear oscillators 46
3.1.1 The autonomous oscillator and the force in the rotating reference frame 46
3.1.2 Phase and frequency locking 49
3.1.3 Synchronization transition 53
3.1.4 An example: entrainment of respiration by a mechanical ventilator 56
3.2 Synchronization by external force: extended discussion 59
3.2.1 Stroboscopic observation 59
3.2.2 An example: periodically stimulated firefly 61
3.2.3 Entrainment by a pulse train 62
3.2.4 Synchronization of higher order. Arnold tongues 65
3.2.5 An example: periodic stimulation of atrial pacemaker cells 67
3.2.6 Phase and frequency locking: general formulation 67
3.2.7 An example: synchronization of a laser 69
3.3 Synchronization of relaxation oscillators: special features 71
3.3.1 Resetting by external pulses. An example: the cardiac pacemaker 71
3.3.2 Electrical model of the heart by van der Pol and van der Mark 72
3.3.3 Variation of the threshold. An example: the electronic relaxation oscillator 73
3.3.4 Variation of the natural frequency 76
3.3.5 Modulation vs. synchronization 77
3.3.6 An example: synchronization of the songs of snowy tree crickets 78
3.4 Synchronization in the presence of noise 79
3.4.1 Phase diffusion in a noisy oscillator 80
3.4.2 Forced noisy oscillators. Phase slips 81
3.4.3 An example: entrainment of respiration by mechanical ventilation 85
3.4.4 An example: entrainment of the cardiac rhythm by weak external stimuli 85
3.5 Diverse examples 86
3.5.2 The menstrual cycle 88
3.5.3 Entrainment of pulsatile insulin secretion by oscillatory glucose infusion 89
3.5.4 Synchronization in protoplasmic strands of Physarum 90
3.6 Phenomena around synchronization 90
3.6.1 Related effects at strong external forcing 91
3.6.2 Stimulation of excitable systems 93
3.6.3 Stochastic resonance from the synchronization viewpoint 94
3.6.4 Entrainment of several oscillators by a common drive 98
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Chapter 4 Synchronization of two and many oscillators 102
4.1 Mutual synchronization of self-sustained oscillators 102
4.1.1 Two interacting oscillators 103
4.1.2 An example: synchronization of triode generators 105
4.1.3 An example: respiratory and wing beat frequency of free-flying barnacle
geese 107
4.1.4 An example: transition between in-phase and anti-phase motion 108
4.1.5 Concluding remarks and related effects 110
4.1.6 Relaxation oscillators. An example: true and latent pacemaker cells in the
sino-atrial node 111
4.1.7 Synchronization of noisy systems. An example: brain and muscle activity of a
Parkinsonian patient 112
4.1.8 Synchronization of rotators. An example: Josephson junctions 114
4.1.9 Several oscillators 117
4.2 Chains, lattices and oscillatory media 119
4.2.1 Synchronization in a lattice. An example: laser arrays 119
4.2.2 Formation of clusters. An example: electrical activity of mammalian
intestine 121
4.2.3 Clusters and beats in a medium: extended discussion 122
4.2.4 Periodically forced oscillatory medium. An example: forced
Belousov–Zhabotinsky reaction 124
4.3 Globally coupled oscillators 126
4.3.1 Kuramoto self-synchronization transition 126
4.3.2 An example: synchronization of menstrual cycles 129
4.3.3 An example: synchronization of glycolytic oscillations in a population of
yeast cells 130
4.3.4 Experimental study of rhythmic hand clapping 131
4.4 Diverse examples 131
4.4.1 Running and breathing in mammals 131
4.4.2 Synchronization of two salt-water oscillators 133
4.4.3 Entrainment of tubular pressure oscillations in nephrons 133
4.4.4 Populations of cells 133
4.4.5 Synchronization of predator–prey cycles 134
4.4.6 Synchronization in neuronal systems 134
Chapter 5 Synchronization of chaotic systems 137
5.1 Chaotic oscillators 137
