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Bifurcation Theory

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Cover of the book “Bifurcation Theory” by Ale Jan Homburg and Jürgen Knobloch, part of the Graduate Studies in Mathematics series, volume 246, published by the American Mathematical Society. The design features a blue rectangle with yellow text over a yellow background with diagonal divisions. At the bottom, the American Mathematical Society logo is featured in dark blue text.
  • Author: Ale Jan Homburg and Jürgen Knobloch
  • Series: Graduate Studies in Mathematics
  • Publisher: AMS
  • Publication Date: 12/05/2024
  • Number of Pages: 521
  • Format: Paperback
  • Price: $89.00
  • ISBN: 978-1-4704-7880-3
  • Category: textbook

[Reviewed by Bill Satzer, on 08/14/2025]

Bifurcation theory is a component of the study of dynamical systems that analyzes the changes in dynamics that occur when there is a qualitative change of behavior arising from varying parameters. Poincaré introduced the term bifurcation in the late nineteenth century to mean the splitting of equilibria for a family of differential equations. A great deal has happened in the field since then.

The focus of this text is on ordinary differential equations and their flows that characterize a dynamical system. The authors’ intention is to capture the principal ideas of bifurcation theory and present them in a detailed, precise, and rigorous fashion. It is by no means an exhaustive treatment, but it covers a considerable amount of material. Readers should be aware that the text moves very rapidly from one topic to another, and the pace never falters.

While there continues to be considerable theoretical interest in bifurcation theory, the subject has become increasingly prominent in applications across the sciences where dynamical systems naturally arise. The authors note that this broader influence of bifurcation theory is evident from the currency of terms like “tipping point” and “critical transition”.

This book is intended for students with backgrounds that include undergraduate courses in ordinary differential equations, analysis, and topology. It is designed as a broad but not overwhelming introduction, emphasizing a lot of basic techniques and demonstrating their value in the modern study of dynamical systems and their utility in applications. It is a challenging book, probably most suitable for graduate students, especially those with a strong background in dynamical systems.

The authors begin by showing the reader two typical examples of bifurcations of equilibria that arise in population models, and then an artificial model designed to show homoclinic bifurcations. All three of these are explored in greater detail later in the book.

The text is broadly divided into three parts consisting of discussions of local bifurcations, nonlocal bifurcations, and global bifurcations. Before this the authors address preliminary background material on flows and invariant sets. This includes subjects like Floquet theory and Poincaré return maps. Since the authors want to focus on bifurcation theory itself, they have moved more detailed discussion of the tools and technical details to appendices. These fill in necessary background material in dynamical systems theory – non-linear analysis, including differential calculus on Banach spaces, invariant manifolds, the implicit function theorem with Lyapunov-Schmidt reduction, and normal forms.

This is a well-written introduction to the subject that is especially attentive to details and provides many examples. Exercises are provided in each chapter, and there is an extensive bibliography.


Bill Satzer (bsatzer@gmail.com), now retired from 3M Company, spent most of his career as a mathematician working in industry on a variety of applications. He did his PhD work in dynamical systems and celestial mechanics.