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There are already a number of requests for textbooks detailing nonlinear stability theory, chaos theory etc. but many of them are more introductory (e.g. Strogatz - Nonlinear Dynamics and Chaos)

I've covered all this material before but I'm prone to forgetting the details. I hoped somebody might be able to point me in the direction of a more formal reference text on this subject. Perhaps a graduate level text, that covers major undergraduate material in a fairly mathematically rigorous way, as well as a little extra?

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Here are two suggestions of the textbooks:

The second one is more or less self-contained, whereas the first one assumes good first ODE course (but reviews the necessary material). These are textbooks and not research monographs, which is more applicable to the texts in the other answer.

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    $\begingroup$ Another vote for Wiggins book. Hard to beat in terms of pedagogical value. Holmes/Guckenheimer is more of a reference text comparatively speaking, and less comprehensive. $\endgroup$ – nonlinearism Mar 12 '14 at 18:09
  • $\begingroup$ Wiggins book does look particularly good, difficult to shell out for when a quick Google returns a full PDF as the second result! But nothing beats the feel of a decent desk reference. $\endgroup$ – oLas Mar 16 '14 at 13:24
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"Nonlinear Oscillations, Dynamical Systems ,and Bifurcation of Vector Fields" by John Guckenheimer and Philip Holmes comes to mind. I took a class on Dynamical Systems with the first author many years ago and this was the text. I see people using the book by Strogatz and always feel that it is just not at the same level.

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    $\begingroup$ I would add the book Geometric theory of dynamical systems by Palis and de Melo, and maybe Dynamical Systems and chaos by Broer and Takens. There are plenty of references of a graduate level, but dynamical systems (in particular nonlinear dynamics) is a very broad topic. There is the encyclopaedia of dynamical systems as well (by V.I. Arnold), which treats many nonlinear phenomena in a rigorous way. Another applied mathematics book would be Nonlinear Differential Equations by Jordan and Smith. $\endgroup$ – PepeToro Mar 12 '14 at 13:49
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    $\begingroup$ Agree V.I. Arnold should not be missed , too many to list. Perhaps Hirsch & Smale should be mentioned; An older classic is "Differential Equations, Dynamical Systems, And Linear Algebra", also more recently "Differential Equations , Dynamical Systems, And An Introduction To Chaos". $\endgroup$ – Alan Mar 12 '14 at 14:50
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    $\begingroup$ The Wiggins book has basically replaced Guckenheimer and Holmes. Wiggins is now THE graduate textbook for ODEs and Dynamical systems, and way more comprehensive. $\endgroup$ – nonlinearism Mar 12 '14 at 17:14

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