Nonlinear ordinary differential equations

540 Pages·2007·5.552 MB·English

by  Dominic Jordan, Peter Smith

Nonlinear Ordinary
Differential Equations
An introduction for Scientists and Engineers
FOURTHEDITION
D. W. Jordan and P. Smith
Keele University
1
3
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Jordan,D.W.(DominicWilliam)
Nonlinearordinarydifferentialequations/D.W.Jordanand
P.Smith.—3rded.
(Oxfordappliedandengineeringmathematics)
1.Differentialequations,Nonlinear.I.Smith,Peter,1935–
II.Title,III.Series.
(cid:1)
QA372.J58 1999 515.352—dc21 99-17648.
TypesetbyNewgenImagingSystems(P)Ltd.,Chennai,India
PrintedinGreatBritain
onacid-freepaperby
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ISBN 978–0–19–920824–1 (Hbk)
ISBN 978–0–19–920825–8 (Pbk)
10 9 8 7 6 5 4 3 2 1
Contents
Prefacetothefourthedition vii
1 Second-orderdifferentialequationsinthephaseplane 1
1.1 Phasediagramforthependulumequation 1
1.2 Autonomousequationsinthephaseplane 5
1.3 Mechanicalanalogyfortheconservativesystemx¨=f(x) 14
1.4 Thedampedlinearoscillator 21
1.5 Nonlineardamping:limitcycles 25
1.6 Someapplications 32
1.7 Parameter-dependentconservativesystems 37
1.8 Graphicalrepresentationofsolutions 40
Problems 42
2 Planeautonomoussystemsandlinearization 49
2.1 Thegeneralphaseplane 49
2.2 Somepopulationmodels 53
2.3 Linearapproximationatequilibriumpoints 57
2.4 Thegeneralsolutionoflinearautonomousplanesystems 58
2.5 Thephasepathsoflinearautonomousplanesystems 63
2.6 Scalinginthephasediagramforalinearautonomoussystem 72
2.7 Constructingaphasediagram 73
2.8 Hamiltoniansystems 75
Problems 79
3 Geometricalaspectsofplaneautonomoussystems 89
3.1 Theindexofapoint 89
3.2 Theindexatinfinity 97
3.3 Thephasediagramatinfinity 100
3.4 Limitcyclesandotherclosedpaths 104
3.5 Computationofthephasediagram 107
3.6 Homoclinicandheteroclinicpaths 111
Problems 113
iv Contents
4 Periodicsolutions;averagingmethods 125
4.1 Anenergy-balancemethodforlimitcycles 125
4.2 Amplitudeandfrequencyestimates:polarcoordinates 130
4.3 Anaveragingmethodforspiralphasepaths 134
4.4 Periodicsolutions:harmonicbalance 138
4.5 Theequivalentlinearequationbyharmonicbalance 140
Problems 143
5 Perturbationmethods 149
5.1 Nonautonomoussystems:forcedoscillations 149
5.2 ThedirectperturbationmethodfortheundampedDuffing’sequation 153
5.3 Forcedoscillationsfarfromresonance 155
5.4 Forcedoscillationsnearresonancewithweakexcitation 157
5.5 Theamplitudeequationfortheundampedpendulum 159
5.6 Theamplitudeequationforadampedpendulum 163
5.7 Softandhardsprings 164
5.8 Amplitude–phaseperturbationforthependulumequation 167
5.9 Periodicsolutionsofautonomousequations(Lindstedt’smethod) 169
5.10 Forcedoscillationofaself-excitedequation 171
5.11 TheperturbationmethodandFourierseries 173
5.12 Homoclinicbifurcation:anexample 175
Problems 179
6 Singularperturbationmethods 183
6.1 Non-uniformapproximationstofunctionsonaninterval 183
6.2 Coordinateperturbation 185
6.3 Lighthill’smethod 190
6.4 Time-scalingforseriessolutionsofautonomousequations 192
6.5 Themultiple-scaletechniqueappliedtosaddlepointsandnodes 199
6.6 Matchingapproximationsonaninterval 206
6.7 Amatchingtechniquefordifferentialequations 211
Problems 217
7 Forcedoscillations:harmonicandsubharmonicresponse,stability,
andentrainment 223
7.1 Generalforcedperiodicsolutions 223
7.2 Harmonicsolutions,transients,andstabilityforDuffing’sequation 225
7.3 Thejumpphenomenon 231
7.4 Harmonicoscillations,stability,andtransientsfortheforcedvanderPolequation 234
7.5 FrequencyentrainmentforthevanderPolequation 239
Contents v
