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What is a a good text on bifurcation theory for mathematicians who haven't seen it before?

I'm looking to get a feel for the intuition behind the subject, major standard theorems, etc. I do not mind some level of rigor and sophistication, although I am not looking for a reference text with maximally generalized results.

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There are many, but these are some of my favorites in this really fun area.

  • Dynamics and Bifurcations, Jack K. Hale, Huseyin Kocak

This book is intended for undergraduate and beginning graduate students in mathematics or science and engineering. It has many chapters in one, two (and three) dimensions and was written with lower formalism and is very accessible. Hale is also one of the authors of Methods of Bifurcation Theory (Grundlehren der mathematischen Wissenschaften) (v. 251), by S.-N. Chow, J. K. Hale, which is a comprehensive book on graduate level bifurcation theory.

  • Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering by Steven Henry Strogatz

This book is aimed at undergraduates to nonlinear dynamics and chaos and is very readable and accessible. It stresses analytical methods, concrete examples and geometric intuition. There are tons of applications, including such oddities as how fireflies synchronize their blinking, how lasers work, and using chaos to send coded messages. Particularly, there is a bifurcation chapter for one-dimensional flows and a different chapter for two-dimensional flows.

  • An Introduction to Dynamical Systems, D. K. Arrowsmith, C. M. Place

This is an advanced text better suited for graduate students in applied math, physics and engineering. This is from the UK system and the style of writing may not be good for all. There are two chapters on bifurcation theory, local bifurcations on planar vector fields on $\mathbb{R}$ and diffeomorphisms on $\mathbb{R^2}$.

  • Differential Equations: A Dynamical Systems Approach: Higher-Dimensional Systems, John H. Hubbard, Beverly H. West

This is a four part series of books and they are an excellent and highly recommended. They are written to be accessible (although part 2 is closer to a graduate level text). Part I is for ODEs and Part II is for Higher Dimensional Systems. Chapter 9 of Part II is dedicated to Bifurcations. This is one of my favorite set of books.

  • Introduction to Applied Nonlinear Dynamical Systems and Chaos, Stephen Wiggins

This book is intended for advanced undergraduates, but is better suited as a graduate level text. Is is full of excellent examples and problems. The main focus of this book is bifurcations and the two largest chapters are Local Bifurcations and Aspects of Global Bifurcations and Chaos. This book has served well as a reference book and should is useful for those who are interested in going into this area.

  • Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, John Guckenheimer, Philip Holmes

This is a graduate level text and requires a fairly advanced mathematical background. This book has three chapters dedicated to bifurcation, Chapter 3 is Local Bifurcations, Ch 6 is Global Bifurcations and Ch 7 is Local Codimension Two Bifurcations of Flows.

  • Elements of Applied Bifurcation Theory, Yuri Kuznetsov

This is a graduate level text and has a pretty moderate mathematical sophistication. What is interesting (as well as the several chapters dedicated to bifurcation) is the numerical analysis and treatment chapter and the development of said numerical methods and tools.

  • Differential Equations and Dynamical Systems, Lawrence Perko

This book is also aimed at advanced undergraduates and graduate level. Perko's book is one of the best books that gives an advanced introduction to dynamical systems from the point of differential equations. It is not well received by many due to the problem set, but has an entire chapter (Ch 4) dedicated to Nonlinear Systems: Bifurcation Theory. This is not my favorite book and I would rather choose Smale, Hirsch and Devaney's Differential Equations, Dynamical Systems, and an Introduction to Chaos, but felt I should mention it as it appears to be decently used and I own it too.

You can peruse these on Amazon and see if any suits the style you like.

Also, by no means is this intended to be an exhaustive list as there were many other books I could have listed and own, but these are mostly from the excellent TAM series of Springer books and I think it is an excellent series.

Regards

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  • $\begingroup$ Honestly, I love Pollard’s, Amzoti. The title is: “Ordinary Differential Equations" and Dover publishes it. I have learnt much about the ODE's from this book. Try it. $\endgroup$ – mrs Mar 22 '13 at 20:14
  • $\begingroup$ Could you include a short description of these books for purposes of comparison? (Doesn't need to be complicated - even short taglines like "This is the applied choice," or "This book is most readable for beginners" would suffice.) $\endgroup$ – Alexander Gruber Jun 17 '14 at 6:41
  • $\begingroup$ Nice summaries, Amzoti (at least I find them useful, especially since I'm hardly a specialist in the subjects.) $\endgroup$ – amWhy Jun 18 '14 at 15:16

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