Table Of ContentVladimirI.Nekorkin
IntroductiontoNonlinearOscillations
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VladimirI.Nekorkin
Introduction to Nonlinear Oscillations
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V
Contents
Preface XI
1 IntroductiontotheTheoryofOscillations 1
1.1 GeneralFeaturesoftheTheoryofOscillations 1
1.2 DynamicalSystems 2
1.2.1 TypesofTrajectories 3
1.2.2 DynamicalSystemswithContinuousTime 3
1.2.3 DynamicalSystemswithDiscreteTime 4
1.2.4 DissipativeDynamicalSystems 5
1.3 Attractors 6
1.4 StructuralStabilityofDynamicalSystems 7
1.5 ControlQuestionsandExercises 8
2 One-DimensionalDynamics 11
2.1 QualitativeApproach 11
2.2 RoughEquilibria 13
2.3 BifurcationsofEquilibria 14
2.3.1 Saddle-nodeBifurcation 14
2.3.2 TheConceptoftheNormalForm 15
2.3.3 TranscriticalBifurcation 16
2.3.4 PitchforkBifurcation 17
2.4 SystemsontheCircle 18
2.5 ControlQuestionsandExercises 19
3 StabilityofEquilibria.AClassificationofEquilibriaofTwo-Dimensional
LinearSystems 21
3.1 DefinitionoftheStabilityofEquilibria 22
3.2 ClassificationofEquilibriaofLinearSystemsonthePlane 24
3.2.1 RealRoots 25
3.2.1.1 Rootsλ andλ oftheSameSign 26
1 2
3.2.1.2 TheRootsλ andλ withDifferentSigns 27
1 2
3.2.1.3 TheRootsλ andλ areMultiplesofλ =λ =λ 28
1 2 1 2
3.2.2 ComplexRoots 29
VI Contents
3.2.3 Oscillationsoftwo-dimensionallinearsystems 30
3.2.4 Two-parameterBifurcationDiagram 30
3.3 ControlQuestionsandExercises 33
4 AnalysisoftheStabilityofEquilibriaofMultidimensionalNonlinear
Systems 35
4.1 LinearizationMethod 35
4.2 TheRouth–HurwitzStabilityCriterion 36
4.3 TheSecondLyapunovMethod 38
4.4 HyperbolicEquilibriaofThree-DimensionalSystems 41
4.4.1 RealRoots 41
4.4.1.1 Rootsλ ofOneSign 41
i
4.4.1.2 Rootsλ ofDifferentSigns 42
i
4.4.2 ComplexRoots 43
4.4.2.1 RealPartsoftheRootsλ ofOneSign 44
i
4.4.2.2 RealPartsofRootsλ ofDifferentSigns 45
i
4.4.3 TheEquilibriaofThree-DimensionalNonlinearSystems 45
4.4.4 Two-ParameterBifurcationDiagram 46
4.5 ControlQuestionsandExercises 49
5 LinearandNonlinearOscillators 53
5.1 TheDynamicsofaLinearOscillator 53
5.1.1 HarmonicOscillator 54
5.1.2 LinearOscillatorwithLosses 57
5.1.3 LinearOscillatorwith“Negative”Damping 60
5.2 DynamicsofaNonlinearOscillator 61
5.2.1 ConservativeNonlinearOscillator 61
5.2.2 NonlinearOscillatorwithDissipation 68
5.3 ControlQuestionsandExercises 69
6 BasicPropertiesofMaps 71
6.1 PointMapsasModelsofDiscreteSystems 71
6.2 PoincaréMap 72
6.3 FixedPoints 75
6.4 One-DimensionalLinearMaps 77
6.5 Two-DimensionalLinearMaps 79
6.5.1 RealMultipliers 79
6.5.1.1 TheStableNodeFixedPoint 80
6.5.1.2 TheUnstableNodeFixedPoint 81
6.5.1.3 TheSaddleFixedPoint 82
6.5.2 ComplexMultipliers 82
6.6 One-DimensionalNonlinearMaps:SomeNotionsand
Examples 84
6.7 ControlQuestionsandExercises 87
Contents VII
7 LimitCycles 89
7.1 IsolatedandNonisolatedPeriodicTrajectories.DefinitionofaLimit