5.1.1 An exemplar: the Lorenz model 138
5.1.2 Sensitive dependence on initial conditions 140
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5.2 Phase synchronization of chaotic oscillators 141
5.2.1 Phase and average frequency of a chaotic oscillator 142
5.2.2 Entrainment by a periodic force. An example: forced chaotic plasma
discharge 144
5.3 Complete synchronization of chaotic oscillators 147
5.3.1 Complete synchronization of identical systems. An example:
synchronization of two lasers 148
5.3.2 Synchronization of nonidentical systems 149
5.3.3 Complete synchronization in a general context. An example: synchronization
and clustering of globally coupled electrochemical oscillators 150
5.3.4 Chaos-destroying synchronization 152
Chapter 6 Detecting synchronization in experiments 153
6.1 Estimating phases and frequencies from data 153
6.1.1 Phase of a spike train. An example: electrocardiogram 154
6.1.2 Phase of a narrow-band signal. An example: respiration 155
6.1.3 Several practical remarks 155
6.2 Data analysis in “active” and “passive” experiments 156
6.2.1 “Active” experiment 156
6.2.2 “Passive” experiment 157
6.3 Analyzing relations between the phases 160
6.3.1 Straightforward analysis of the phase difference. An example: posture control
in humans 160
6.3.2 High level of noise 163
6.3.3 Stroboscopic technique 163
6.3.4 Phase stroboscope in the case n1 ≈ m2. An example: cardiorespiratory
interaction 164
6.3.5 Phase relations in the case of strong modulation. An example: spiking of
6.4 Concluding remarks and bibliographic notes 168
6.4.1 Several remarks on “passive” experiments 168
6.4.2 Quantification and significance of phase relation analysis 170
6.4.3 Some related references 171
Part II: Phase locking and frequency entrainment
Chapter 7 Synchronization of periodic oscillators by periodic external
action 175
7.1 Phase dynamics 176
7.1.1 A limit cycle and the phase of oscillations 176
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7.1.2 Small perturbations and isochrones 177
7.1.3 An example: complex amplitude equation 179
7.1.4 The equation for the phase dynamics 180
7.1.5 An example: forced complex amplitude equations 181
7.1.6 Slow phase dynamics 182
7.1.7 Slow phase dynamics: phase locking and synchronization region 184
7.1.8 Summary of the phase dynamics 187
7.2 Weakly nonlinear oscillator 189
7.2.1 The amplitude equation 189
7.2.2 Synchronization properties: isochronous case 192
7.2.3 Synchronization properties: nonisochronous case 198
7.3 The circle and annulus map 199
7.3.1 The circle map: derivation and examples 201
7.3.2 The circle map: properties 204
7.3.3 The annulus map 210
7.3.4 Large force and transition to chaos 213
7.4 Synchronization of rotators and Josephson junctions 215
7.4.1 Dynamics of rotators and Josephson junctions 215
7.4.2 Overdamped rotator in an external field 217
7.5 Phase locked loops 218
7.6 Bibliographic notes 221
Chapter 8 Mutual synchronization of two interacting periodic
oscillators 222
8.1 Phase dynamics 222
8.1.1 Averaged equations for the phase 224
8.1.2 Circle map 226
8.2 Weakly nonlinear oscillators 227
8.2.1 General equations 227
8.2.2 Oscillation death, or quenching 229
8.2.3 Attractive and repulsive interaction 230
8.3 Relaxation oscillators 232
8.4 Bibliographic notes 235
Chapter 9 Synchronization in the presence of noise 236
9.1 Self-sustained oscillator in the presence of noise 236
9.2 Synchronization in the presence of noise 237
9.2.1 Qualitative picture of the Langevin dynamics 237
9.2.2 Quantitative description for white noise 240
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9.2.3 Synchronization by a quasiharmonic fluctuating force 244
