SYNCHRONIZATION

By JOHN DAVIS

CONTENTS CHAPTER INTRODUCTION SYNCHRONIZATION IN HISTORICAL PERSPECTIVE SYNCHRONIZATION: JUST A DESCRIPTION WHAT IS SYNCHRONIZATION? WHAT IS NOT SYNCHRONIZATION? SYNCHRONIZATION: AN OVERVIEW OF DIFFERENT CASES TERMINOLOGICAL REMARKS MAIN BIBLIOGRAPHY PART I: SYNCHRONIZATION WITHOUT FORMULAE CHAPTER BASIC NOTIONS: THE SELF-SUSTAINED OSCILLATOR AND ITS PHASE SELF-SUSTAINED OSCILLATORS: MATHEMATICAL MODELS OF NATURAL SYSTEMS SELF-SUSTAINED OSCILLATIONS ARE TYPICAL IN NATURE GEOMETRICAL IMAGE OF PERIODIC SELF-SUSTAINED OSCILLATIONS: LIMIT CYCLE PHASE: DE?NITION AND PROPERTIES PHASE AND AMPLITUDE OF A QUASILINEAR OSCILLATOR AMPLITUDE IS STABLE, PHASE IS FREE GENERAL CASE: LIMIT CYCLE OF ARBITRARY SHAPE SELF-SUSTAINED OSCILLATORS: MAIN FEATURES DISSIPATION, STABILITY AND NONLINEARITY AUTONOMOUS AND FORCED SYSTEMS: PHASE OF A FORCED SYSTEM IS NOT FREE! SELF-SUSTAINED OSCILLATORS: FURTHER EXAMPLES AND DISCUSSION TYPICAL SELF-SUSTAINED SYSTEM: INTERNAL FEEDBACK LOOP RELAXATION OSCILLATORS CHAPTER SYNCHRONIZATION OF A PERIODIC OSCILLATOR BY EXTERNAL FORCE WEAKLY FORCED QUASILINEAR OSCILLATORS THE AUTONOMOUS OSCILLATOR AND THE FORCE IN THE ROTATING REFERENCE FRAME PHASE AND FREQUENCY LOCKING SYNCHRONIZATION TRANSITION AN EXAMPLE: ENTRAINMENT OF RESPIRATION BY A MECHANICAL VENTILATOR SYNCHRONIZATION BY EXTERNAL FORCE: EXTENDED DISCUSSION STROBOSCOPIC OBSERVATION AN EXAMPLE: PERIODICALLY STIMULATED ?RE?Y ENTRAINMENT BY A PULSE TRAIN SYNCHRONIZATION OF HIGHER ORDER ARNOLD TONGUES AN EXAMPLE: PERIODIC STIMULATION OF ATRIAL PACEMAKER CELLS PHASE AND FREQUENCY LOCKING: GENERAL FORMULATION AN EXAMPLE: SYNCHRONIZATION OF A LASER SYNCHRONIZATION OF RELAXATION OSCILLATORS: SPECIAL FEATURES RESETTING BY EXTERNAL PULSES AN EXAMPLE: THE CARDIAC PACEMAKER ELECTRICAL MODEL OF THE HEART BY VAN DER POL AND VAN DER MARK VARIATION OF THE THRESHOLD AN EXAMPLE: THE ELECTRONIC RELAXATION OSCILLATOR VARIATION OF THE NATURAL FREQUENCY MODULATION VS SYNCHRONIZATION AN EXAMPLE: SYNCHRONIZATION OF THE SONGS OF SNOWY TREE CRICKETS SYNCHRONIZATION IN THE PRESENCE OF NOISE PHASE DIFFUSION IN A NOISY OSCILLATOR FORCED NOISY OSCILLATORS PHASE SLIPS AN EXAMPLE: ENTRAINMENT OF RESPIRATION BY MECHANICAL VENTILATION AN EXAMPLE: ENTRAINMENT OF THE CARDIAC RHYTHM BY WEAK EXTERNAL STIMULI DIVERSE