Differential Equations, Dynamical Systems, and Linear Algebra (Pure and Applied Mathematics Book 60) by Morris W. Hirsch (PDF)

Description

This book is about dynamical aspects of ordinary differential equations and the relations between dynamical systems and certain fields outside pure mathematics. A prominent role is played by the structure theory of linear operators on finite-dimensional vector spaces; the authors have included a self-contained treatment of that subject.

User’s Reviews

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐As a senior undergrad majoring in math and economics, this book is everything but an easy read. To all fellow undergrads who are not math superheroes (that should about 75% of us), if you happen to come across this book in an upcoming course description, it may be a good idea to look for alternative. Currently, I’m looking for another book that I may be able to use as a supplement to get me through this course with a passing grade. Up to this point in my math career, I have never come across a text as ungraspable as this one; this is unfortunate since it appears that there is a lot of knowledge and content on the pages.

⭐This is an excellent book with a rigorous mathematical treatment of differential equations. Important topics such as stability of dynamical systems and operator theory are covered in great detail. I recommend this book for an introductory graduate course on differential equations and dynamical systems.Carlos Frederico Trotta Matt, Ph.D., Mechanical EngineerElectric Power Research Center, Rio de Janeiro, Brazil

⭐Good

⭐The preface: “this book can be used as early as the sophomore year” and “our goal is to develop nonlinear ordinary differential equations in open subsets of real Cartesian space in such a way that extension to manifolds is simple and natural.” My review concentrates upon those two aims:(1) Yes, portions of the book can profitably be utilized in sophomore year. With the prerequisite of Serge Lang’s Calculus texts, much here is within reach. Glance at exercises ! I would be surprised (shocked) if any of those in chapter one (1 to 8, page 12) posed an issue for the student who has studied Lang’s textbooks.(2) Initial chapters, one and two, are painless. Second chapter is a synopsis of Newtonian mechanics.A typo mars the discussion (first edition, page 2, Figure A, last graph is mislabeled ” ‘a’ greater than zero,” change that to “less than”. Read (bottom of page two): “the sign of ‘a’ is crucial here”…. so it pays to label the accompanying figures properly. As is evident, the chain rule will be utilized over and over again.(3) Third chapter is linear algebra: Linearity (page 30) then gives way to “natural correspondence between operators and n-X-n matrices” (page 31). Read: “Eigenvalues and eigenvectors are very important.” (page 42). Theorem two is capstone of the chapter: “By using this theorem we get much information about the general solutions directly from knowledge of the eigenvalues, without solving the differential equation.” (page 50). Before continuing with the text, re-read the first three chapters. Finish every problem (at the least re-do every example). The concluding section, complex eigenvalues (pages 55-59), will be important for the next chapter:(4) Linear systems with constant coefficients and complex e: This will be a short interlude of ten pages.It is prelude to the abstract chapter five: Linear Systems and Exponentials of Operators. There you will get a survey of introductory topology. Norms and operators, follow. Note: correspondence between complex numbers and 2X2 matrices (page 85, only square matrices are utilized in the text). Now, series expansions of exponential and trigonometric functions should be second nature. As usual, homogeneous followed by nonhomogeneous equations (traditional). Autonomous (that is, no explicit time-dependence) followed by non-autonomous (page 99, also traditional). Take-away: variation of parameters. Another clue: Leibniz. Read: “The eigenvector theory of real linear operator is rarely treated in texts, and is important for theory of linear differential equations.” That is what you get: eigenvalues, nilpotence, Cayley-Hamilton, semi-simple, canonical forms and more: “Operators on function spaces have many uses for both theoretical and practical work in differential equations.” (page 143).(5) Theory for following two chapters (seven and eight). Equilibrium states defined. Glance at problem #4 (page 150). Many physics students have already been exposed to this equation. That equation placed into mathematical context. Learn the meaning of ‘dense.’ (page 153, also problem #1, page 157). Of chapter eight, read it is “more difficult” and authors suggest “omitting the proofs.” That is good advice ! If you are unfamiliar with things such as Lipschitz condition (or, manipulating epsilons and deltas), then a review of such is recommended. Uniform convergence should be firmly grasped (glance at problems #1, 4, and 5, page 177). Serge Lang is the prime reference.(6) Stability, again. Physics students will find this material interesting and useful (pendulums in gravity-field, Maxwell on Saturn’s rings). We continue with stability and equilibria: Glance at example (page 201), graphing by hand a multivariable function. True, it is easy if you use software for that task (Mathematica), but, do it once by hand ! My initial exposure to differential equations occurred by way of “computer labs” (this was late 1990’s). More time spent with computer graphs meant less time learning the mathematics. Terms such as kinetic energy, potential energy, and conservative forces should already be firmly grasped.(7) Chapters ten to twelve, applications: circuits (background in Feynman Volume One), periodic solutions, ecology (predator and prey). Theory in chapter thirteen (glance at Problem #1, page 285, certainly that is not difficult).(8) An introduction to Hamiltonian mechanics. Read: “Our present goal is to put Newton’s equations into the framework of this book.” This will be an introduction to the advanced text of Arnold: Mathematical Methods Of Classical Mechanics and Ordinary Differential Equations. Arnold to be consulted in any event !(9) Final chapters: theory. That is, perturbations. Read: “this book is only an introduction to the subject of dynamical systems” and “appendix one (elementary facts) should have been seen before” If appendix one is material new to you, learn it first (before tackling the book). If searching for “manifolds,” see Arnold).(10) Concluding thoughts: As with textbooks that attempt a trilogy of topics (similar to Weinberger’s Partial Differential Equations) it is difficult to please every group of readers. The book encompasses three topics: differential equations, dynamical systems, linear algebra. You run the risk of pleasing no one when you attempt to please everyone. Any one of those topics can fill a book. Here, in only 350 pages, we get all those topics. If you are unaccustomed to mathematical proof , then the later chapters will be difficult. Hubbard and West write (of Hirsch and Smale): “this is the first book bringing modern developments of differential equations to a broad audience” and “Smale has profoundly influenced the authors.” (Differential Equations, part one, page 307, 1991). We should all be so profoundly influenced !.

