Ordinary Differential Equations and Dynamical Systems (Graduate Studies in Mathematics) by Gerald Teschl (PDF)

Description

This book provides a self-contained introduction to ordinary differential equations and dynamical systems suitable for beginning graduate students. The first part begins with some simple examples of explicitly solvable equations and a first glance at qualitative methods. Then the fundamental results concerning the initial value problem are proved: existence, uniqueness, extensibility, dependence on initial conditions. Furthermore, linear equations are considered, including the Floquet theorem, and some perturbation results. As somewhat independent topics, the Frobenius method for linear equations in the complex domain is established and Sturm-Liouville boundary value problems, including oscillation theory, are investigated. The second part introduces the concept of a dynamical system. The Poincaré-Bendixson theorem is proved, and several examples of planar systems from classical mechanics, ecology, and electrical engineering are investigated. Moreover, attractors, Hamiltonian systems, the KAM theorem, and periodic solutions are discussed. Finally, stability is studied, including the stable manifold and the Hartman-Grobman theorem for both continuous and discrete systems. The third part introduces chaos, beginning with the basics for iterated interval maps and ending with the Smale-Birkhoff theorem and the Melnikov method for homoclinic orbits. The text contains almost three hundred exercises. Additionally, the use of mathematical software systems is incorporated throughout, showing how they can help in the study of differential equations.

User’s Reviews

Editorial Reviews: Review It’s easy to build all sorts of courses from this book — a classical one-semester course with a brief introduction to dynamical systems, a one-semester dynamical systems course with just brief coverage of the existence and linear systems theory, or a rather nice two-semester course based on most (if not all) of the material. –MAA Reviews

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

⭐good book

⭐In my opinion, authors and publishers confident enough to give us extensive look insides like this does want purchasers like you and I that are pleased with our choice, so do check it out, and thanks to the author! This happens more often with older books, but I’m always pleased when they do it with a very up to date title like this.I use this book in conjunction with a free Astrodynamics course I teach online at Phoenix U. There are many other much less expensive (especially Dover) books on ODE’s, like the classic

⭐. These have a lot more pages for far less money.The problem is that, especially in fields like Astrodynamics (eg. satellite orbits), there are the classic and older, still valid ODEs that are time tested and proven, and take 400 pages, and then 400 more pages of recent work! For example, we can break AD into 1609, 1709, 1809, 1909 and 2009 like I do for my class. The inexpensive ODE texts are fine for all the classics from Kepler through Newton and even Gauss. However, once you transition into the computer era, numerical methods, Hamiltonians, deterministic functions treated as stochastic due to number of variables, big O, and many others are not covered.This is one of the few “general” texts that covers developments in the whole field. To get this broad a survey would take at least 5 other specialty books, which is why I recommend this for my students, many of whom are autodidacts. Although this text is outstanding for self study, it IS GRADUATE level (as you’ll see in the look inside) and requires a good grounding in both calculus and (IMO) linear algebra. You CAN (as many of my students do) get this first, then fill in details from the lead up topics as needed, but realize it is necessarily fast paced due to what it covers. There are a LOT of outstanding examples and exercises from numerous applied fields.As you likely know if you are reading this, there is an entire settled body of ODEs in many fields, with proven constants and functions. Many of these are already in the free GNU Octave and other (expensive) CAS software programs. The new HP Prime can even graph many of them! This book deviates from just the practical and does cover many more areas of current research, and, compared to other up to date texts, is about $200 less than some Springer texts that are up to date on the same topics. The author is an excellent teacher, and transitions seamlessly between high proof level accuracy and numerical/ big O “good enough” topics for those of us on the applied side.Also a great refresher and updater for older Engineers, Physicists, Coyote vs. rabbit folk, etc. Has a surprisingly good treatment of chaos, given the dozens of dynamical systems books out there. If like me you were skeptical of chaos applications to astrodynamics, you might want to check out Belbruno:

⭐. For more detail on Hamiltonians in multiple particle motion, here is a gem available at decent prices used:

⭐. Enjoy!

⭐The book is in pristine condition, thanks a lot

⭐For context, I’m a second year grad student in general mathematics, with a B.S. in applied mathematics, so do with that what you will.This is, without a doubt, the worst math text I have ever encountered. The lack of specification in the proofs; the complete absence of explanation; the random use of the same variable for numerous things in the same proof without comment; the disaster that is this books’ lack of notational rigor when necessary and useful in some places, and the over complication of incredibly simple concepts in others. All of these things combine to make this book a nightmare.My professor for this course, with decades of experience and whose research area is differential equations, made numerous comments on how confusing this book can be. When I asked at one point what the book meant by some statement, the answer was, and I quote, “I have no idea.” Why this book was assigned to us, I will never cease to wonder.This book is a graduate level text meant for mathematically mature audiences, no doubt. But to be absolutely clear: this is a text for people who are already VERY familiar with the material, and absolutely NOT for those who are seeing this material for the first time. And even then, based on style and lack of precision alone, I couldn’t see myself recommending this to anyone in good conscience.

⭐Un clásico

⭐

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