Insightful books on differential equations?

Question

What are some recommendations for insightful books on differential equations and difference equations? These books don't need to be in the format of a textbook, and don't need to provide the same topics as a textbook on ODEs might.

After reading such a book, one should be somewhat in awe of the subject. They shouldn't be pop-science-y however.

See this question for inspiration on what answers should look like.

Answer 1

Here is a very subjective list:

• Arnold, Mathematical Methods of Classical Mechanics This book treats classical mechanics from mathematical point of view and introduces a wealth of various mathematical concepts (such as a differential manifold or differential form) in a right context. The main mathematical tool is ODE and it is really inspiring to see how they work to describe the physical reality.
• Lectures on Analytic Differential Equations is written by two leading experts in the theory of analytic ODEs and brings you right to the frontier of the current research. This is not exactly a textbook, but a very well written research monograph that covers a lot of various topics pertinent to the ordered part of ODE theory (as opposed to the real of chaos).
• Evolutionary Games and Population Dynamics. Do not be mislead by the title. This is quite an advanced book on applied ODE theory which treats a lot of mathematical techniques applied to genetics, population dynamics, and evolutionary problems. (There is a first edition, which is quite different, and which, to my taste, makes even better read).