Reference book for "Dynamical Systems"

The book you started reading is good. However, despite that it treats some of more involved topics (as proof by exercises of Sharkovskii's theorem or stable and unstable manifold theorem), I still think that it is too wordy at the initial stage and skips a few very relevant points later. As a first introduction it is perfectly fine, but probably you'd like to see something more comprehensive.

There is another introductory book, which is quite rigorous and still accessible, and which goes really patient by introducing relevant concepts and notions: Hale and Kocak, Dynamics and Bifurcations. I think this book gives an ideal exposition to start reading graduate texts after it. This book however assumes that you already were exposed to differential equations (this is not a prerequisite in the book you are currently reading).

My favorites:

• Differential Dynamical Systems - Meiss;
• Differential Equations, Dynamical Systems and an Introduction to Chaos - Hirsch, Smale, Devaney;

Other good (and perhaps more rigorous) books:

• Differential Equations and Dynamical Systems - Perko;
• Introduction to Applied Nonlinear Dynamical Systems and Chaos - Wiggins;

Reference containing plenty of solved examples and exercises:

• Nonlinear Ordinary Differential Equations - An Introduction for Scientists and Engineers - Jordan, Smith;

and the respective problem book

• Nonlinear Ordinary Differential Equations - Problems and Solutions - A Sourcebook for Scientists and Engineers - Jordan, Smith;

PS: I second Artem's opinion about Hale and Kocak's book; it's really nice.