Ordinary and partial differential equations. An introduction to dynamical systems

416 Pages·2010·2.124 MB·English

by  Cain J.W., Reynolds A.M.

Ordinary and Partial Differential Equations
AnIntroductiontoDynamicalSystems
John W. Cain, Ph.D. and Angela M. Reynolds, Ph.D.
MathematicsTextbookSeries.Editor:LonMitchell
1. BookofProof byRichardHammack
2. LinearAlgebrabyJimHefferon
3. AbstractAlgebra:TheoryandApplicationsbyThomasJudson
4. OrdinaryandPartialDifferentialEquationsbyJohnW.CainandAngelaReynolds
DepartmentofMathematics&AppliedMathematics
VirginiaCommonwealthUniversity
Richmond,Virginia,23284
PublicationofthiseditionsupportedbytheCenterforTeachingExcellenceatvcu
OrdinaryandPartialDifferentialEquations:AnIntroductiontoDynamicalSystems
Edition1.0
©2010byJohnW.CainandAngelaReynolds
This work is licensed under the Creative Commons Attribution-NonCommercial-No Derivative
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Acknowledgements
John W. Cain expresses profound gratitude to his advisor, Dr. David G. Scha-
effer, James B. Duke Professor of Mathematics at Duke University. The first
fivechaptersarebasedinpartuponProfessorSchaeffer’sintroductorygradu-
ate course on ordinary differential equations. The material has been adapted
to accommodate upper-level undergraduate students, essentially by omitting
technicalproofsofthemajortheoremsandincludingadditionalexamples. Other
majorinfluencesonthisbookincludetheexcellenttextsofPerko[8],Strauss[10],
andStrogatz[11]. Inparticular,thematerialpresentedinthelastfivechapters
(includingtheorderingofthetopics)isbasedheavilyonStrauss’book. Onthe
other hand, our exposition, examples, and exercises are more “user-friendly”,
makingourtextmoreaccessibletoreaderswithlessbackgroundinmathematics.
Dr.Reynoldsdedicatesherportionofthistextbooktohermother,fatherand
sisters,shethanksthemforalltheirsupportandlove.
Finally,Dr.CaindedicateshisportionofthistextbooktohisparentsJeanette
andHarry,whohelovesmorethanwordscanexpress.
iii
Contents
Acknowledgements iii
Contents iv
1 Introduction 1
1.1 InitialandBoundaryValueProblems . . . . . . . . . . . . . . . . . 4
2 Linear,Constant-CoefficientSystems 8
2.1 HomogeneousSystems . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.1.1 DiagonalizableMatrices. . . . . . . . . . . . . . . . . . . . . 12
2.1.2 AlgebraicandGeometricMultiplicitiesofEigenvalues. . . . 21
2.1.3 ComplexEigenvalues. . . . . . . . . . . . . . . . . . . . . . . 29
2.1.4 RepeatedEigenvaluesandNon-DiagonalizableMatrices. . 37
2.2 PhasePortraitsandPlanarSystems . . . . . . . . . . . . . . . . . . 45
2.3 Stable,Unstable,andCenterSubspaces . . . . . . . . . . . . . . . . 57
2.4 TraceandDeterminant . . . . . . . . . . . . . . . . . . . . . . . . . . 65
2.5 InhomogeneousSystems . . . . . . . . . . . . . . . . . . . . . . . . . 67
3 NonlinearSystems: LocalTheory 78
3.1 LinearApproximationsofFunctionsofSeveralVariables . . . . . . 81
3.2 FundamentalExistenceandUniquenessTheorem . . . . . . . . . . 84
3.3 GlobalExistence,DependenceonInitialConditions . . . . . . . . . 86
3.4 EquilibriaandLinearization . . . . . . . . . . . . . . . . . . . . . . . 94
3.5 TheHartman-GrobmanTheorem . . . . . . . . . . . . . . . . . . . . 98
3.6 TheStableManifoldTheorem . . . . . . . . . . . . . . . . . . . . . . 100
3.7 Non-HyperbolicEquilibriaandLyapunovFunctions . . . . . . . . 105
4 Periodic,Heteroclinic,andHomoclinicOrbits 122
4.1 PeriodicOrbitsandthePoincaré-BendixonTheorem . . . . . . . . 122
iv
contents v
4.2 HeteroclinicandHomoclinicOrbits . . . . . . . . . . . . . . . . . . 130