EXAMPLES CIRCADIAN RHYTHMS THE MENSTRUAL CYCLE ENTRAINMENT OF PULSATILE INSULIN SECRETION BY OSCILLATORY GLUCOSE INFUSION SYNCHRONIZATION IN PROTOPLASMIC STRANDS OF PHYSARUM PHENOMENA AROUND SYNCHRONIZATION RELATED EFFECTS AT STRONG EXTERNAL FORCING STIMULATION OF EXCITABLE SYSTEMS STOCHASTIC RESONANCE FROM THE SYNCHRONIZATION VIEWPOINT ENTRAINMENT OF SEVERAL OSCILLATORS BY A COMMON DRIVE SYNCHRONIZATION OF TWO AND MANY OSCILLATORS MUTUAL SYNCHRONIZATION OF SELF-SUSTAINED OSCILLATORS TWO INTERACTING OSCILLATORS AN EXAMPLE: SYNCHRONIZATION OF TRIODE GENERATORS AN EXAMPLE: RESPIRATORY AND WING BEAT FREQUENCY OF FREE-?YING BARNACLE GEESE AN EXAMPLE: TRANSITION BETWEEN IN-PHASE AND ANTI-PHASE MOTION CONCLUDING REMARKS AND RELATED EFFECTS RELAXATION OSCILLATORS AN EXAMPLE: TRUE AND LATENT PACEMAKER CELLS IN THE SINO-ATRIAL NODE SYNCHRONIZATION OF NOISY SYSTEMS AN EXAMPLE: BRAIN AND MUSCLE ACTIVITY OF A PARKINSONIAN PATIENT SYNCHRONIZATION OF ROTATORS AN EXAMPLE: JOSEPHSON JUNCTIONS SEVERAL OSCILLATORS CHAINS, LATTICES AND OSCILLATORY MEDIA SYNCHRONIZATION IN A LATTICE AN EXAMPLE: LASER ARRAYS FORMATION OF CLUSTERS AN EXAMPLE: ELECTRICAL ACTIVITY OF MAMMALIAN INTESTINE CLUSTERS AND BEATS IN A MEDIUM: EXTENDED DISCUSSION PERIODICALLY FORCED OSCILLATORY MEDIUM AN EXAMPLE: FORCED BELOUSOV–ZHABOTINSKY REACTION GLOBALLY COUPLED OSCILLATORS KURAMOTO SELF-SYNCHRONIZATION TRANSITION AN EXAMPLE: SYNCHRONIZATION OF MENSTRUAL CYCLES AN EXAMPLE: SYNCHRONIZATION OF GLYCOLYTIC OSCILLATIONS IN A POPULATION OF YEAST CELLS EXPERIMENTAL STUDY OF RHYTHMIC HAND CLAPPING RUNNING AND BREATHING IN MAMMALS SYNCHRONIZATION OF TWO SALT-WATER OSCILLATORS ENTRAINMENT OF TUBULAR PRESSURE OSCILLATIONS IN NEPHRONS POPULATIONS OF CELLS SYNCHRONIZATION OF PREDATOR–PREY CYCLES SYNCHRONIZATION IN NEURONAL SYSTEMS SYNCHRONIZATION OF CHAOTIC SYSTEMS CHAOTIC OSCILLATORS AN EXEMPLAR: THE LORENZ MODEL SENSITIVE DEPENDENCE ON INITIAL CONDITIONS PHASE SYNCHRONIZATION OF CHAOTIC OSCILLATORS PHASE AND AVERAGE FREQUENCY OF A CHAOTIC OSCILLATOR ENTRAINMENT BY A PERIODIC FORCE AN EXAMPLE: FORCED CHAOTIC PLASMA DISCHARGE COMPLETE SYNCHRONIZATION OF CHAOTIC OSCILLATORS COMPLETE SYNCHRONIZATION OF IDENTICAL SYSTEMS AN EXAMPLE: SYNCHRONIZATION OF TWO LASERS SYNCHRONIZATION OF NONIDENTICAL SYSTEMS COMPLETE SYNCHRONIZATION IN A GENERAL