FIRST A COURSE IN THE
QUALITATIVE THEORY
OF
DIFFERENTIAL EQUATIONS
James Hetao Liu
James Madison University
Pearson Education, Inc., Upper Saddle River, New Jersey 07458
Library of Congress Cataloging-in-Publication Data Liu, James Hetao. A rst course in the qualitative theory of dierential equations / James Hetao Liu. p. cm. Includes bibliographical references and index. ISBN 0-13-008380-1 1. Dierential equations, Nonlinear. I. Title. QA372.L765 515.355dc21 2003 2002032370
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TO YOU the readers who would like to start a journey with me
Contents
Preface Chapter 1. A Brief Description 1. Linear Dierential Equations 2. The Need for Qualitative Analysis 3. Description and Terminology Chapter 2. Existence and Uniqueness 1. 2. 3. 4. 5. Introduction Existence and Uniqueness Dependence on Initial Data and Parameters Maximal Interval of Existence Fixed Point Method 1 1 10 16 41 41 46 72 89 100 106 106 111 132 150
2
Chapter 3. Linear Dierential Equations 1. 2. 3. 4. Introduction General Nonhomogeneous Linear Equations Linear Equations with Constant Coecients Periodic Coecients and Floquet Theory
Chapter 4. Autonomous Dierential Equations in 1. Introduction 2. Linear Autonomous Equations in 2 3. Perturbations on Linear Equations in iv
163 163 178 194
2
Contents 4. An Application: A Simple Pendulum 5. Hamiltonian and Gradient Systems Chapter 5. Stability. Part I 1. 2. 3. 4. 5. 6. Introduction Linear Equations with Constant Coecients Perturbations on Linear Equations Linear Equations with Periodic Coecients Liapunovs Method for Autonomous Equations Some Applications
v 206 209 221 221 229 236 243 250 265 272 272 278 288 296 306 317 317 324 336 356 361 370 378 378 380 399 413 418
Chapter 6. Bifurcation 1. 2. 3. 4. 5. Introduction Saddle-Node Bifurcation Transcritical Bifurcation Pitchfork Bifurcation Poincar-Andronov-Hopf Bifurcation e
Chapter 7. Chaos 1. 2. 3. 4. 5. 6. Introduction Maps and Their Bifurcations Period-Doubling Bifurcations: Route to Chaos Universality The Lorenz System and Strange Attractors The Smale Horseshoe
Chapter 8. Dynamical Systems 1. 2. 3. 4. 5. Introduction Poincar-Bendixson Theorem in 2 e Limit Cycles An Application: Lotka-Volterra Equation Manifolds and the Hartman-Grobman Theorem
vi Chapter 9. Stability. Part II 1. Introduction 2. General Linear Dierential Equations 3. Liapunovs Method for General Equations Chapter 10. Bounded Solutions 1. 2. 3. 4. 5. Introduction General Linear Dierential Equations Linear Equations with Constant Coecients Linear Equations with Periodic Coecients Liapunovs Method for General Equations
Contents 443 443 446 455 470 470 474 486 488 489 499 499 502 511 520 520 524 526 528 531 533 534 534 539 547 556
Chapter 11. Periodic Solutions 1. Introduction 2. Linear Dierential Equations 3. Nonlinear Dierential Equations Chapter 12. Some New Types of Equations 1. 2. 3. 4. 5. 6. 7. 8. Introduction Finite Delay Dierential Equations Innite Delay Dierential Equations Integrodierential Equations Impulsive Dierential Equations Equations with Nonlocal Conditions Impulsive Equations with Nonlocal Conditions Abstract Dierential Equations
Appendix References Index
Preface
Why should we learn some qualitative theory of dierential equations? Dierential equations are mainly used to describe the change of quantities or behavior of certain systems in applications, such as those governed by Newtons laws in physics. When the dierential equations under study are linear, the conventional methods, such as the Laplace transform method and the power series solutions, can be used to solve the dierential equations analytically, that is, the solutions can be written out using formulas. When the dierential equations under study are nonlinear, analytical solutions cannot, in general, be found; that is, solutions cannot be written out using formulas. In those cases, one approach is to use numerical approximations. In fact, the recent advances in computer technology make the numerical approximation classes very popular because powerful software allows students to quickly approximate solutions of nonlinear dierential equations and visualize, even in 3-D, their properties. However, in most applications in biology, chemistry, and physics modeled by nonlinear dierential equations where analytical solutions may be unavailable, people are interested in the questions related to the so-called qualitative properties, such as: will the system have at least one solution? will the system have at most one solution? can certain behavior of the system be controlled or stabilized? or will the system exhibit some periodicity? If these questions can be answered without solving the dierential equations, especially when analytical solutions are unavailable, we can still get a very good understanding of the system. Therefore, besides learning some numerical methods, it is also important and benecial to learn how to analyze some qualitative properties, such as the existence and uniqueness of vii
