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I would be very glad if someone could point me to modern reviews/surveys on these topics. To be concrete, I'll provide some examples:

So, I'm looking for texts that talk about current development of theory, milestone ideas, possible directions and open questions.

As I've already said, my topics of interest are dynamical systems and differential equations (primarily ODE). But other texts will be interesting too if their topics are closely related to these; e.g. PDEs, chaotic dynamics, symbolic dynamics, Hamiltonian systems, etc.

So, thanks in advance!

LATER ADDITIONS:

If no one doesn't mind, I would like to add from time to time interesting findings to this list.

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Smale also wrote a little book Mathematics of Time which, as I recall, is also a pretty good survey.

Another favorite of mine is Foundations of Mechanics by Abraham and Marsden.

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  • $\begingroup$ Aha, so Smale's "Differentiable dynamical systems" is a part of Mathematics of time? That's curious, hope that I'll find this book soon $\endgroup$
    – Evgeny
    Oct 1 '13 at 4:13
  • $\begingroup$ @Evgeny: I don't think Mathematics of Time includes all of Differentiable Dynamical Systems, but it might. Pretty sure MoT has some of DDS, though. It's a pretty cool little book, and inspired much of my early interest in dynamical systems. Hope you find it! $\endgroup$ Oct 1 '13 at 4:23
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On ODEs and Chaos, I have the following recommendations:

1). This is a ~50 page document which describes some of the modern techniques of detecting chaos

http://docenti.lett.unisi.it/files/4/2/13/1/Wiggins_CHAOS.pdf

If this short document intrigues you, here are some detailed texts:

2). Global bifurcations and chaos: Wiggins

3). Chaos near resonance by G. Haller

The above two are monographs, and quite recent (by mathematical time scales).

4). Symplectic maps, variational principles, and transport JD Meiss Reviews of Modern Physics 64 (3), 795

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  • $\begingroup$ I had a look at Wiggins and Meiss: Wiggins' book seems very comprehensive and Meiss seems to be a quite great dive in for symplectic science. Thank you! $\endgroup$
    – Evgeny
    Oct 2 '13 at 17:35

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