7.6 SubharmonicsofDuffing’sequationbyperturbation 242
7.7 StabilityandtransientsforsubharmonicsofDuffing’sequation 247
Problems 251
8 Stability 259
8.1 Poincaréstability(stabilityofpaths) 260
8.2 Pathsandsolutioncurvesforgeneralsystems 265
8.3 Stabilityoftimesolutions:Liapunovstability 267
8.4 Liapunovstabilityofplaneautonomouslinearsystems 271
8.5 Structureofthesolutionsofn-dimensionallinearsystems 274
8.6 Structureofn-dimensionalinhomogeneouslinearsystems 279
8.7 Stabilityandboundednessforlinearsystems 283
8.8 Stabilityoflinearsystemswithconstantcoefficients 284
8.9 Linearapproximationatequilibriumpointsforfirst-ordersystemsinnvariables 289
8.10 Stabilityofaclassofnon-autonomouslinearsystemsinndimensions 293
8.11 Stabilityofthezerosolutionsofnearlylinearsystems 298
Problems 300
9 Stabilitybysolutionperturbation:Mathieu’sequation 305
9.1 Thestabilityofforcedoscillationsbysolutionperturbation 305
9.2 Equationswithperiodiccoefficients(Floquettheory) 308
9.3 Mathieu’sequationarisingfromaDuffingequation 315
9.4 TransitioncurvesforMathieu’sequationbyperturbation 322
9.5 Mathieu’sdampedequationarisingfromaDuffingequation 325
Problems 330
10 Liapunovmethodsfordeterminingstabilityofthezerosolution 337
10.1 IntroducingtheLiapunovmethod 337
10.2 TopographicsystemsandthePoincaré–Bendixsontheorem 338
10.3 Liapunovstabilityofthezerosolution 342
10.4 Asymptoticstabilityofthezerosolution 346
10.5 ExtendingweakLiapunovfunctionstoasymptoticstability 349
10.6 Amoregeneraltheoryforautonomoussystems 351
10.7 Atestforinstabilityofthezerosolution:ndimensions 356
10.8 Stabilityandthelinearapproximationintwodimensions 357
10.9 Exponentialfunctionofamatrix 365
10.10 Stabilityandthelinearapproximationfornthorderautonomoussystems 367
10.11 Specialsystems 373
Problems 377
vi Contents
11 Theexistenceofperiodicsolutions 383
11.1 ThePoincaré–Bendixsontheoremandperiodicsolutions 383
11.2 Atheoremontheexistenceofacentre 390
11.3 Atheoremontheexistenceofalimitcycle 394
11.4 VanderPol’sequationwithlargeparameter 400
Problems 403
12 Bifurcationsandmanifolds 405
12.1 Examplesofsimplebifurcations 405
12.2 Thefoldandthecusp 407
12.3 Furthertypesofbifurcation 411
12.4 Hopfbifurcations 419
12.5 Higher-ordersystems:manifolds 422
12.6 Linearapproximation:centremanifolds 427
Problems 433
13 Poincarésequences,homoclinicbifurcation,andchaos 439
13.1 Poincarésequences 439
13.2 Poincarésectionsfornonautonomoussystems 442
13.3 Subharmonicsandperioddoubling 447
13.4 Homoclinicpaths,strangeattractorsandchaos 450
13.5 TheDuffingoscillator 453
13.6 Adiscretesystem:thelogisticdifferenceequation 462
13.7 Liapunovexponentsanddifferenceequations 466
13.8 Homoclinicbifurcationforforcedsystems 469
13.9 Thehorseshoemap 476
13.10 Melnikov’smethodfordetectinghomoclinicbifurcation 477
13.11 Liapunovexponentsanddifferentialequations 483
13.12 Powerspectra 491
13.13 Somefurtherfeaturesofchaoticoscillations 492
Problems 494
Answerstotheexercises 507
Appendices 511
A Existenceanduniquenesstheorems 511
B Topographicsystems 513
C Normsforvectorsandmatrices 515
D Acontourintegral 517
E Usefulresults 518
Referencesandfurtherreading 521
Index 525
Preface to the fourth edition
ThisbookisarevisedandreseteditionofNonlinearordinarydifferentialequations,published
inpreviouseditionsin1977,1987,and1999.Additionalmaterialreflectingthegrowthinthe
literatureonnonlinearsystemshasbeenincluded,whilstretainingthebasicstyleandstructure
ofthetextbook.Thewideapplicabilityofthesubjecttothephysical,engineering,andbiological
sciencescontinuestogenerateasupplyofnewproblemsofpracticalandtheoreticalinterest.