⭐We used this text in the second quarter of my frosh honors calculus class in college back in 1986. I found it very difficult. Given all the positive reviews, I’ll venture this is simply because I’m an idiot.

⭐This is the book from which I was introduced to dynamical systems some twenty-odd years ago. It’s a thorough introduction that presumes a basic knowledge of multivariate differential calculus but is pretty well self-contained as far as linear algebra is concerned. Rigorous but readable, it provides a foundational understanding of n-dimensional linear dynamical systems and their basic exponential solution.But my opinions won’t be as helpful to the Amazon math shopper as a simple listing of what’s in the book. So here’s the table of contents.Chapter 1: First ExamplesChapter 2: Newton’s Equation and Kepler’s LawChapter 3: Linear Systems with Constant Coefficiants and Real EigenvaluesChapter 4: Linear Systems with Constant Coefficients and Complex EigenvaluesChapter 5: Linear Systems and Exponentials of OperatorsChapter 6: Linear Systems and Canonical Forms of OperatorsChapter 7: Contractions and Generic Properties of OperatorsChapter 8: Fundamental TheoryChapter 9: Stability of EquilibriaChapter 10: Differential Equations for Electric CircuitsChapter 11: The Poincare-Bendixson TheoremChapter 12: EcologyChapter 13: Periodic AttractorsChapter 14: Classical MechanicsChapter 15: Nonautonomous Equations and Differentiability of FlowsChapter 16: Perturbation Theory and Structural StabilityAfterwordAppendix I: Elementary FactsAppendix II: PolynomialsAppendix III: On Canonical FormsAppendix IV: The Inverse Function TheoremReferencesAnswers to Selected Problems

⭐I can see that this is not the book for you if you want to solve a particular differential equation. But in terms of understanding the field of dynamical systems, there is no rival. This book is a pleasure to read, for the first time I understood the importance and beauty of linear algebra. Academic Press says that this is their most successful mathematics text, and it is not hard to see why. I wish more texts were as clearly written and as beautiful to read.

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