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
"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.