Cycle 89
7.2 OrbitalStability.StableandUnstableLimitCycles 91
7.2.1 DefinitionofOrbitalStability 91
7.2.2 CharacteristicsofLimitCycles 92
7.3 RotationalandLibrationalLimitCycles 94
7.4 RoughLimitCyclesinThree-DimensionalSpace 94
7.5 TheBendixson–DulacCriterion 96
7.6 ControlQuestionsandExercises 98
8 BasicBifurcationsofEquilibriainthePlane 101
8.1 BifurcationConditions 101
8.2 Saddle-NodeBifurcation 102
8.3 TheAndronov–HopfBifurcation 104
8.3.1 TheFirstLyapunovCoefficientisNegative 105
8.3.2 TheFirstLyapunovCoefficientisPositive 106
8.3.3 “Soft”and“Hard”GenerationofPeriodicOscillations 107
8.4 StabilityLossDelayfortheDynamicAndronov–Hopf
Bifurcation 108
8.5 ControlQuestionsandExercises 110
9 BifurcationsofLimitCycles.SaddleHomoclinicBifurcation 113
9.1 Saddle-nodeBifurcationofLimitCycles 113
9.2 SaddleHomoclinicBifurcation 117
9.2.1 MapintheVicinityoftheHomoclinicTrajectory 117
9.2.2 LibrationalandRotationalHomoclinicTrajectories 121
9.3 ControlQuestionsandExercises 122
10 TheSaddle-NodeHomoclinicBifurcation.DynamicsofSlow–Fast
SystemsinthePlane 123
10.1 HomoclinicTrajectory 123
10.2 FinalRemarksonBifurcationsofSystemsinthePlane 126
10.3 DynamicsofaSlow-FastSystem 127
10.3.1 SlowandFastMotions 128
10.3.2 SystemswithaSingleRelaxation 129
10.3.3 RelaxationalOscillations 130
10.4 ControlQuestionsandExercises 133
11 DynamicsofaSuperconductingJosephsonJunction 137
11.1 StationaryandNonstationaryEffects 137
11.2 EquivalentCircuitoftheJunction 139
11.3 DynamicsoftheModel 140
11.3.1 ConservativeCase 140
11.3.2 DissipativeCase 141
VIII Contents
11.3.2.1 AbsorbingDomain 141
11.3.2.2 EquilibriaandTheirLocalProperties 142
11.3.2.3 TheLyapunovFunction 144
11.3.2.4 ContactlessCurvesandControlChannelsforSeparatrices 146
11.3.2.5 HomoclinicOrbitsandTheirBifurcations 150
11.3.2.6 LimitCyclesandtheBifurcationDiagram 153
11.3.2.7 I–VCurveoftheJunction 156
11.4 ControlQuestionsandExercises 158
12 TheVanderPolMethod.Self-SustainedOscillationsandTruncated
Systems 159
12.1 TheNotionofAsymptoticMethods 159
12.1.1 ReducingtheSystemtotheGeneralForm 160
12.1.2 Averaged(Truncated)System 160
12.1.3 AveragingandStructurallyStablePhasePortraits 161
12.2 Self-SustainedOscillationsandSelf-Oscillatory
Systems 162
12.2.1 DynamicsoftheSimplestModelofaPendulumClock 163
12.2.2 Self-SustainedOscillationsintheSystemwithanActive
Element 166
12.3 ControlQuestionsandExercises 173
13 ForcedOscillationsofaLinearOscillator 175
13.1 DynamicsoftheSystemandtheGlobalPoincaréMap 175
13.2 ResonanceCurve 180
13.3 ControlQuestionsandExercises 183
14 ForcedOscillationsinWeaklyNonlinearSystemswithOneDegreeof
Freedom 185
14.1 ReductionofaSystemtotheStandardForm 185
14.2 ResonanceinaNonlinearOscillator 187
14.2.1 DynamicsoftheSystemofTruncatedEquations 188
14.2.2 ForcedOscillationsandResonanceCurves 192
14.3 ForcedOscillationRegime 194
