Ordinary Differential Equations And Dynamical Systems

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Ordinary Differential Equationsand Dynamical SystemsGerald TeschlThis is a preliminary version of the book Ordinary Differential Equations and Dynamical Systemspublished by the American Mathematical Society (AMS). This preliminary version is made available withthe permission of the AMS and may not be changed, edited, or reposted at any other website withoutexplicit written permission from the author and the AMS.Author's preliminary version made available with permission of the publisher, the American Mathematical Society

To Susanne, Simon, and JakobAuthor's preliminary version made available with permission of the publisher, the American Mathematical Society

Author's preliminary version made available with permission of the publisher, the American Mathematical Society

ContentsPrefacexiPart 1. Classical theoryChapter 1.Introduction3§1.1.Newton’s equations3§1.2.Classification of differential equations6§1.3.First order autonomous equations9§1.4.Finding explicit solutions13§1.5.Qualitative analysis of first-order equations20§1.6.Qualitative analysis of first-order periodic equations28Chapter 2.Initial value problems33§2.1.Fixed point theorems33§2.2.The basic existence and uniqueness result36§2.3.Some extensions39§2.4.Dependence on the initial condition42§2.5.Regular perturbation theory48§2.6.Extensibility of solutions50§2.7.Euler’s method and the Peano theorem54Chapter 3.Linear equations59§3.1.The matrix exponential59Linear autonomous first-order systems66§3.3.Linear autonomous equations of order n74§3.2.viiAuthor's preliminary version made available with permission of the publisher, the American Mathematical Society

viii§3.4.§3.5.§3.6.§3.7.§3.8.ContentsGeneral linear first-order systemsLinear equations of order nPeriodic linear systemsPerturbed linear first order systemsAppendix: Jordan canonical form80879197103Chapter 4. Differential equations in the complex domain§4.1. The basic existence and uniqueness result§4.2. The Frobenius method for second-order equations111111116Chapter 5. Boundary value problems§5.1. Introduction§5.2. Compact symmetric 5.6.Linear systems with singularitiesThe Frobenius method130134Sturm–Liouville equationsRegular Sturm–Liouville problemsOscillation theory153155166Periodic Sturm–Liouville equations175Part 2. Dynamical systemsChapter 6. Dynamical systems§6.1. Dynamical he flow of an autonomous equationOrbits and invariant setsThe Poincaré map188192196Stability of fixed pointsStability via Liapunov’s methodNewton’s equation in one dimension198200203Chapter 7. Planar dynamical systems§7.1. Examples from ecology209209Chapter 8.229§7.2.§7.3.§8.1.§8.2.Examples from electrical engineeringThe Poincaré–Bendixson theoremHigher dimensional dynamical systemsAttracting setsThe Lorenz equation215220229234Author's preliminary version made available with permission of the publisher, the American Mathematical Society

Contents§8.3.§8.4.§8.5.§8.6.ixHamiltonian mechanicsCompletely integrable Hamiltonian systems238242The Kepler problemThe KAM theorem247249Chapter 9.§9.1.§9.2.§9.3.§9.4.Local behavior near fixed points253Stability of linear systemsStable and unstable manifoldsThe Hartman–Grobman theorem253255262Appendix: Integral equations268Part 3. ChaosChapter 10. Discrete dynamical systems§10.1. The logistic equation279279Chapter 11. Discrete dynamical systems in one dimension§11.1. Period �11.4.§11.5.§11.6.§11.7.Fixed and periodic pointsLinear difference equationsLocal behavior near fixed pointsSarkovskii’s theoremOn the definition of chaosCantor sets and the tent map294295298Symbolic dynamicsStrange attractors/repellors and fractal setsHomoclinic orbits as source for chaos301307311Chapter 12. Periodic solutions§12.1. Stability of periodic solutions§12.2.§12.3.§12.4.§12.5.315315The Poincaré mapStable and unstable manifoldsMelnikov’s method for autonomous perturbations317319322Melnikov’s method for nonautonomous perturbations327Chapter 13. Chaos in higher dimensional systems§13.1. The Smale horseshoe§13.2.§13.3.282285286The Smale–Birkhoff homoclinic theoremMelnikov’s method for homoclinic orbits331331333334Author's preliminary version made available with permission of the publisher, the American Mathematical Society

xContentsBibliographical notes339Bibliography343Glossary of notation347Index349Author's preliminary version made available with permission of the publisher, the American Mathematical Society