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solutions, phase portraits analysis, dynamics of systems, stability, bifurcations, chaos, boundedness, and periodicity of dierential equations without solving them analytically or numerically. This makes learning the qualitative theory of dierential equations very valuable, as it helps students get well equipped with tools they can use when they apply the knowledge of dierential equations in their future studies and careers. For example, when taking a numerical methods course, before a numerical approximation is carried out, the existence and uniqueness of solutions should be checked to make sure that there does exist one and only one solution to be approximated. Otherwise, how does one know what one is approximating? A related remark is that even though numerical solutions can be carried out to suggest certain properties, they are obtained through discretization on nite intervals. They reveal certain properties that are only valid for the limited numerical solutions on nite intervals and, therefore, cannot be used to determine the qualitative properties on the whole interval of all solutions of the original dierential equations. Based on the above remarks, we conclude that in order to have a more complete knowledge of dierential equations, and be able to analyze dierential equations without solving them analytically or numerically, we should learn some qualitative theory of dierential equations. To whom is this book written? This book is written for upper level undergraduates (second undergraduate course in ODEs) and beginning graduate students. To be more specic, Chapters 17 are for upper level undergraduate students, where the basic qualitative properties concerning existence and uniqueness, structures of solutions, phase portraits, stability, bifurcation, and chaos are discussed. Chapters 812, together with Chapters 17, are for beginning graduate students, where some additional subjects on stability, dynamical systems, bounded and periodic solutions are covered. Another reason for writing this book is that nowadays it is a popular trend for upper level undergraduate students and beginning graduate students to get involved in some research activity. Compared to other more abstract subjects in mathematics, qualitative analysis of dierential equations is readily accessible to upper level undergraduates and beginning graduate students. It is a vast hunting eld in which students will get an opportunity to combine and apply their knowledge in linear algebra, elementary dierential equations, advanced calculus, and others to hunt some prey.
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Furthermore, the qualitative analysis of dierential equations is on the border line of applied mathematics and pure mathematics, so it can attract students interested in either discipline. How does this book dier from other ODE books? It is often the case that in a book written at the graduate level or even at the upper undergraduate level, there are jumps in the reasonings or in the proofs, as evidenced by words like obviously or clearly. However, inexperienced undergraduate and beginning graduate students need more careful and detailed guidance to help them learn the material and gain maturity on the subject. In this book, I selected only the subjects that are of fundamental importance, that are accessible to upper level undergraduate and beginning graduate students, and that are related to current research in the eld. Then, for each selected topic, I provided a complete analysis that is suitable for the targeted audience, and lled in the details and gaps which are missing from some other books. Sometimes, I produced elementary proofs using calculus and linear algebra for certain results that are treated in a more abstract frame in other books. Also, examples and reasons are given before introducing many concepts and results. Therefore, this book is dierent from other ODE books in that it is more detailed, and, as the title of this book indicates, the level of this book is lower than most books for graduate students, and higher than the books for elementary dierential equations. Moreover, this book contains many interesting pure and applied topics that can be used for one or two semesters. What topics are covered in this book? Chapter 1. A Brief Description. We rst give a brief treatment of some subjects covered in an elementary dierential equations course. Then we introduce some terminology and describe some qualitative properties of dierential equations that we are going to study in this book. We use the geometric and physical arguments to show why certain qualitative properties are plausible and why sometimes we pursue a qualitative analysis rather than solving dierential equations analytically or numerically. This will give readers an opportunity to become familiar with the objective and terminology of qualitative analysis in a somewhat familiar setting.
