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I am a first year graduate student in math. I am taking a graduate course on ODE which covers the topics listed below. I feel that the lecture notes of my instructor are great. However, like with any math course, there is no way for me to have a full understanding of the topic without solving problems on each topic. I really appreciate if anyone can tell me about a good textbook that has lots of solved examples/problems about all/some of the topics listed below so that I can go over all these problems/examples to "master" the material. I also included at the end of this post a bit about my background so that you list only books that I can read. Thanks in advance to anyone who answers this post!

Chapter 2: Linear Systems:

Chapter 3: Stability: Basic theory

Chapter 4: 4. Nonlinear Systems: Local Theory

Chapter 5: Nonlinear Systems: Global Theory

Chapter 6: Bifurcations

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"Dynamics and Bifurcations" by Hale and Koçak covers most of your chapters 2 through 6. That book was one of my best friends for a while. I first met that book as a first semester grad student. There are some exercises in the book at the end of each chapter, but it is far from a book of exercises.

"Nonlinear Dynamics and Chaos" by Steven Strogatz is a go to upper level undergraduate book covering much of this material. There are plenty of exercises there. That was my favorite book for a while as an undergrad, and still one of my favorite books on nonlinear dynamics. It is easy to understand, but it is not a trivial read, as it covers much of what you wrote here in a thorough manner.

As you might expect, both of these books cover material not in your list.

If there were a way for you to ask either your instructor, or a graduate student who has taken courses with him, this very same question, and also get their opinion on ones that have been suggested to you, it would probably be a good idea.

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There are some nice cheap Dover books on this subject which I find very useful, even though they are quite old. However, the facts of ODEs haven't changed that much. Even the more recent books say the same thing as the old books, just with better typesetting and slightly more modern notations.

Morris Tenenbaum, Harry Pollard (1963)
Ordinary differential equations: An elementary textbook for students of mathematics, engineering, and the sciences
ISBN: 978-0-486-64940-5

Earl Alexander Coddington (1961)
An introduction to ordinary differential equations
ISBN: 978-0-486-65942-8

Francis Joseph Murray, Kenneth Sielke Miller (1954)
Existence theorems for ordinary differential equations
ISBN: 978-0-486-45810-6

Herbert Stanley Bear (1962)
Differential equations: A concise course
ISBN: 978-0-486-40678-7

You can see previews of these on the web site of a certain market-dominant online book-vendor. These books are so inexpensive, I bought them all. They are all useful. Some are best for existence/uniqueness, while others are best for methods for specific kinds of equations.

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