CONTEXT AN EXAMPLE: SYNCHRONIZATION AND CLUSTERING OF GLOBALLY COUPLED ELECTROCHEMICAL OSCILLATORS CHAOS-DESTROYING SYNCHRONIZATION DETECTING SYNCHRONIZATION IN EXPERIMENTS ESTIMATING PHASES AND FREQUENCIES FROM DATA PHASE OF A SPIKE TRAIN AN EXAMPLE: ELECTROCARDIOGRAM PHASE OF A NARROW-BAND SIGNAL AN EXAMPLE: RESPIRATION SEVERAL PRACTICAL REMARKS DATA ANALYSIS IN “ACTIVE” AND “PASSIVE” EXPERIMENTS “ACTIVE” EXPERIMENT “PASSIVE” EXPERIMENT ANALYZING RELATIONS BETWEEN THE PHASES STRAIGHTFORWARD ANALYSIS OF THE PHASE DIFFERENCE AN EXAMPLE: POSTURE CONTROL IN HUMANS HIGH LEVEL OF NOISE STROBOSCOPIC TECHNIQUE PHASE STROBOSCOPE IN THE CASE N1 ?M2 AN EXAMPLE: CARDIORESPIRATORY INTERACTION PHASE RELATIONS IN THE CASE OF STRONG MODULATION AN EXAMPLE: SPIKING OF ELECTRORECEPTORS OF A PADDLE?SH CONCLUDING REMARKS AND BIBLIOGRAPHIC NOTES SEVERAL REMARKS ON “PASSIVE” EXPERIMENTS QUANTI?CATION AND SIGNI?CANCE OF PHASE RELATION ANALYSIS SOME RELATED REFERENCES PART II: PHASE LOCKING AND FREQUENCY ENTRAINMENT CHAPTER SYNCHRONIZATION OF PERIODIC OSCILLATORS BY PERIODIC EXTERNAL ACTION PHASE DYNAMICS A LIMIT CYCLE AND THE PHASE OF OSCILLATIONS SMALL PERTURBATIONS AND ISOCHRONES AN EXAMPLE: COMPLEX AMPLITUDE EQUATION THE EQUATION FOR THE PHASE DYNAMICS AN EXAMPLE: FORCED COMPLEX AMPLITUDE EQUATIONS SLOW PHASE DYNAMICS SLOW PHASE DYNAMICS: PHASE LOCKING AND SYNCHRONIZATION REGION SUMMARY OF THE PHASE DYNAMICS WEAKLY NONLINEAR OSCILLATOR THE AMPLITUDE EQUATION SYNCHRONIZATION PROPERTIES: ISOCHRONOUS CASE SYNCHRONIZATION PROPERTIES: NONISOCHRONOUS CASE THE CIRCLE AND ANNULUS MAP THE CIRCLE MAP: DERIVATION AND EXAMPLES THE CIRCLE MAP: PROPERTIES THE ANNULUS MAP LARGE FORCE AND TRANSITION TO CHAOS SYNCHRONIZATION OF ROTATORS AND JOSEPHSON JUNCTIONS DYNAMICS OF ROTATORS AND JOSEPHSON JUNCTIONS OVERDAMPED ROTATOR IN AN EXTERNAL ?ELD PHASE LOCKED LOOPS BIBLIOGRAPHIC NOTES MUTUAL SYNCHRONIZATION OF TWO INTERACTING PERIODIC OSCILLATORS AVERAGED EQUATIONS FOR THE PHASE CIRCLE MAP WEAKLY NONLINEAR OSCILLATORS GENERAL EQUATIONS OSCILLATION DEATH, OR QUENCHING ATTRACTIVE AND REPULSIVE INTERACTION SELF-SUSTAINED OSCILLATOR IN THE PRESENCE OF NOISE QUALITATIVE PICTURE OF THE LANGEVIN DYNAMICS QUANTITATIVE DESCRIPTION FOR WHITE NOISE SYNCHRONIZATION BY A QUASIHARMONIC ?UCTUATING FORCE MUTUAL SYNCHRONIZATION OF