# 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 1 offers a quick introduction to dynamical systems and bifurcation analysis. It introduces stability of equilibria, transcritical bifurcation points, fold points, the cusp catastrophe and Hopf bifurcations using examples from population dynamics. It also contains an introduction to finite elements and finite difference discretisations of PDEs over a one-dimensional domain. A PDE model illustrates symmetry breaking. Finally, the chapter introduces the notion of linear and nonlinear stability.
• Chapter 2 introduces manifold theory. After a short discussion of branches and limit points, it continues with an extensive discussion of numerical continuation methods.
• Chapter 3 introduces one of the most essential elements of this book: bordered matrices. To study the rank deficiency of a matrix A, Govaerts suggests extending A with an additional k rows and l columns such that the extended matrix is square and nonsingular. The lower right lxk-block of the inverse then has the same rank deficiency as A. Of most practical importance for the book is the case where A is square and k=l. The singular value decomposition is a very useful tool in this chapter. The author also presents his BEM method (Block Elimination Mixed) for solving bordered linear systems in a backward stable way given a backward stable solver for the matrices A and AT.
• Chapter 4 contains an extensive discussion of numerical methods for computation and continuation of quadratic turning points and Hopf bifurcation points. The most important methods for these codimension-one phenomena are presented, including minimal defining systems (involving only one equation) and methods using large defining systems, typically involving eigenvectors and eigenvalues at the bifurcation point. The book presents a thorough introduction to the bialternate product matrix and a complete study of its Jordan normal form. The bialternate product matrix of a n×n-matrix is a sparse matrix of size n(n-1)/2 whose eigenvalues are the sums of eigenvalue pairs of the original matrix. It is singular at a Hopf point, allowing for an easy extension of bordered matrix methods.
• Chapter 5 discusses higher-codimension linearly determined bifurcations, including all codimension-two cases (Bogdanov-Takens, zero-Hopf and double Hopf bifurcations) and several higher-codimension cases. It also introduces the ideas of unfolding and transversality of manifolds.
• Chapter 6 covers singularity theory in the case where there is no distinguished parameter. First, the numerical Lyapunov-Schmidt reduction is introduced. The chapter further covers singularities from R->R, R2->R and R2->R2. At its end, numerical methods for several cases are presented.
• Chapter 7 covers singularity theory with a distinguished parameter. Besides an extensive classification of singularities of codimension up to four, it also discusses numerical methods and contains a section on numerical branching.
• Chapter 8 discusses symmetry-breaking bifurcations. Most of the chapter is restricted to the easiest case, that of Z2 symmetry. It illustrates the numerical problems that can arise when a numerical method that does not take into account the symmetry is used for a system with symmetry. Phenomena that are of higher codimension in systems without symmetry become generic in one- or two-parameter families of solutions. At the end of the chapter, extensions to other symmetry groups are also discussed.
• Chapter 9 covers methods based on center manifold theory. Both minimally and large defining systems for cusp points and generalised Hopf points are discussed in this chapter.
• Chapter 10 is a limited introduction to large dynamical systems resulting from the space discretisation of systems of partial differential equations. This chapter essentially discusses the method implemented in the continuation package CONTENT  for a class of reaction-diffusion-convection problems in one space dimension. It also briefly discusses an experiment of the author with dimension reduction: the determining system for a bifurcation point is based on a projection of the system in the subspace of the rightmost eigenvectors. This is particularly important for methods for detecting Hopf bifurcations based on the bialternate product matrix. It finishes with a larger "Notes and Further Reading" section than the other chapters, with references to some other useful techniques.

Chapter 6, 7 and 8 make use extensively of results from  and , while chapters 9 and 10 use material from . 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.

### Bibliography

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. (  ftp://ftp.cwi.nl/pub/CONTENT ).

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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