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I want to do my thesis about oscillations. I am a math student so I enjoy rigorous texts and hate sketchy ones. I am looking for a textbook or a good source that could help me with dynamical systems. What I mean is an introductory book for it.

For example I have enjoyed Real Mathematical Analysis by C.C. Pugh. I would greatly appreciate if someone could introduce me a book that could put everything about dynamical systems in perspective as good as it has been done in this book by Pugh (Pugh's is about analysis of course!).

Right now I started "Chaos: An Introduction to Dynamical Systems" by Alligood et al.

But I don't know if I am on the right track. So as a final sum-up:

  1. To be rigorous
  2. To cover introductory stuff(or maybe even advanced too!!)
  3. Good text: this probably means the ability of writer to intuitively pass the message beside being rigorous.

Thanks in advance.

P.S. I already see this but I find it irrelevant to my question.

Community
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Cupitor
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2 Answers2

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The book you started reading is good. However, despite that it treats some of more involved topics (as proof by exercises of Sharkovskii's theorem or stable and unstable manifold theorem), I still think that it is too wordy at the initial stage and skips a few very relevant points later. As a first introduction it is perfectly fine, but probably you'd like to see something more comprehensive.

There is another introductory book, which is quite rigorous and still accessible, and which goes really patient by introducing relevant concepts and notions: Hale and Kocak, Dynamics and Bifurcations. I think this book gives an ideal exposition to start reading graduate texts after it. This book however assumes that you already were exposed to differential equations (this is not a prerequisite in the book you are currently reading).

Artem
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  • Thank you very much for explanations. I had an introductory course on ODE but I am very novice to it. I don't know what you mean by being exposed to ODE. In what detail? Do you mean like knowing the details of mathematical analysis of ODEs or in general? In any case I will take a look at it. Thank you very much. Vote up. – Cupitor Feb 03 '14 at 13:27
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    @Naji The test is very simple. If you are OK to read Section 1.1 of Hale and Kocak, you'll be fine. If you think too many things unclear, you should first work through the book you are reading currently. – Artem Feb 03 '14 at 16:02
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My favorites:

  • Differential Dynamical Systems - Meiss;
  • Differential Equations, Dynamical Systems and an Introduction to Chaos - Hirsch, Smale, Devaney;

Other good (and perhaps more rigorous) books:

  • Differential Equations and Dynamical Systems - Perko;
  • Introduction to Applied Nonlinear Dynamical Systems and Chaos - Wiggins;

Reference containing plenty of solved examples and exercises:

  • Nonlinear Ordinary Differential Equations - An Introduction for Scientists and Engineers - Jordan, Smith;

and the respective problem book

  • Nonlinear Ordinary Differential Equations - Problems and Solutions - A Sourcebook for Scientists and Engineers - Jordan, Smith;

PS: I second Artem's opinion about Hale and Kocak's book; it's really nice.

etothepitimesi
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  • I am really surprised how different people have different tastes though I am really happy you second the other book there. I will take a look at all of those that I can access. Thank you very much for your "reference LIST" :) vote up. – Cupitor Feb 03 '14 at 13:28