The final prices may differ from the prices shown due to specifics of VAT rules About this Textbook This textbook presents a systematic study of the qualitative and geometric theory of nonlinear differential equations and dynamical systems. Although the main topic of the book is the local and global behavior of nonlinear systems and their bifurcations, a thorough treatment of linear systems is given at the beginning of the text. All the material necessary for a clear understanding of the qualitative behavior of dynamical systems is contained in this textbook, including an outline of the proof and examples illustrating the proof of the Hartman-Grobman theorem, the use of the Poincare map in the theory of limit cycles, the theory of rotated vector fields and its use in the study of limit cycles and homoclinic loops, and a description of the behavior and termination of one-parameter families of limit cycles. The text succeeds admiraby Examples abound, figures are used to advantage, and a reasonable balance is maintained between what is proved in detail and what is asserted with supporting references Each section closes with a set of problems, many of which are quite interesting and round out the text material
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Marsden L. Sirovich M. Golubitsky Advisors G. Holmes D. Barkley M. Dellnitz P. Sirovich: Introduction to Applied Mathematics. Perko: Differential Equations and Dynamical Systems, 3rd ed. Seaborn: Hypergeometric Functions and Their Applications. Pipkin: A Course on Integral Equations. Braun: Differential Equations and Their Applications, 4th ed. Van de Velde: Concurrent Scientific Computing. Holmes: Introduction to Perturbation Methods.
Merkin: Introduction to the Theory of Stability of Motion. Perko Series Editors J. ISBN alk. Differential equations, Nonlinear. Differentiable dynamical systems. P47 All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher Springer-Verlag New York, Inc.
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Series Preface Mathematics is playing an ever more important role in the physical and biological sciences, provoking a blurring of boundaries between scientific disciplines and a resurgence of interest in the modern as well as the clas- sical techniques of applied mathematics. This renewal of interest, both in research and teaching, has led to the establishment of the series: Texts in Applied Mathematics TAM.
The development of new courses is a natural consequence of a high level of excitement on the research frontier as newer techniques, such as numerical and symbolic computer systems, dynamical systems, and chaos, mix with and reinforce the traditional methods of applied mathematics. Thus, the purpose of this textbook series is to meet the current and future needs of these advances and encourage the teaching of new courses. TAM will publish textbooks suitable for use in advanced undergraduate and beginning graduate courses, and will complement the Applied Math- ematical Sciences AMS series, which will focus on advanced textbooks and research level monographs.
Pasadena, California J. Marsden Providence, Rhode Island L. Sirovich Houston, Texas M. Golubitsky Preface to the Third Edition This book covers those topics necessary for a clear understanding of the qualitative theory of ordinary differential equations and the concept of a dynamical system. It is written for advanced undergraduates and for beginning graduate students. It begins with a study of linear systems of ordinary differential equations, a topic already familiar to the student who has completed a first course in differential equations.
An efficient method for solving any linear system of ordinary differential equations is presented in Chapter 1. The major part of this book is devoted to a study of nonlinear sys- tems of ordinary differential equations and dynamical systems. Since most nonlinear differential equations cannot be solved, this book focuses on the qualitative or geometrical theory of nonlinear systems of differential equa- tions originated by Henri Poincarc in his work on differential equations at the end of the nineteenth century as well as on the functional properties inherent in the solution set of a system of nonlinear differential equations embodied in the more recent concept of a dynamical system.
Our primary goal is to describe the qualitative behavior of the solution set of a given system of differential equations including the invariant sets and limiting behavior of the dynamical system or flow defined by the system of dif- ferential equations.
In order to achieve this goal, it is first necessary to develop the local theory for nonlinear systems. This is done in Chapter 2 which includes the fundamental local existence-uniqueness theorem, the Hartman-Grobman Theorem and the Stable Manifold Theorem.
These lat- ter two theorems establish that the qualitative behavior of the solution set of a nonlinear system of ordinary differential equations near an equilibrium point is typically the same as the qualitative behavior of the solution set of the corresponding linearized system near the equilibrium point. After developing the local theory, we turn to the global theory in Chap- ter 3. This includes a study of limit sets of trajectories and the behavior of trajectories at infinity.
Some unresolved problems of current research inter- est are also presented in Chapter 3. For example, the Poincare-Bendixson Theorem, established in Chapter 3, describes the limit sets of trajectories of two-dimensional systems; however, the limit sets of trajectories of three- dimensional and higher dimensional systems can be much more compli- cated and establishing the nature of these limit sets is a topic of current x Preface to the Third Edition research interest in mathematics.
In particular, higher dimensional systems can exhibit strange attractors and chaotic dynamics. All of the preliminary material necessary for studying these more advance topics is contained in this textbook.