14.4 ControlQuestionsandExercises 195
15 ForcedSynchronizationofaSelf-OscillatorySystemwithaPeriodic
ExternalForce 197
15.1 DynamicsofaTruncatedSystem 198
15.1.1 DynamicsintheAbsenceofDetuning 202
15.1.2 DynamicswithDetuning 203
15.2 ThePoincaréMapandSynchronousRegime 205
15.3 Amplitude-FrequencyCharacteristic 207
15.4 ControlQuestionsandExercises 208
Contents IX
16 ParametricOscillations 209
16.1 TheFloquetTheory 210
16.1.1 GeneralSolution 210
16.1.2 PeriodMap 213
16.1.3 StabilityofZeroSolution 214
16.2 BasicRegimesofLinearParametricSystems 216
16.2.1 ParametricOscillationsandParametricResonance 217
16.2.2 ParametricOscillationsofaPendulum 220
16.2.2.1 PendulumOscillationsintheConservativeCase 220
16.2.2.2 PendulumOscillationswiththeLossesTakenintoAccount 223
16.3 PendulumDynamicswithaVibratingSuspensionPoint 228
16.4 OscillationsofaLinearOscillatorwithSlowlyVariable
Frequency 230
17 AnswerstoSelectedExercises 233
Bibliography 245
Index 247
XI
Preface
Atthefoundationofthiscoursematerialarelecturesonageneralcourseinthe
theoryofoscillations,whichweretaughtbytheauthorformorethan20yearsat
theFacultyofRadiophysicsatNizhnyNovgorodStateUniversity(NNSU).
The aim of the course was not only to express fundamental ideas and meth-
odsofthetheoryofoscillationsasascienceofevolutionaryprocesses,butalso
toteachtheaudiencethemethodsandtechniquesofsolvingspecific(practical)
problems.
Thekeyroleinformingthislecturecourseisplayedbyqualitativemethodsof
thetheoryofdynamicalsystemsandmethodsofthetheoryofbifurcations,which
followthetraditionofNizhnyNovgorodschoolofnonlinearoscillations.These
methodsareevenusedwhensolvingsimpleproblems,where,inprinciple,their
useisnotnecessary.Suchawayofpresentingthefollowingmaterialallowsus,
firstofall,torevealtheessenceandfundamentalprinciplesofthemethods,and,
secondly,forthereadertodeveloptheskillsnecessarytoputthemtouse,which
appearstobeimportantforthetransitiontostudyingmorecomplexproblems.
Thebookisconstructedintheformoflecturesinaccordancewiththesyllabus
ofthecourse“TheoryofOscillations”fortheFacultyofRadiophysicsatNNSU.
Yet,thecontentofnearlyeverylectureinthisbookisexpandedfurtherthanitis
usuallypresentedduringthereadingofaformallecture.Thismakesitpossiblefor
thereadertogainadditionalknowledgeonthesubject.Attheendofeachlecture,
therearetestquestionsandproblemsforrevisionandindependentstudy.
Thistextcouldalsoproveusefultoundergraduateandgraduatestudentsspe-
cializinginthefieldofnonlineardynamics,informationsystems,controltheory,
biophysics,andsoon.
Theauthorisgratefultothecolleaguesatthedepartmentof“TheoryofOscilla-
tionsandAutomatedControl”formanyusefuldiscussionsonthetopicsofthis
text and to the colleagues from the department of Nonlinear Dynamics at the
InstituteofAppliedPhysicsoftheRussianAcademyofSciences.
NizhnyNovgorod VladimirI.Nekorkin
October2014