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Chapter 2. Existence and Uniqueness. In Section 1, we use examples from applications to dene general rst order dierential equations in n . In Section 2, we study existence and uniqueness of solutions, that is, we examine under what conditions a dierential equation has solutions and how the solutions are uniquely determined, without solving the dierential equation analytically. A condition called Lipschitz condition is utilized. In Section 3, we show under certain conditions that solutions are continuous and dierentiable with respect to initial data and parameters. In Section 4, we determine structures of the maximal intervals of existence for solutions, and study properties of solutions with respect to the maximal intervals of existence. In Section 5 (which may be optional), we introduce the Fixed Point Method. We use the contraction mapping principle to derive existence and uniqueness of solutions if a local Lipschitz condition is satised. Then, when a local Lipschitz condition is not assumed, we use Schauders second xed point theorem to obtain existence of solutions, in which case, uniqueness is not guaranteed. Chapter 3. Linear Dierential Equations. In Section 1, we make some denitions concerning linear dierential equations. In Section 2, we study general nonhomogeneous linear dierential equations and obtain the fundamental matrix solutions and verify that they satisfy the evolution system property. Then we derive the variation of parameters formula using the fundamental matrix solutions and observe what these solutions should look like. In Section 3, we look at equations with constant coecients and examine detailed structure of solutions in terms of eigenvalues of the leading constant matrix, using the Jordan canonical form theorem. In addition we derive the Putzer algorithm that can be used to actually solve or compute solutions for equations with constant coecients. This result will appeal to readers interested in computation. In Section 4, we look at equations with periodic coecients and study Floquet theory, which allows us to transform equations with periodic coecients into equations with constant coecients. The results of Section 3 can then be applied to the transformed equations. The concept of Liapunov exponents is also briey introduced in Section 4. Chapter 4. Autonomous Dierential Equations in 2 . In Section 1, we introduce the concept of dynamical systems, discuss possible trajectories in phase planes for two-dimensional autonomous equations, and outline the relationship between nonlinear dierential equations and their linearizations. In Section 2, we provide a complete analysis for linear autonomous
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dierential equations in 2 and draw all phase for portraits the dierent cases according to eigenvalues of the coecient matrix. We also introduce some terminology, including stability of solutions, according to the properties revealed, which leads us to detailed study of the same subject later for general dierential equations in n , n 1. In Section 3, we examine the conditions which ensure that solutions of autonomous dierential equations and their linearizations have essentially the same local geometric and qualitative properties near the origin. In Section 4, we apply the results to analyze an equation of a simple pendulum. In Section 5, we generalize the ideas of a simple pendulum and study the Hamiltonian systems and gradient systems. Chapter 5. Stability. Part I. In Section 1, we introduce the notion of stabilities in the sense of Liapunov for general dierential equations in n , which are based on some consideration in physics and the planar dierential equations studied in Chapter 4. In Section 2, we study stabilities for linear dierential equations with constant coecients and show that eigenvalues of the coecient matrices determine stability properties. In Section 3, stabilities of linear equations with linear or nonlinear perturbations are studied using the variation of parameters formula and Gronwalls inequality. The results include some planar autonomous nonlinear dierential equations studied in Chapter 4 as special cases. Therefore, some unproven results in Chapter 4 can now get a partial proof. In Section 4, linear periodic dierential equations are treated. The Floquet theory from Chapter 3 is used to transform linear periodic equations into linear equations with constant coecients and the results from Section 2 can then be applied. In Section 5, we introduce Liapunovs method for autonomous nonlinear dierential equations and prove their stability properties under the assumption that there exist appropriate Liapunov functions. Thus, we can obtain stabilities without explicitly solving dierential equations. In Section 6, we provide examples to demonstrate how the Liapunov theory is applied by constructing Liapunov functions in specic applications. Liapunovs method for general (nonautonomous) dierential equations will be given in Chapter 9. Chapter 6. Bifurcation. In Section 1, we use examples, including Eulers buckling beam, to introduce the concept of bifurcation of critical points of dierential equations when some parameters are varied. In Section 2, we study saddle-node bifurcations and use examples to explain why saddle and node appear for this type of bifurcations. We analyze the geometric aspects of some scalar dierential equations that undergo saddle-node bifur-