PrefaceAboutWhen you publish a textbook on such a classical subject the first question you will be faced with is: Why the heck another book? Well, everythingstarted when I was supposed to give the basic course on Ordinary Differential Equations in Summer 2000 (which at that time met 5 hours per week).While there were many good books on the subject available, none of themquite fitted my needs. I wanted a concise but rigorous introduction with fullproofs also covering classical topics such as Sturm–Liouville boundary valueproblems, differential equations in the complex domain as well as modernaspects of the qualitative theory of differential equations. The course wascontinued with a second part on Dynamical Systems and Chaos in Winter2000/01 and the notes were extended accordingly. Since then the manuscripthas been rewritten and improved several times according to the feedback Igot from students over the years when I redid the course. Moreover, since Ihad the notes on my homepage from the very beginning, this triggered a significant amount of feedback as well. Beginning from students who reportedtypos, incorrectly phrased exercises, etc. over colleagues who reported errorsin proofs and made suggestions for improvements, to editors who approachedme about publishing the notes. Last but not least, this also resulted in achinese translation. Moreover, if you google for the manuscript, you can seethat it is used at several places worldwide, linked as a reference at varioussites including Wikipedia. Finally, Google Scholar will tell you that it iseven cited in several publications. Hence I decided that it is time to turn itinto a real book.xiAuthor's preliminary version made available with permission of the publisher, the American Mathematical Society

xiiPrefaceContentIts main aim is to give a self contained introduction to the field of ordinary differential equations with emphasis on the dynamical systems pointof view while still keeping an eye on classical tools as pointed out before.The first part is what I typically cover in the introductory course forbachelor students. Of course it is typically not possible to cover everythingand one has to skip some of the more advanced sections. Moreover, it mightalso be necessary to add some material from the first chapter of the secondpart to meet curricular requirements.The second part is a natural continuation beginning with planar examples (culminating in the generalized Poincaré–Bendixon theorem), continuing with the fact that things get much more complicated in three and moredimensions, and ending with the stable manifold and the Hartman–Grobmantheorem.The third and last part gives a brief introduction to chaos focusing ontwo selected topics: Interval maps with the logistic map as the prime example plus the identification of homoclinic orbits as a source for chaos andthe Melnikov method for perturbations of periodic orbits and for findinghomoclinic orbits.PrerequisitesIt only requires some basic knowledge from calculus, complex functions,and linear algebra which should be covered in the usual courses. In addition,I have tried to show how a computer system, Mathematica, can help withthe investigation of differential equations. However, the course is not tiedto Mathematica and any similar program can be used as well.UpdatesThe AMS is hosting a web page for this book athttp://www.ams.org/bookpages/gsm-XXX/where updates, corrections, and other material may be found, including alink to material on my own web site:http://www.mat.univie.ac.at/ gerald/ftp/book-ode/There you can also find an accompanying Mathematica notebook with thecode from the text plus some additional material. Please do not put aAuthor's preliminary version made available with permission of the publisher, the American Mathematical Society

Prefacexiiicopy of this file on your personal webpage but link to the pageabove.AcknowledgmentsI wish to thank my students, Ada Akerman, Kerstin Ammann, JörgArnberger, Alexander Beigl, Paolo Capka, Jonathan Eckhardt, Michael Fischer, Anna Geyer, Ahmed Ghneim, Hannes Grimm-Strele, Tony Johansson,Klaus Kröncke, Alice Lakits, Simone Lederer, Oliver Leingang, JohannaMichor, Thomas Moser, Markus Müller, Andreas Németh, Andreas Pichler, Tobias Preinerstorfer, Jin Qian, Dominik Rasipanov, Martin Ringbauer,Simon Rößler, Robert Stadler, Shelby Stanhope, Raphael Stuhlmeier, Gerhard Tulzer, Paul Wedrich, Florian Wisser, and colleagues, Edward Dunne,Klemens Fellner, Giuseppe Ferrero, Ilse Fischer, Delbert Franz, Heinz Hanßmann, Daniel Lenz, Jim Sochacki, and Eric Wahlén, who have pointed outseveral typos and made useful suggestions for improvements. Finally, I alsolike to thank the anonymous referees for valuable suggestions improving thepresentation of the material.If you also find an error or if you have comments or suggestions(no matter how small), please let me know.I have been supported by the Austrian Science Fund (FWF) during muchof this writing, most recently under grant Y330.Gerald TeschlVienna, AustriaApril 2012Gerald TeschlFakultät für MathematikNordbergstraße 15Universität Wien1090 Wien, AustriaE-mail: Gerald.Teschl@univie.ac.atURL: http://www.mat.univie.ac.at/ gerald/Author's preliminary version made available with permission of the publisher, the American Mathematical Society

Author's preliminary version made available with permission of the publisher, the American Mathematical Society

Part 1Classical theoryAuthor's preliminary version made available with permission of the publisher, the American Mathematical Society