5 Bifurcations 140
5.1 ThreeBasicBifurcations . . . . . . . . . . . . . . . . . . . . . . . . . 140
5.2 DependenceofSolutionsonParameters . . . . . . . . . . . . . . . . 148
5.3 Andronov-HopfBifurcations . . . . . . . . . . . . . . . . . . . . . . 151
6 IntroductiontoDelayDifferentialEquations 166
6.1 InitialValueProblems . . . . . . . . . . . . . . . . . . . . . . . . . . 168
6.2 SolvingConstant-CoefficientDelayDifferentialEquations . . . . . 169
6.3 CharacteristicEquations . . . . . . . . . . . . . . . . . . . . . . . . . 171
6.4 TheHutchinson-WrightEquation . . . . . . . . . . . . . . . . . . . . 172
7 IntroductiontoDifferenceEquations 180
7.1 BasicNotions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180
7.2 Linear,Constant-CoefficientDifferenceEquations . . . . . . . . . . 181
7.3 First-OrderNonlinearEquationsandStability . . . . . . . . . . . . 191
7.4 SystemsofNonlinearEquationsandStability . . . . . . . . . . . . 195
7.5 Period-DoublingBifurcations . . . . . . . . . . . . . . . . . . . . . . 200
7.6 Chaos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204
7.7 HowtoControlChaos . . . . . . . . . . . . . . . . . . . . . . . . . . 208
8 IntroductiontoPartialDifferentialEquations 218
8.1 BasicClassificationofPartialDifferentialEquations . . . . . . . . . 221
8.2 SolutionsofPartialDifferentialEquations . . . . . . . . . . . . . . . 227
8.3 InitialConditionsandBoundaryConditions . . . . . . . . . . . . . 228
8.4 VisualizingSolutionsofPartialDifferentialEquations . . . . . . . . 233
9 Linear,First-OrderPartialDifferentialEquations 236
9.1 DerivationandSolutionoftheTransportEquation . . . . . . . . . 239
9.2 MethodofCharacteristics: MoreExamples . . . . . . . . . . . . . . 241
10 TheHeatandWaveEquationsonanUnboundedDomain 250
10.1 DerivationoftheHeatandWaveEquations . . . . . . . . . . . . . . 250
10.2 CauchyProblemfortheWaveEquation . . . . . . . . . . . . . . . . 255
10.3 CauchyProblemfortheHeatEquation . . . . . . . . . . . . . . . . 265
10.4 Well-PosednessandtheHeatEquation . . . . . . . . . . . . . . . . 276
10.5 InhomogeneousEquationsandDuhamel’sPrinciple . . . . . . . . 284
vi
11 Initial-BoundaryValueProblems 297
11.1 HeatandWaveEquationsonaHalf-Line . . . . . . . . . . . . . . . 297
11.2 SeparationofVariables . . . . . . . . . . . . . . . . . . . . . . . . . . 306
11.2.1 WaveEquation,DirichletProblem. . . . . . . . . . . . . . . . 307
11.2.2 HeatEquation,DirichletProblem. . . . . . . . . . . . . . . . 313
11.2.3 WaveEquation,NeumannProblem. . . . . . . . . . . . . . . 318
11.2.4 HeatEquation,NeumannProblem. . . . . . . . . . . . . . . 324
11.2.5 MixedBoundaryConditions: AnExample. . . . . . . . . . . 324
12 IntroductiontoFourierSeries 330
12.1 Fourierseries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 332
12.1.1 Fouriersineseries. . . . . . . . . . . . . . . . . . . . . . . . . 332
12.1.2 Fouriercosineseries. . . . . . . . . . . . . . . . . . . . . . . . 337
12.1.3 Fourierseries. . . . . . . . . . . . . . . . . . . . . . . . . . . . 342
12.2 ConvergenceofFourierSeries . . . . . . . . . . . . . . . . . . . . . . 344
12.2.1 Norms,distances,innerproducts,andconvergence. . . . . 347
12.2.2 Convergencetheorems. . . . . . . . . . . . . . . . . . . . . . 359
13 TheLaplaceandPoissonEquations 367
13.1 DirchletandNeumannProblems . . . . . . . . . . . . . . . . . . . . 370
13.2 Well-posednessandtheMaximumPrinciple . . . . . . . . . . . . . 372
13.3 TranslationandRotationInvariance . . . . . . . . . . . . . . . . . . 375
13.4 Laplace’sEquationonBoundedDomains . . . . . . . . . . . . . . . 383