9.2.4 Mutual synchronization of noisy oscillators 245
9.3 Bibliographic notes 246
Chapter 10 Phase synchronization of chaotic systems 247
10.1 Phase of a chaotic oscillator 248
10.1.1 Notion of the phase 248
10.1.2 Phase dynamics of chaotic oscillators 254
10.2 Synchronization of chaotic oscillators 255
10.2.1 Phase synchronization by external force 256
10.2.2 Indirect characterization of synchronization 258
10.2.3 Synchronization in terms of unstable periodic orbits 260
10.2.4 Mutual synchronization of two coupled oscillators 262
10.3 Bibliographic notes 263
Chapter 11 Synchronization in oscillatory media 266
11.1 Oscillator lattices 266
11.2 Spatially continuous phase profiles 269
11.2.1 Plane waves and targets 269
11.2.2 Effect of noise: roughening vs. synchronization 271
11.3 Weakly nonlinear oscillatory medium 273
11.3.1 Complex Ginzburg–Landau equation 273
11.3.2 Forcing oscillatory media 276
11.4 Bibliographic notes 278
Chapter 12 Populations of globally coupled oscillators 279
12.1 The Kuramoto transition 279
12.2 Noisy oscillators 283
12.3 Generalizations 286
12.3.1 Models based on phase approximation 286
12.3.2 Globally coupled weakly nonlinear oscillators 289
12.3.3 Coupled relaxation oscillators 290
12.3.4 Coupled Josephson junctions 291
12.3.5 Finite-size effects 294
12.3.6 Ensemble of chaotic oscillators 294
12.4 Bibliographic notes 296
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Part III: Synchronization of chaotic systems
Chapter 13 Complete synchronization I: basic concepts 301
13.1 The simplest model: two coupled maps 302
13.2 Stability of the synchronous state 304
13.3 Onset of synchronization: statistical theory 307
13.3.1 Perturbation is a random walk process 307
13.3.2 The statistics of finite-time Lyapunov exponents determine diffusion 308
13.3.3 Modulational intermittency: power-law distributions 310
13.3.4 Modulational intermittency: correlation properties 316
13.4 Onset of synchronization: topological aspects 318
13.4.1 Transverse bifurcations of periodic orbits 318
13.4.2 Weak vs. strong synchronization 319
13.4.3 Local and global riddling 322
13.5 Bibliographic notes 323
Chapter 14 Complete synchronization II: generalizations and complex
systems 324
14.1 Identical maps, general coupling operator 324
14.1.1 Unidirectional coupling 325
14.1.2 Asymmetric local coupling 327
14.1.3 Global (mean field) coupling 328
14.2 Continuous-time systems 329
14.3 Spatially distributed systems 331
14.3.1 Spatially homogeneous chaos 331
14.3.2 Transverse synchronization of space–time chaos 332
14.3.3 Synchronization of coupled cellular automata 334
14.4 Synchronization as a general symmetric state 335
14.4.1 Replica-symmetric systems 336
14.5 Bibliographic notes 337
Chapter 15 Synchronization of complex dynamics by external forces 340
15.1 Synchronization by periodic forcing 341
15.2 Synchronization by noisy forcing 341
15.2.1 Noisy forced periodic oscillations 343
15.2.2 Synchronization of chaotic oscillations by noisy forcing 345
15.3 Synchronization of chaotic oscillations by chaotic forcing 346
15.3.1 Complete synchronization 346
15.3.2 Generalized synchronization 347
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15.3.3 Generalized synchronization by quasiperiodic driving 352
15.4 Bibliographic notes 353
Appendices
Appendix A1: Discovery of synchronization by Christiaan Huygens 357
A1.1 A letter from Christiaan Huygens to his father, Constantyn Huygens 357
A1.2 Sea clocks (sympathy of clocks). Part V 358
Appendix A2: Instantaneous phase and frequency of a signal 362
A2.1 Analytic signal and the Hilbert transform 362
A2.2 Examples 363
A2.3 Numerics: practical hints and know-hows 366
A2.4 Computation of the instantaneous frequency