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cations and use them to formulate and prove a result concerning saddle-node bifurcations for scalar dierential equations. In Section 3, we study transcritical bifurcations and apply them to a solid-state laser in physics. Again, the geometric aspects of some examples are analyzed and used to formulate and prove a result concerning transcritical bifurcations for scalar dierential equations. In Section 4, we study pitchfork bifurcations and apply them to Eulers buckling beam and calculate Eulers rst buckling load, which is the value the buckling takes place. The hysteresis eect with applications in physics is also discussed. A result concerning pitchfork bifurcations for scalar dierential equations is formulated using the geometric interpretation. In Section 5, we analyze the situations where a pair of two conjugate complex eigenvalues cross the pure imaginary axis when some parameters are varied. We introduce the Poincar-Andronov-Hopf bifurcation theorem and e apply it to van der Pols oscillator in physics. Chapter 7. Chaos. In Section 1, we use examples, such as some discrete maps and the Lorenz system, to introduce the concept of chaos. In Section 2, we study recursion relations, also called maps, and their bifurcation properties by nding the similarities to the bifurcations of critical points of dierential equations, hence the results in Chapter 6 can be carried over. In Section 3, we look at a phenomenon called period-doubling bifurcations cascade, which provides a route to chaos. In Section 4, we introduce some universality results concerning one-dimensional maps. In Section 5, we study some properties of the Lorenz system and introduce the notion of strange attractors. In Section 6, we study the Smale horseshoe which provides an example of a strange invariant set possessing chaotic dynamics. Chapter 8. Dynamical Systems. In Section 1, we discuss the need to study the global properties concerning the geometrical relationship between critical points, periodic orbits, and nonintersecting curves. In Section 2, we e study the dynamics in 2 and prove the Poincar-Bendixson theorem. In Section 3, we use the Poincar-Bendixson theorem, together with other e results, to obtain existence and nonexistence of limit cycles, which in turn help us determine the global properties of planar systems. In Section 4, we apply the results to a Lotka-Volterra competition equation. In Section 5, we study invariant manifolds and the Hartman-Grobman theorem, which generalize certain results for planar equations in Chapter 4 to dierential equations in n .
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Chapter 9. Stability. Part II. In Section 1, we prove a result concerning the equivalence of stability (or asymptotic stability) and uniform stability (or uniform asymptotic stability) for autonomous dierential equations. In Section 2, we use the results from Chapter 3 to derive stability properties for general linear dierential equations, and prove that they are determined by the fundamental matrix solutions. The results here include those derived in Chapter 5 for linear dierential equations with constant or periodic coecients as special cases. Stability properties of general linear dierential equations with linear or nonlinear perturbations are also studied using the variation of parameters formula and Gronwalls inequality. In Section 3, we introduce Liapunovs method for general (nonautonomous) differential equations and derive their stability properties, which extends the study of stabilities in Chapter 5 for autonomous dierential equations. Chapter 10. Bounded Solutions. In Section 1, we make some denitions and discuss the relationship between boundedness and ultimate boundedness. In Section 2, we derive boundedness results for general linear dierential equations by using the results from Chapter 9. It will be seen that stability and boundedness are almost equivalent for linear homogeneous differential equations, and they are determined by the fundamental matrix solutions. For nonlinear dierential equations, examples will be given to show that the concepts of stability and boundedness are not equivalent. In Section 3, we look at the case when the coecient matrix is a constant matrix, and verify that the eigenvalues of the coecient matrix determine boundedness properties. In Section 4, the case of a periodic coecient matrix is treated. The Floquet theory from Chapter 3 is used to transform the equation with a periodic coecient matrix into an equation with a constant coecient matrix. Therefore, the results from Section 3 can be applied. In Section 5, we use Liapunovs method to study boundedness properties for general nonlinear dierential equations. Chapter 11. Periodic Solutions. In Section 1, we give some basic results concerning the search of periodic solutions and indicate that it is appropriate to use a xed point approach. In Section 2, we derive the existence of periodic solutions for general linear dierential equations. First, we derive periodic solutions using the eigenvalues of U (T, 0), where U (t, s) is the fundamental matrix solution of linear homogeneous dierential equations. Then we derive periodic solutions from the bounded solutions. Periodic solutions of linear dierential equations with linear and nonlinear perturbations are
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also given. In Section 3, we look at general nonlinear dierential equations. Since using eigenvalues is not applicable now, we extend the idea of deriving periodic solutions using the boundedness. First, we present some Masseratype results for one-dimensional and two-dimensional dierential equations, whose proofs are generally not ext...