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:
- S. Smale, Differentiable dynamical systems
- D. V. Anosov, On the development of the theory of dynamical systems during the past quarter century
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.
- P. Holmes, Ninety plus thirty years of nonlinear dynamics: less is more and more is different
- M. Golubitskiy, I. Stewart, Recent Advances in Symmetric and Network Dynamics
- P. Ashwin, S. Coombes, R. Nicks, Mathematical frameworks for oscillatory network dynamics in neuroscience
- M. Viana, Dynamical Systems: Moving into the Next Century
- A. Pikovsky, M. Rosenblum, Dynamics of globally coupled oscillators: Progress and perspectives
- P. Holmes, Some Joys and Trials of Mathematical Neuroscience
- B. Krauskopf, H. M. Osinga, E. J. Doedel, M. E. Henderson, J. Guckenheimer,
A. Vladimirsky, M. Dellnitz, O. Junge, A survey of methods for computing (un)stable
manifolds of vector fields - É. Ghys, Knots in dynamics
- A. J. Homburg, B. Sandstede, Homoclinic and Heteroclinic Bifurcations in Vector Fields
- D. Eroglu, J. Lamb, T. Pereira, Synchronization of Chaos
- J.-M. Ginoux, C. Letellier, Van der Pol and the history of relaxation
oscillations: toward the emergence of a concept - M. B. Sevryuk, The classical KAM theory at the dawn of the twenty-first century
- D. Aubin, Writing the History of Dynamical Systems and Chaos:
Longue Duree and Revolution, Disciplines and Cultures - J. Chavarriga, M. Sabatini, A Survey of Isochronous Centers
- A. Champneys, A Twenty-First Century Guidebook for Applied Dynamical Systems
- C. Mira, Some Historical Aspects of Nonlinear Dynamics Possible Trends for the Future
Best Answer
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.