Author's preliminary version made available with permission of the publisher, the American Mathematical Society

Chapter 1Introduction1.1. Newton’s equationsLet us begin with an example from physics. In classical mechanics a particleis described by a point in space whose location is given by a functionx : R R3 .(1.1).rx(t) v(t)The derivative of this function with respect to time is the velocity of theparticlev ẋ : R R3(1.2)and the derivative of the velocity is the accelerationa v̇ : R R3 .(1.3)In such a model the particle is usually moving in an external force fieldF : R3 R3(1.4)which exerts a force F (x) on the particle at x. Then Newton’s secondlaw of motion states that, at each point x in space, the force acting onthe particle must be equal to the acceleration times the mass m (a positive3Author's preliminary version made available with permission of the publisher, the American Mathematical Society

41. Introductionconstant) of the particle, that is,m ẍ(t) F (x(t)),for all t R.(1.5)Such a relation between a function x(t) and its derivatives is called a differential equation. Equation (1.5) is of second order since the highestderivative is of second degree. More precisely, we have a system of differential equations since there is one for each coordinate direction.In our case x is called the dependent and t is called the independentvariable. It is also possible to increase the number of dependent variablesby adding v to the dependent variables and considering (x, v) R6 . Theadvantage is, that we now have a first-order systemẋ(t) v(t)1v̇(t) F (x(t)).m(1.6)This form is often better suited for theoretical investigations.For given force F one wants to find solutions, that is functions x(t) thatsatisfy (1.5) (respectively (1.6)). To be more specific, let us look at themotion of a stone falling towards the earth. In the vicinity of the surfaceof the earth, the gravitational force acting on the stone is approximatelyconstant and given by 0 F (x) m g 0 .(1.7)1Here g is a positive constant and the x3 direction is assumed to be normalto the surface. Hence our system of differential equations readsm ẍ1 0,m ẍ2 0,m ẍ3 m g.(1.8)The first equation can be integrated with respect to t twice, resulting inx1 (t) C1 C2 t, where C1 , C2 are the integration constants. Computingthe values of x1 , ẋ1 at t 0 shows C1 x1 (0), C2 v1 (0), respectively.Proceeding analogously with the remaining two equations we end up with 0g 2x(t) x(0) v(0) t 0 t .(1.9)21Hence the entire fate (past and future) of our particle is uniquely determinedby specifying the initial location x(0) together with the initial velocity v(0).Author's preliminary version made available with permission of the publisher, the American Mathematical Society

1.1. Newton’s equations5From this example you might get the impression, that solutions of differential equations can always be found by straightforward integration. However, this is not the case in general. The reason why it worked here is thatthe force is independent of x. If we refine our model and take the realgravitational forcexF (x) γ m M 3 ,γ, M 0,(1.10) x our differential equation readsγ m M x1, x22 x23 )3/2γ m M x2,m ẍ2 2(x1 x22 x23 )3/2γ m M x3m ẍ3 2(x1 x22 x23 )3/2m ẍ1 (x21(1.11)and it is no longer clear how to solve it. Moreover, it is even unclear whethersolutions exist at all! (We will return to this problem in Section 8.5.)Problem 1.1. Consider the case of a stone dropped from the height h.Denote by r the distance of the stone from the surface. The initial conditionreads r(0) h, ṙ(0) 0. The equation of motion readsr̈ γM(R r)2(exact model)respectivelyr̈ g(approximate model),where g γM/R2 and R, M are the radius, mass of the earth, respectively.(i) Transform both equations into a first-order system.(ii) Compute the solution to the approximate system corresponding tothe given initial condition. Compute the time it takes for the stoneto hit the surface (r 0).(iii) Assume that the exact equation also has a unique solution corresponding to the given initial condition. What can you say aboutthe time it takes for the stone to hit the surface in comparisonto the approximate model? Will it be longer or shorter? Estimatethe difference between the solutions in the exact and in the approximate case. (Hints: You should not compute the solution to theexact equation! Look at the minimum, maximum of the force.)(iv) Grab your physics book from high school and give numerical valuesfor the case h 10m.Author's preliminary version made available with permission of the publisher, the American Mathematical Society