Thebookdevelopedfromcoursesonnonlineardifferentialequationsgivenovermanyyears
in the Mathematics Department of Keele University. It presents an introduction to dynamical
systemsinthecontextofordinarydifferentialequations,andisintendedforstudentsofmathe-
matics,engineeringandthesciences,andworkersintheseareaswhoaremainlyinterestedinthe
more direct applications of the subject. The level is about that of final-year undergraduate, or
master’sdegreecoursesintheUK.IthasbeenfoundthatselectedmaterialfromChapters1to5,
and8,10,and11canbecoveredinaone-semestercoursebystudentshavingabackgroundof
techniques in differential equations and linear algebra. The book is designed to accommodate
courses of varying emphasis, the chapters forming fairly self-contained groups from which a
coherentselectioncanbemadewithoutusingsignificantpartsoftheargument.
From the large number of citations in research papers it appears that although it is mainly
intended to be a textbook it is often used as a source of methods for a wide spectrum of
applicationsinthescientificandengineeringliterature.Wehopethatresearchworkersinmany
disciplineswillfindtheneweditionequallyhelpful.
Generalsolutionsofnonlineardifferentialequationsarerarelyobtainable,thoughparticular
solutions can be calculated one at a time by standard numerical techniques. However, this
book deals with qualitative methods that reveal the novel phenomena arising from nonlinear
equations, and produce good numerical estimates of parameters connected with such general
featuresasstability,periodicityandchaoticbehaviourwithouttheneedtosolvetheequations.
Weillustratethereliabilityofsuchmethodsbygraphicalornumericalcomparisonwithnumer-
ical solutions. For this purpose the Mathematica™software was used to calculate particular
exact solutions; this was also of great assistance in the construction of perturbation series,
trigonometricidentities,andforotheralgebraicmanipulation.However,experiencewithsuch
softwareisnotnecessaryforthereader.
Chapters1to4mainlytreatplaneautonomoussystems.Thetreatmentiskeptatanintuitive
level,butwetrytoencouragethereadertofeelthat,almostimmediately,usefulnewinvestiga-
tivetechniquesarereadilyavailable.Themainfeaturesofthephaseplane—equilibriumpoints,
linearization, limit cycles, geometrical aspects—are investigated informally. Quantitative esti-
mates for solutions are obtained by energy considerations, harmonic balance, and averaging
methods.
viii Prefacetothefourthedition
Variousperturbationtechniquesfordifferentialequationswhichcontainasmallparameter
aredescribedinChapter5,andsingularperturbationsfornon-uniformexpansionsaretreated
extensively in Chapter 6. Chapter 7 investigates harmonic and subharmonic responses, and
entrainment,usingmainlythevanderPolplanemethod.Chapters8,9,and10dealmorefor-
mallywithstability.InChapter9itsisshownthatsolutionperturbationtoteststabilitycanlead
to linear equations with periodic coefficients including Mathieu’s equation, and Floquet the-
oryisincludedChapter10presents.Liapunovmethodsforstabilityforpresented.Chapter11
includescriteriafortheexistenceofperiodicsolutions.Chapter12containsanintroductionto
bifurcation methods and manifolds. Poincaré sequences, homoclinic bifurcation; Melnikov’s
methodandLiapunovexponentsareexplained,mainlythroughexamples,inChapter13.