13.4.1 Dirichletproblemonarectangle.. . . . . . . . . . . . . . . . 383
13.4.2 Dirichletproblemonadisc. . . . . . . . . . . . . . . . . . . . 390
GuidetoCommonlyUsedNotation 404
References 406
Index 407
CHAPTER 1
Introduction
Themathematicalsub-disciplineofdifferentialequationsanddynamicalsystems
isfoundationalinthestudyofappliedmathematics. Differentialequations
arise in a variety of contexts, some purely theoretical and some of practical
interest. As you read this textbook, you will find that the qualitative and
quantitativestudyofdifferentialequationsincorporatesanelegantblendoflinear
algebraandadvancedcalculus. Forthisreason,itisexpectedthatthereaderhas
alreadycompletedcoursesin(i)linearalgebra;(ii)multivariablecalculus;and
(iii)introductorydifferentialequations. Familiaritywiththefollowingtopicsis
especiallydesirable:
(cid:43) Frombasicdifferentialequations: separabledifferentialequationsandsepara-
tionof variables; andsolvinglinear, constant-coefficient differentialequations
usingcharacteristicequations.
(cid:43) From linear algebra: solving systems of m algebraic equations with n un-
knowns;matrixinversion;linearindependence;andeigenvalues/eigenvectors.
(cid:43) From multivariable calculus: parametrized curves; partial derivatives and
gradients;andapproximatingasurfaceusingatangentplane.
Someofthesetopicswillbereviewedasweencounterthemlater—inthis
chapter,wewillrecallafewbasicnotionsfromanintroductorycoursein
differentialequations. Readersareencouragedtosupplementthisbookwiththe
excellenttextbooksofHubbardandWest[5],Meiss[7],Perko[8],Strauss[10],
andStrogatz[11].
Question: Whystudydifferentialequations?
1
2
Answer: When scientists attempt to mathematically model various natural
phenomena,theyofteninvokephysical“laws”orbiological“principles”which
governtheratesofchangeofcertainquantitiesofinterest. Hence,theequations
in mathematical models tend to include derivatives. For example, suppose
thatahotcupofcoffeeisplacedinaroomofconstantambienttemperature α.
Newton’sLawofCoolingstatesthattherateofchangeofthecoffeetemperature
T(t)isproportionaltothedifferencebetweenthecoffee’stemperatureandthe
room temperature. Mathematically, this can be expressed as dT = k(T−α),
dt
where k isaproportionalityconstant.
Solutiontechniquesfordifferentialequations(des)dependinpartuponhow
manyindependentvariablesanddependentvariablesthesystemhas.
Example 1.0.1. One independent variable and one independent variable. In
writingtheequation
d2y
+cos(xy) = 3,
dx2
it is understood that y is the dependent variable and x is the independent
variable.
Whenadifferentialequationinvolvesasingleindependentvariable,werefer
totheequationasanordinarydifferentialequation(ode).
Example1.0.2. Ifthereareseveraldependentvariablesandasingleindependent
variable,wemighthaveequationssuchas
dy dz
= x2y−xy2+z, = z−ycosx.
dx dx
This is a system of two odes, and it is understood that x is the independent
variable.
Example1.0.3. Onedependentvariable,severalindependentvariables. Consider
thede
∂u ∂2u ∂2u
= + .
∂t ∂x2 ∂y2
Thisequationinvolvesthreeindependentvariables(x, y,and t)andonedepen-
dentvariable, u. Thisisanexampleofapartialdifferentialequation(pde). Ifthere
areseveralindependentvariablesandseveraldependentvariables,onemayhave
systemsofpdes.
introduction 3
Althoughtheseconceptsareprobablyfamiliartothereader,wegiveamore
exactdefinitionforwhatwemeanbyode. Supposethatxandyareindependent
anddependentvariables,respectively,andlet y(k)(x) denotethe kthderivative
of y withrespectto x. (If k ≤3,wewilluseprimes.)