NOISY OSCILLATORS CHAPTER 10 PHASE SYNCHRONIZATION OF CHAOTIC SYSTEMS PHASE OF A CHAOTIC OSCILLATOR NOTION OF THE PHASE PHASE DYNAMICS OF CHAOTIC OSCILLATORS SYNCHRONIZATION OF CHAOTIC OSCILLATORS PHASE SYNCHRONIZATION BY EXTERNAL FORCE INDIRECT CHARACTERIZATION OF SYNCHRONIZATION SYNCHRONIZATION IN TERMS OF UNSTABLE PERIODIC ORBITS MUTUAL SYNCHRONIZATION OF TWO COUPLED OSCILLATORS CHAPTER 11 SYNCHRONIZATION IN OSCILLATORY MEDIA OSCILLATOR LATTICES SPATIALLY CONTINUOUS PHASE PRO?LES PLANE WAVES AND TARGETS EFFECT OF NOISE: ROUGHENING VS SYNCHRONIZATION WEAKLY NONLINEAR OSCILLATORY MEDIUM COMPLEX GINZBURG–LANDAU EQUATION FORCING OSCILLATORY MEDIA CHAPTER 12 POPULATIONS OF GLOBALLY COUPLED OSCILLATORS THE KURAMOTO TRANSITION NOISY OSCILLATORS GENERALIZATIONS MODELS BASED ON PHASE APPROXIMATION GLOBALLY COUPLED WEAKLY NONLINEAR OSCILLATORS COUPLED RELAXATION OSCILLATORS COUPLED JOSEPHSON JUNCTIONS FINITE-SIZE EFFECTS ENSEMBLE OF CHAOTIC OSCILLATORS CONTENTS PART III: SYNCHRONIZATION OF CHAOTIC SYSTEMS CHAPTER 13 COMPLETE SYNCHRONIZATION I: BASIC CONCEPTS THE SIMPLEST MODEL: TWO COUPLED MAPS STABILITY OF THE SYNCHRONOUS STATE ONSET OF SYNCHRONIZATION: STATISTICAL THEORY PERTURBATION IS A RANDOM WALK PROCESS THE STATISTICS OF ?NITE-TIME LYAPUNOV EXPONENTS DETERMINE DIFFUSION MODULATIONAL INTERMITTENCY: POWER-LAW DISTRIBUTIONS MODULATIONAL INTERMITTENCY: CORRELATION PROPERTIES ONSET OF SYNCHRONIZATION: TOPOLOGICAL ASPECTS TRANSVERSE BIFURCATIONS OF PERIODIC ORBITS WEAK VS STRONG SYNCHRONIZATION LOCAL AND GLOBAL RIDDLING CHAPTER 14 COMPLETE SYNCHRONIZATION II: GENERALIZATIONS AND COMPLEX SYSTEMS IDENTICAL MAPS, GENERAL COUPLING OPERATOR UNIDIRECTIONAL COUPLING ASYMMETRIC LOCAL COUPLING GLOBAL (MEAN ?ELD) COUPLING CONTINUOUS-TIME SYSTEMS SPATIALLY DISTRIBUTED SYSTEMS SPATIALLY HOMOGENEOUS CHAOS TRANSVERSE SYNCHRONIZATION OF SPACE–TIME CHAOS SYNCHRONIZATION OF COUPLED CELLULAR AUTOMATA SYNCHRONIZATION AS A GENERAL SYMMETRIC STATE REPLICA-SYMMETRIC SYSTEMS CHAPTER 15 SYNCHRONIZATION OF COMPLEX DYNAMICS BY EXTERNAL FORCES SYNCHRONIZATION BY PERIODIC FORCING SYNCHRONIZATION BY NOISY FORCING NOISY FORCED PERIODIC OSCILLATIONS SYNCHRONIZATION OF CHAOTIC OSCILLATIONS BY NOISY FORCING SYNCHRONIZATION OF CHAOTIC OSCILLATIONS BY CHAOTIC FORCING COMPLETE SYNCHRONIZATION GENERALIZED SYNCHRONIZATION