Ordinary Differential Equations (Universitext)

There are dozens of books on ODEs, but none with the elegant geometric insight of Arnol'd's book. Arnol'd puts a clear emphasis on the qualitative and geometric properties of ODEs and their solutions, rather than on theroutine presentation of algorithms for solving special classes of equations.Of course, the reader learns how to solve equations, but with much more understanding of the systems, the solutions and the techniques. Vector fields and one-parameter groups of transformations come right from the startand Arnol'd uses this "language" throughout the book. This fundamental difference from the standard presentation allows him to explain some of the real mathematics of ODEs in a very understandable way and without hidingthe substance. The text is also rich with examples and connections with mechanics. Where possible, Arnol'd proceeds by physical reasoning, using it as a convenient shorthand for much longer formal mathematical reasoning. This technique helps the student get a feel for the subject. Following Arnol'd's guiding geometric and qualitative principles, there are 272 figures in the book, but not a single complicated formula. Also, the text is peppered with historicalremarks, which put the material in context, showing how the ideas have developped since Newton and Leibniz. This book is an excellent text for a course whose goal is a mathematical treatment of differential equations and the related physical systems.

Both are excellent, are are all the other
books & papers I've seen by V.I. Arnol'd.


A beauty; a struggle   March 11, 2004
Raman (Chicago, IL USA)
36 out of 38 found this review helpful

This has to be one of the most amazing math books I've ever read. Arnol'd seems to do the impossible here - he blends abstract theory with an intuitive exposition while avoiding any tendency to become verbose. By the end of Arnol'd, it's hard not to have a deep understanding of the way that ODEs and their solutions behave.

Arnol'd accomplishes this feat through an intense parsimony of words and topics. Everything in this book builds on the central theme of the relations between vector fields and one-parameter groups of diffeomorphisms, and the topics are illustrated (and often motivated) almost exclusively through problems in classical mechanics, most notably the plane pendulum. Almost no solution techniques are given in this book - expect no mention of integrating factors or Bessel functions. One of the main reasons that the book does so much without bogging down is that the mathematical formalism is minimal and terse - proofs are often one or two lines long, merely mentioning the conceptual justification of a result without detailed, formal constructions.

But the result of this parsimony is that Arnol'd is a very difficult book. To understand every detail and to be able to attempt every problem, I think, basically requires a math degree - lots of linear algebra (for his monumental 116-page chapter on linear systems), a solid background in analysis and topology, and a bit of differential geometry and abstract algebra are prerequsites for a full understanding. (I found the section on the "topological classification of singular points," in particular, nearly incomprehensible with my thin chemistry-major math background.) There are foibles, too, including proofs that satisfy the requirements for some theorem or definition without actually stating what theorem or definition is now applicable. One can detect some mild arrogance in places (after an arduous two-page proof, he mentions "As always in proving obvious theorems, it is easier to carry out the proof of the extension theorem than to read through it.") Also, a few typos can be found here and there, which sometimes result in confusion.

One very curious thing about Arnol'd is that my most brilliant math-major friends find it impenetrable, whereas I know biologists who got through it with no problem. So I guess that, for a mere mortal, reading Arnol'd demands a willingness to have a feel for a big picture without worrying about every epsilon and delta.

So grab a copy of this book, let it flow, and learn about ODEs. It's well worth the effort.


wow! differential equations made appealing   December 20, 2005
mathwonk
18 out of 19 found this review helpful

I had always hated d.e.'s until this book made me see the geometry. And I have only read a few pages.

I never realized before that the existence and uniqueness theorem defines an equivalence relation on the compact manifold, where two points are equivalent iff they lie on the same flow curve. This instantly renders a d.e. visible, and not just some ugly formulas.

He also made me understand for the first time the proof of Reeb's theorem that a compact manifold with a function having only 2 critical points is a sphere. If they are non degenerate at least, the proof is simple. Each critical point has a nbhd looking like a disc. In between, the lack of critical points means there is a one parameter flow from the boundary circle of one disc to the other, i.e. thus the in between stuff is a cylinder.

Hence gluing a disc into each end of a cylinder gives a sphere! It also makes it clear why the sphere may have a non standard differentiable structure, because the diff. structure depends on how you glue in the discs.

What a book. I bought the cheaper older version, thanks to a reviewer here, and I love it. No other book gives me the geometry this forcefully and quickly. Of course I am a mathematician so the vector field and manifold language are familiar to me. But I guess this is a great place for beginners to learn it.

One tiny remark. He does not mind "deceiving you" in the sense of making plausible statements that are actually deep theorems in mathematics to prove. E.g. the fact that in a rectangle it is impossible to join two pairs of opposite corners by continuous curves that do not intersect, is non trivial to prove.

Hence the staement on page 2 that the problem is "solved" merely by introducing the phase plane, is not strictly true, until you prove the intersection statement above. All the phase plane version does for me is render the problem's solution highly plausible, and show the way to solving it. You still have to do it. But it was huge fun thiunking up a fairly elementary winding number argument for this fact.

Good teachers know how to deceive you instructively by making plausible statements that a beginner is willing to accept. I presume a physicist, e.g., would not quarrel with the statement above about curves intersecting.

This is the best differential, equaitons book I know of if you want to understand what they are, as opposed to learn to calculate canned solution fornmulas for special ones. He even makes clear what it is that is special about the special ones, e.g. linear equations are nice not just because the solutions are familiar exponential functions, but because the flow curves exist for all time,...


A geometrical approach to differential equations.   November 10, 1995
17 out of 20 found this review helpful

This is one of the few original books in the area of differential equations. In his clear style, Arnold presents the basics of differential equations. He is more interested in understanding the solutions than in deriving them by analytical methods. The text is well organized and there seem to be more figures than proofs (although all proofs are there, it just that they do not get in the way). A must, if you are in the area of chaos and dynamical systems. (RM)


An understanding-oriented mathematical textbook on ODEs.   August 8, 2000
Ambrosini Walter (Italia)
10 out of 13 found this review helpful

It is hardly needed to add words to the existing positive reviews of the book. In the line of previous comments, I just mention that it is an enjoyable book on a basic subject of great interest also for engineers and physicists. The matter is treated with the evident purpose to make the reader fully aware of the interesting geometrical and dynamic implications of the conclusions reached at each step. It is a nice counterexample for those who believe that, to be rigorous, a mathematical book needs to be very hard to read.