Thetexthasbeensubjectedtoathoroughrevisiontoimprove,wehope,theunderstandingof
nonlinearsystemsforawidereadership.Themainnewfeaturesofthesubjectmatterincludean
extendedexplanationofMathieu’sequationwithparticularreferencetodampedsystems,more
on the exponential matrix and a detailed account of Liapunov exponents for both difference
anddifferentialequations.
Many of the end-of-chapter problems, of which there are over 500, contain significant
applications and developments of the theory in the chapter. They provide a way of indicat-
ing developments for which there is no room in the text, and of presenting more specialized
material.Wehavehadmanyrequestssincethefirsteditionforasolutionsmanual,andsimul-
taneouslywiththepublicationofthefourthedition,thereisnowavailableacompanionbook,
NonlinearOrdinaryDifferentialEquations:ProblemsandSolutionsalsopublishedbyOxford
UniversityPress,whichpresents,indetail,solutionsofallend-of-chapterproblems.Thisoppor-
tunity has resulted in a re-working and revision of these problems. In addition there are 124
fullyworkedexamplesinthetext.Wefeltthatweshouldincludesomeroutineproblemsinthe
textwithselectedanswersbutnofullsolutions.Thereare88ofthesenew“Exercises”,which
canbefoundattheendofmostsections.Inalltherearenowover750examplesandproblems
inthebook.
Onthewholewehavewehavetriedtokeepthetextfreefromscientifictechnicalityandto
presentequationsinasimplereducedfromwherepossible,believingthatstudentshaveenough
todotofollowtheunderlyingarguments.
Wearegratefultomanycorrespondentsforkindwords,fortheirqueries,observationsand
suggestionsforimprovements.WewishtoexpressourappreciationtoOxfordUniversityPress
for giving us this opportunity to revise the book, and to supplement it with the new solutions
handbook.
DominicJordan
PeterSmith
Keele
June2007
Second-order differential
1 equations in the phase
plane
Very few ordinary differential equations have explicit solutions expressible in finite terms.
This is not simply because ingenuity fails, but because the repertory of standard functions
(polynomials,exp,sin,andsoon)intermsofwhichsolutionsmaybeexpressedistoolimited
toaccommodatethevarietyofdifferentialequationsencounteredinpractice.Evenifasolution
can be found, the ‘formula’ is often too complicated to display clearly the principal features
of the solution; this is particularly true of implicit solutions and of solutions which are in the
formofintegralsorinfiniteseries.
The qualitative study of differential equations is concerned with how to deduce important
characteristics of the solutions of differential equations without actually solving them. In this
chapter we introduce a geometrical device, the phase plane, which is used extensively for
obtaining directly from the differential equation such properties as equilibrium, periodicity,
unlimited growth, stability, and so on. The classical pendulum problem shows how the phase
plane may be used to reveal all the main features of the solutions of a particular differential
equation.
1.1 Phase diagram for the pendulum equation
The simple pendulum (see Fig. 1.1) consists of a particle P of mass m suspended from a fixed
point O by a light string or rod of length a, which is allowed to swing in a vertical plane. If
thereisnofrictiontheequationofmotionis
x¨ +ω2sinx =0, (1.1)
wherexistheinclinationofthestringtothedownwardvertical,gisthegravitationalconstant,
andω2 =g/a.
Weconverteqn(1.1)intoanequationconnectingx˙ andx bywriting
dx˙ dx˙ dx
x¨ = =
dt dx dt
(cid:1) (cid:2) (1.2)
d 1
= x˙2 .
dx 2
Thisrepresentaionofx¨ iscalledtheenergytransformation.Equation(1.1)thenbecomes
(cid:1) (cid:2)
d 1
x˙2 +ω2sinx =0.
dx 2