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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.

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    $\begingroup$ "Qualitative Theory of Planar Differential Systems" by Dumortier, Freddy, Llibre, Jaume, Artés, Joan if you want to have some fun. For a complete and dense exposition (the best one as far as I know) see Coddington's "Theory of Ordinary Differential Equations". There's Chicone's book too for the same purpose of the later one. For an Eulerian style of math, there's always Arnold's book. $\endgroup$ – user40276 Jul 20 '14 at 5:36
  • $\begingroup$ @user40276 What is an Eulerian style of math? I am not sure what "Arnold's book" is either. (sorry, don't have too much familiarity with existing mathematical literature) $\endgroup$ – user89 Jul 20 '14 at 19:22
  • $\begingroup$ I was just kidding. By Eulerian I meant in the way that Euler used to write, pretty incomplete and informal, but very insightful and inspiring. $\endgroup$ – user40276 Jul 20 '14 at 22:25
  • $\begingroup$ @user40276 You have read some of Euler's works first hand? P.S. "Arnold" = "V. I. Arnold"? $\endgroup$ – user89 Jul 22 '14 at 4:45
  • $\begingroup$ The answer is no and yes. $\endgroup$ – user40276 Jul 22 '14 at 16:31
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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).
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