Definition 1.0.4. Any equation of the form F(x,y,y(cid:48),y(cid:48)(cid:48),...,y(n)) = 0 is called
anordinarydifferentialequation. If y(n) isthehighestderivativeappearinginthe
equation,wesaythattheodeisofordern.
Example1.0.5.
(cid:18)d3y(cid:19)2 dy d2y
−(cosx) = y
dx3 dx dx2
canbewrittenas(y(cid:48)(cid:48)(cid:48))2−yy(cid:48)(cid:48)−(cosx)y(cid:48) =0,sousingthenotationintheabove
Definition,wewouldhave F(x,y,y(cid:48),y(cid:48)(cid:48),y(cid:48)(cid:48)(cid:48)) = (y(cid:48)(cid:48)(cid:48))2−yy(cid:48)(cid:48)−(cosx)y(cid:48). Thisisa
third-orderode.
Definition1.0.6. Asolutionoftheode F(x,y,y(cid:48),y(cid:48)(cid:48),...,y(n)) =0onaninterval
I isanyfunction y(x) whichis n-timesdifferentiableandsatisfiestheequation
on I.
Example1.0.7. Foranychoiceofconstant A,thefunction
Aex
y(x) =
1+Aex
isasolutionofthefirst-orderode y(cid:48) = y−y2 forallreal x. Toseewhy,weuse
thequotientruletocalculate
Aex(1+Aex)−(Aex)2 Aex
y(cid:48) = = .
(1+Aex)2 (1+Aex)2
Bycomparison,wecalculatethat
Aex (Aex)2 Aex
y−y2 = − = .
(1+Aex) (1+Aex)2 (1+Aex)2
Therefore, y(cid:48) = y−y2,asclaimed.
The definition of a solution of an ode is easily extended to systems of odes
(seebelow). Inwhatfollows,wewillfocussolelyonsystemsoffirst-orderodes.
Thismayseemoverlyrestrictive,untilwemakethefollowingobservation.
4 initial and boundary value problems
Observation. Any nth-order ode can be written as a system of n first-order
odes. Theprocessofdoingsoisstraightforward,asillustratedinthefollowing
example:
Example1.0.8. Considerthesecond-orderodey(cid:48)(cid:48)+(cosx)y(cid:48)+y2 =ex. Toavoid
usingsecondderivatives,weintroduceanewdependentvariable z = y(cid:48) sothat
z(cid:48) = y(cid:48)(cid:48). Our ode can be re-written as z(cid:48)+(cosx)z+y2 = ex. Thus, we have
obtainedasystemoftwofirst-orderodes:
dy dz
= z, = −(cosx)z−y2+ex.
dx dx
A solution of the above system of odes on an open interval I is any vector
of differentiable functions [y(x),z(x)] which simultaneously satisfy both odes
when x ∈ I.
Example1.0.9. Considerthesystem
dy dz
= z, = −y.
dt dt
WeclaimthatforanychoicesofconstantsC andC ,
1 2
(cid:34) (cid:35) (cid:34) (cid:35)
y(t) C cost+C sint
= 1 2
z(t) −C sint+C cost
1 2
is a solution of the system. To verify this, assume that y and z have this form.
Differentiationrevealsthat y(cid:48) = −C sint+C cost and z(cid:48) = −C cost−C sint.
1 2 1 2
Thus, y(cid:48) = z and z(cid:48) = −y,asrequired.
1.1. InitialandBoundaryValueProblems
Inthepreviousexample,thesolutionofthesystemof odescontainsarbitrary
constants C and C . Therefore, the system has infinitely many solutions. In
1 2
practice, one often has additional information about the underlying system,
allowing us to select a particular solution of practical interest. For example,
supposethatacupofcoffeeiscoolingoffandobeysNewton’sLawofCooling.
Inordertopredictthecoffee’stemperatureatfuturetimes,wewouldneedto
specifythetemperatureofthecoffeeatsomereferencetime(usuallyconsidered
tobethe“initial”time). Byspecifyingauxiliaryconditionsthatsolutionsofan