Reddit reviews Differential Equations, Dynamical Systems, and an Introduction to Chaos (Pure and Applied Mathematics)

I'll try to link to where there's a preview available. Check Amazon for reviews.

Set Theory

Halmos, Naive Set Theory (ignore the incorrect cover) - I read this in the bathroom. I like to describe it as a haiku on the basic principles of set theory. A classic, and my overall favorite math book.

Differential Equations

Hirsch, Smale, and Devaney, Differential Equations, Dynamical Systems and an Introduction to Chaos - Concerns itself with the qualitative study of differential equations, written by a hugely influential dynamicist (Smale). This book won't teach you how to solve differential equations but god damn you'll be able to understand them.

Elementary Number Theory

Dudley, Elementary Number Theory - Very fun read, exercises integrated into the exposition, natural progression of topics and ideas. A good book to take on the bus.

Elementary Abstract Algebra

Pinter, A Book of Abstract Algebra - The course in abstract algebra I took in undegrad was just plain shitty. Decided to pick this up a couple weeks ago to give myself a better education and ended up breezing through the whole damn thing, including every exercise. This book is simply incredible for self teaching. It is broken up into short chapters (usually 4 to 6 pages) followed by exercises (another 4 to 6 pages). Often the exercises are grouped to allow the reader to prove bigger results step-by-step. Had an absolute blast with this one.

Topology

Alexandroff, Elementary Concepts of Topology - This book blew my mind. It's about 50 pages long and focuses on building the machinery necessary to bridge the gap from point-set topology to algebraic topology. The book climaxes with the statement that questions about homeomorphisms between manifolds can be asked and answered in terms of homomorphisms between groups. Never before has the motivation for an entire field of mathematics been made so evident to me. This should be on everyone's bookshelf.

Well, you accused a perfectly rigorous (presumably, I haven't read it) classic text of being insufficiently rigorous for pedantic reasons. Your post is analogous to someone accusing a chef of not making a dish from scratch because he or she did not grow the vegetables and slaughter the meat.

I know this isn't what you asked for, but if you're looking to understand differential equations at an advanced undergrad level, I cannot recommend this book more highly. It is one of the best math books I've ever read or taught from. For beginning grad level with a bit more rigor, this book is quite nice.

But if you're really looking for a formal development that broaches topic of differential equations, I'll reluctantly point you to chapter 30 of this book. I warn you that it is far less pleasant to read than the other links above.

The second book that gerschgorin listed is very good, though a little old fashioned.

Since you are finishing up your math major, I'd recommend Hirsch & Smale & Devaney, an excellent book if you have a little bit of mathematical background.

There is also a video series I'm making meant to be a quick overview of many of the key topics. Maybe useful, maybe not. Also, the MIT lectures are excellent.

My favourite used to be Calculus on Manifolds until I started reading Munkres' Analysis on Manifolds. It covers the same material and then some and does a better job at explaining it. Spivak's purpose was a graduate reference book, and I think it does a good job at that. But in terms of learning Multivariable Analysis from it, it is very dense, and leaves out some stuff which I feel hinders it.

In terms of DE you could look at this one by Hirsh. It has some humour like Spivak, and is very theoretical, it has some applications in it but we skipped them when we took DE at my uni. There's also the dover book Advanced Ordinary Differential Equations (I think) which was used for the same course. However DE/Dynamical systems/chaos isn't a really concrete subject as opposed to analysis, so there are many ways of approaching it.

What level of dynamical systems are we talking here? Graduate or undergraduate. In the former case I would recommend: http://www.amazon.com/Introduction-Dynamical-Encyclopedia-Mathematics-Applications/dp/0521575575
and for an undergraduate approach I would recommend:
http://www.amazon.com/Differential-Equations-Dynamical-Introduction-Mathematics/dp/0123497035
Both are pretty fun introductions to the subject. Good luck in your search