I am looking for a good book on Geometric Theory of Dynamical Systems . I found Geometric Theory of Dynamical Systems by Jr. Palis myself,but it's very old, anyway i would like to find a pure topological book about Dynamical systems. On the other hand about: local stability, The kupka-Smale Theorm, Genericity and stability of Morse-Smale Vector Field,
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1$\begingroup$ Your question is too vague: what kind of result are you mostly interested in? What quality do you expect? What reading level? $\endgroup$– Benoît KloecknerCommented Nov 25, 2013 at 14:36
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$\begingroup$ I'm a Phd student and my interesting is Differential Topology and Dynamical systems $\endgroup$– aliCommented Nov 25, 2013 at 14:42
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2$\begingroup$ You should be little bit more specific, I guess I have at least two books on my desk alone on these topics. Are you interested by lower or higher dimensional systems or both? By purely topological, do you mean the ergodic point of view does not interest you? or smooth systems do not interest you? Or both? Is there a result in particular you want to aim? Anything that could narrow down the list of the dozens of good books that certainly exist. $\endgroup$– Benoît KloecknerCommented Nov 25, 2013 at 14:47
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$\begingroup$ Thanks for attention yes my point of view was Smooth Dynamics too and Geometric theory of dynamical systems $\endgroup$– aliCommented Nov 25, 2013 at 14:50
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6 Answers
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Pick up (almost) anything by Ethan Akin. I particularly recommend "The General Topology of Dynamical Systems" available on Amazon. Although it is somewhat older than what you indicate you are looking for, he has many newer surveys on the subject that are downloadable on his website.
In particular, the topological aspects of dynamics and stability are the main focus of his book and his survey articles. For instance, consider reading "Tourist's Guide to The General Topology of Dynamical Systems". (PDF download link.)
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A. Katok, B. Hasselblatt, Introduction to the Modern Theory of Dynamical Systems, 1997.
M. Brin, G. Stuck, Introduction to Dynamical Systems, 2002.
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4$\begingroup$ trying to learn dynamics form Katok-Hassenblatt is like trying to learn history by reading the 1911 edition of the Encyclopaediea Brittanica: i.e.: forget it. $\endgroup$ Commented Jun 29, 2015 at 2:55
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Such books are abundant. My favorite ones are Z. Nitecki, Differentiable dynamics, and J. Palis, W. de Melo, Geometric theory of dynamical systems. An introduction, Springer-Verlag, New York-Berlin, 1982.
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$\begingroup$ Thanks but i want a new book at least for 2000 $\endgroup$– aliCommented Nov 25, 2013 at 15:50
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1$\begingroup$ New books are listed in the answer of Misha. I agree with him about Katok Hasselblatt, but have not read the second one. (Actually the requirement >2000 looks weird to me. Cannot imagine any reason for this restriction:-) $\endgroup$ Commented Nov 25, 2013 at 18:25
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$\begingroup$ The second one is more suitable as a textbook. $\endgroup$– PengfeiCommented Nov 30, 2013 at 15:34
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Take a look at Bob Easton's book Geometric Methods for Discrete Dynamical Systems'.
He was a student of Conley's, who took a very topological/geometric perspective. Conley's tiny monograph,Isolated Invariant Sets and the Morse Index' is also a gem
but perhaps too dense.
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I guess "Geometrical Methods in the Theory of Ordinary Differential Equations" by Arnold should be in the list too, although it doesn't satisfy the "purely" topological criteria.
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Mainly from the Hamiltonian point of view:
Zehnder: Lectures on Dynamical Systems