UK Nonlinear News, August 2000

Numerical Methods for Bifurcations of Dynamical Equilibria

Willy Govaerts

Reviewed by Kurt Lust

This book focuses on the numerical study of bifurcations of equilibria of systems of ordinary differential equations (ODEs). In its 10 chapters it provides a fairly complete overview of issues arising in the field. The first three chapters are introductory in nature. From then on, a distinction is made between bifurcations that can be studied by looking only at a linearised system (higher-order derivatives show up in the nondegeneracy conditions) and bifurcations that involve the nonlinear terms. Chapters 4 and 5 discuss the former, while the chapters 6 to 9 are concerned with the latter. Finally, chapter 10 gives a brief introduction to methods for partial differential equations (PDEs). The detailed contents of each chapter are as follows:

Chapter 6, 7 and 8 make use extensively of results from [1] and [2], while chapters 9 and 10 use material from [3]. Some familiarity with the material covered by these books will help to make the respective chapters easier to read.

As can be seen from the above list, the book gives a fairly complete overview bifurcations of equilibria of ODEs and their computation. It discusses both large and minimal extended systems and mentions different ideas that are used to construct extended systems. It discusses singularity theory, methods based on center manifold theory and symmetry. One main feature of the book is the extensive use of bordered matrices. In fact, the book is even a bit biased towards such methods. Methods based on bordered matrices are typically very good, but it is not clear whether they are that much better than other approaches. The numerical aspects of the book remain mostly limited to the construction of defining systems for various bifurcations and the proof of their regularity. The book does include a nice discussion of continuation methods and it does discuss the author's method to solve bordered systems, but it lacks any discussion of alternative methods nor does it discuss how the structure arising in large extended systems can be exploited. The book also avoids a discussion of the relative merits of the various methods. The many links between related material in different chapters are not always clearly expressed.

The chapter on large-scale systems is rather weak. It fails to mention many of the problems arising when the methods discussed in the book are extended to really large systems. What if our solver for A is an iterative solver? Bordered matrix methods become a lot more expensive! What if we can only compute matrix-vector products with A and not with its transpose? Then the BEM method cannot be applied, and obtaining singular vectors or left eigenvectors is also impossible. Also, dimension reduction is a very expensive process. If the determining system is based on projected information, the accuracy to which the bifurcation point can be computed will depend strongly on the accuracy of the subspace. Moreover, derivatives of the basis will also be needed.

The book contains plenty of examples, both analytical and numerical. I would have liked more results illustrated graphically rather than presented as long tables of data, and more complete and annotated bifurcation diagrams. In its current format it is too easy to lose the thread when reading through a 5 or 10 page example.

The book is not really meant intended as a textbook, although the cover suggests that it can be used in a graduate course. A reader requires more than just the basic knowledge of linear algebra, numerical linear algebra, calculus and differential equations suggested on the back cover and in the preface. More advanced concepts are only briefly explained and, although the exercises at the end of each chapter would help, the amount of material and the fast pace of the text would be too overwhelming for most students. The first four or five chapters are useful for a course on numerical methods for bifurcation analysis. Even here the book would have to be supplemented, to cover both the computation of periodic solutions of ODEs and, perhaps briefly, basic methods for computing homoclinic orbits, rather than covering bifurcations of codimension higher than one or two. At first, the proofs are a bit hard, e.g. it is not always easy to see how one matrix was obtained from another. After a while things get easier since the same strategies return all the time. The book contains plenty of exercises at the end of each chapter. Some of them contain mostly analytical work, while others challenge the reader to solve a problem with any available continuation code or to write their own code. The author mostly used CONTEN. I recommend using this, since many of the algorithms presented in the book are implemented in CONTENT and since the on-line help of the package offers additional information about the numerical methods used. I was surprised that the author frequently suggested using Lapack, implying Fortran or C programming, rather than making an implementation in Matlab or some similar package.

In all, I liked the book because of its breadth, despite the restriction to bifurcations of equilibria of ODEs. From it I learnt more about singularity theory and the influence of symmetry on the performance of numerical methods. It is a good book for people active in the field who require background material on numerical methods. Due to the extensive analysis of the methods, the book is more appealing to people with a very strong mathematical background than to engineers or scientists looking for some algorithms to implement quickly in a custom code.


  1. M. Golubitsky and D. G. Schaeffer, Singularities and Groups in Bifurcation Theory, Vol. I, Springer-Verlag, Berlin, New York, 1985.
  2. M. Golubitsky, I. Stuart and D. G. Schaeffer, Singularities and Groups in Bifurcation Theory, Vol. II, Springer-Verlag, Berlin, 1988.
  3. Yu. A. Kuznetsov, Elements of Applied Bifurcation Theory, Apll. Math. Sci. 112, 2nd ed., Springer-Verlag, New York, 1998.
  4. Yu. A. Kuznetsov and V. V. Levitin, CONTENT: A Multiplatform Environment for Analyzing Dynamical Systems, Dynamical Systems Laboratory, CWI, Amsterdam, 1995-1997. ( ).

A listing of books reviewed in UK Nonlinear News is available.

UK Nonlinear News thanks SIAM for providing a review copy of this book.

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Last Updated: 18th July 2000.