This book can therefore serve as a springboard for those stu- dents interested in continuing their study of ordinary differential equations and dynamical systems and doing research in these areas. Chapter 3 ends with a technique for constructing the global phase portrait of a dynami- cal system. The global phase portrait describes the qualitative behavior of the solution set for all time. In general, this is as close as we can come to "solving" nonlinear systems.
In Chapter 4, we study systems of differential equations depending on pa- rameters. The question of particular interest is: For what parameter values does the global phase portrait of a dynamical system change its qualitative structure?
The answer to this question forms the subject matter of bifurca- tion theory. An introduction to bifurcation theory is presented in Chapter 4 where we discuss bifurcations at nonhyperbolic equilibrium points and periodic orbits as well as Hopf bifurcations.
Chapter 4 ends with a dis- cussion of homoclinic loop and Takens-Bogdanov bifurcations for planar systems and an introduction to tangential homoclinic bifurcations and the resulting chaotic dynamics that can occur in higher dimensional systems. The prerequisites for studying differential equations and dynamical sys- tems using this book are courses in linear algebra and real analysis.
For example, the student should know how to find the eigenvalues and cigenvec- tors of a linear transformation represented by a square matrix and should be familiar with the notion of uniform convergence and related concepts.
In using this book, the author hopes that the student will develop an appre- ciation for just how useful the concepts of linear algebra, real analysis and geometry are in developing the theory of ordinary differential equations and dynamical systems. The heart of the geometrical theory of nonlinear differential equations is contained in Chapters of this book and in or- der to cover the main ideas in those chapters in a one semester course, it is necessary to cover Chapter 1 as quickly as possible.
In addition to the new sections on center manifold and normal form theory, higher codimension bifurcations, higher order Melnikov theory, the Takens-Bogdanov bifurcation and bounded quadratic systems in R2 that were added to the second edition of this book, the third edition contains two new sections, Section 4.
Also, some new results on the structural stability of polynomial systems on R2 have been added at the end of Section 4. Preface to the Third Edition xi A solutions manual for this book has been prepared by the author and is now available under separate cover from Springer-Verlag at no additional cost.
I would like to express my sincere appreciation to my colleagues Freddy Dumortier, Iliya Iliev, Doug Shafer and especially to Terence Blows and Jim Swift for their many helpful suggestions which substantially improved this book. I would also like to thank Louella Holter for her patience and precision in typing the original manuscript for this book.
A good portion of this chapter is concerned with the computation of the matrix eAt in terms of the eigenvalues and eigenvectors of the square matrix A. Throughout this book all vectors will be written as column vectors and AT will denote the transpose of the matrix A. The general solution of the above uncoupled linear system can once again be found by the method of separation of variables.
This motion can be described geometrically by drawing the solution curves 2 in the x1i x2 plane, referred to as the phase plane, and by using arrows to indicate the direction of the motion along these curves with increasing time t; cf.
Figure 1. Note that solutions starting on the xl-axis approach the origin as t - oo and that solutions starting on the x2-axis approach the origin as t -. Figure 1 gives a geometrical representation of the phase portrait of the uncoupled linear system considered above. This mapping which need not be linear defines a vector field on 1. Uncoupled Linear Systems 3 Figure 1 R2; i.
If we draw each vector f x with its initial point at the point x E R2, we obtain a geometrical representation of the vector field as shown in Figure 2. Note that at each point x in the phase space R2, the solution curves 2 are tangent to the vectors in the vector field Ax. Consider the following uncoupled linear system in R And the phase portrait for this system is shown in Figure 3 above.
The x1, x2 plane is referred to as the unstable subspace of the system 3 and 1. Uncoupled Linear Systems 5 the x3 axis is called the stable subspace of the system 3. Precise definitions of the stable and unstable subspaces of a linear system will be given in the next section. Find the general solution and draw the phase portraits for the fol- lowing three-dimensional linear systems: Hint: In c , show that the solution curves lie on right circular cylin- ders perpendicular to the x1, x2 plane.
Identify the stable and unsta- ble subspaces in a and b. The x3-axis is the stable subspace in c and the x1, x2 plane is called the center subspace in c ; cf. Section 1. Linear Systems 3. Describe this relationship both for k positive and k negative. We first consider the case when A has real, distinct eigenvalues. The following theorem from linear algebra then allows us to solve the linear system 1. If the eigenvalues Al, A2, A proof of this theorem can be found, for example, in Lowenthal [Lol.
Under the hypotheses of the above theorem, the solution of the linear system 1 is given by the function x t defined by 2.
Differential Equations and Dynamical Systems, Third Edition
Differential Equations and Dynamical Systems, Third Edition
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Differential Equations and Dynamical Systems