61. IntroductionProblem 1.2. Consider again the exact model from the previous problemand writeγM ε21r̈ ,ε .2(1 εr)RIt can be shown that the solution r(t) r(t, ε) to the above initial conditionsis C (with respect to both t and ε). Show that1γMh t2g 2.r(t) h g(1 2 ) O( 4 ),R 2RR23(Hint: Insert r(t, ε) r0 (t) r1 (t)ε r2 (t)ε r3 (t)ε O(ε4 ) into thedifferential equation and collect powers of ε. Then solve the correspondingdifferential equations for r0 (t), r1 (t), . . . and note that the initial conditionsfollow from r(0, ε) h respectively ṙ(0, ε) 0. A rigorous justification forthis procedure will be given in Section 2.5.)1.2. Classification of differential equationsLet U Rm , V Rn and k N0 . Then C k (U, V ) denotes the set offunctions U V having continuous derivatives up to orderT k. kIn addition,0 we will abbreviate C(U, V ) C (U, V ), C (U, V ) k N C (U, V ), andC k (U ) C k (U, R).A classical ordinary differential equation (ODE) is a functional relation of the formF (t, x, x(1) , . . . , x(k) ) 0(1.12)for the unknown function x C k (J), J R, and its derivativesdj x(t),j N0 .(1.13)dtjHere F C(U ) with U an open subset of Rk 2 . One frequently calls tthe independent and x the dependent variable. The highest derivativeappearing in F is called the order of the differential equation. A solutionof the ODE (1.12) is a function φ C k (I), where I J is an interval, suchthatF (t, φ(t), φ(1) (t), . . . , φ(k) (t)) 0,for all t I.(1.14)x(j) (t) This implicitly implies (t, φ(t), φ(1) (t), . . . , φ(k) (t)) U for all t I.Unfortunately there is not too much one can say about general differential equations in the above form (1.12). Hence we will assume that one cansolve F for the highest derivative, resulting in a differential equation of theformx(k) f (t, x, x(1) , . . . , x(k 1) ).(1.15)By the implicit function theorem this can be done at least locally near somepoint (t, y) U if the partial derivative with respect to the highest derivativeAuthor's preliminary version made available with permission of the publisher, the American Mathematical Society

1.2. Classification of differential equations7 Fdoes not vanish at that point, y(t, y) 6 0. This is the type of differentialkequations we will consider from now on.We have seen in the previous section that the case of real-valued functions is not enough and we should admit the case x : R Rn . This leadsus to systems of ordinary differential equations(k)x1 f1 (t, x, x(1) , . . . , x(k 1) ),.(1)(k 1)x(k)).n fn (t, x, x , . . . , x(1.16)Such a system is said to be linear, if it is of the form(k)xi gi (t) n Xk 1X(j)fi,j,l (t)xl .(1.17)l 1 j 0It is called homogeneous, if gi (t) 0.Moreover, any system can always be reduced to a first-order system bychanging to the new set of dependent variables y (x, x(1) , . . . , x(k 1) ).This yields the new first-order systemẏ1 y2 ,.ẏk 1 yk ,ẏk f (t, y).(1.18)We can even add t to the dependent variables z (t, y), making the righthand side independent of tż1 1,ż2 z3 ,.żk zk 1 ,żk 1 f (z).(1.19)Such a system, where f does not depend on t, is called autonomous. Inparticular, it suffices to consider the case of autonomous first-order systemswhich we will frequently do.Of course, we could also look at the case t Rm implying that wehave to deal with partial derivatives. We then enter the realm of partialdifferential equations (PDE). However, we will not pursue this case here.Author's preliminary version made available with permission of the publisher, the American Mathematical Society

81. IntroductionFinally, note that we could admit complex values for the dependentvariables. It will make no difference in the sequel whether we use real orcomplex dependent variables. However, we will state most results only forthe real case and leave the obvious changes to the reader. On the otherhand, the case where the independent variable t is complex requires morethan obvious modifications and will be considered in Chapter 4.Problem 1.3. Classify the following differential equations. Is the equationlinear, autonomous? What is its order?(i) y ′ (x) y(x) 0.(iii)d2dt2 u(t) t sin(u(t)).y(t)2 2y(t) 0.(iv) 2u(x, y) x2(ii) 2u(x, y) y 2 0.(v) ẋ y, ẏ x.Problem 1.4. Which of the following differential equations for y(x) arelinear?(i) y ′ sin(x)y cos(y).(ii) y ′ sin(y)x cos(x).(iii) y ′ sin(x)y cos(x).Problem 1.5. Find the most general form of a second-order linear equation.Problem 1.6. Transform the following differential equations into first-ordersystems.(i) ẍ t sin(ẋ) x.(ii) ẍ y, ÿ x.The last system is linear. Is the corresponding first-order system also linear?Is this always the case?Problem 1.7. Transform the following differential equations into autonomousfirst-order systems.(i) ẍ t sin(

Ordinary Differential Equations . and Dynamical Systems . Gerald Teschl . This is a preliminary version of the book Ordinary Differential Equations and Dynamical Systems. published by the American Mathematical Society (AMS). This preliminary version is made available with

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