by Yu. A. Kuznetsov

**Alan Champneys Department of Engineering Mathematics University of
Bristol**

**Wed Aug 23 18:39:35 BST 1995
**

Springer-Verlag Applied Mathematical Sciences Vol 112

If you had a graduate or masters student who wanted to learn the core theory of dynamical systems relevant for applications, what single book would you recommend? 10 years ago, it would almost certainly have been Guckenheimer & Holmes[3]. 5 years ago you might have replaced this recommendation with the books by Wiggins[9] or Arrowsmith & Place [1]. In recent years there has been a proliferation of books aimed more at undergraduates (e.g. Strogatz[8], Glendinning [2], Ott [5] and Hale & Koçak [4] to name but a few), at specific applied audiences (e.g. Moon [6]) and books covering specific mathematical aspects (e.g. Palis & Takens [7]), but in my view there has been nothing recently that seeks to cover the whole of dynamical systems theory at an level accessible to both mathematicians and applied scientists. My belief is that Kuznetsov's book fulfills this role.

The book is systematically ordered into 10 chapters, ranging in length from about 30 pages in Chapter 1 which introduces the concept of a dynamical system, to over 100 pages in Chapter 8 on two-parameter bifurcations of equilibria in continuous-time dynamical systems. This latter chapter, and Chapter 9 on analogous results for discrete-time dynamical systems, represent the most comprehensive, up-to-date review of codimension-two local bifurcations available, correcting and bringing up to date the treatment in other texts. Another important chapter is the final one on the numerical computation of bifurcation diagrams. Here the author's research experience is used to present specific numerical techniques which reflect the theoretical material presented in previous sections. References are given to the available state-of-the-art software for bifurcation analysis. Other chapters cover structural stability, one-parameter local bifurcations of continuous and discrete-time dynamical systems in low dimensions and in arbitrary dimensions, homoclinic and heteroclinic bifurcations and ``other'' codimension-one bifurcation phenomena, including bifurcations on tori, and a brief introduction to bifurcation with symmetry.

The style of the book is that of a teaching manual. Theory is presented systematically. Only a modest mathematical background is required in keeping with the target audience. Where a simple proof is available, it is given --- often relying on non-traditional approaches. Otherwise, where possible, an indication of the method of proof is given. More advanced topics, like infinite-dimensional dynamical systems (Chapter 1), and Feigenbaum's theory (Chapter 4), are left to appendices. Each chapter contains illustrative examples mostly derived from recent research papers, notably in the biological sciences. The book is full of well-constructed illustrations which in many cases provide insight that is not available elsewhere. For example, a picture on p.11 nicely depicts the continuous deformation of a vertical Smale horseshoe into its image, a horizontal horseshoe, and vice-versa. The pictorial approach is especially useful when providing a comprehensive catalogue of phase portraits near codim 2 local bifurcations in Chapters 8 and 9. A particular strength of the book is the well-written Bibliographical Notes section at the end of each Chapter. These both give some historical background (sometimes redressing imbalances with regard to the Russian literature) and also provide references to recent results which are beyond the scope of the book.

Despite its primary intention as an introduction for an applied
audience, there are gems here of interest to
researchers in dynamical systems. For example, the material on
saddle-node homoclinic bifurcations in Chapter 7 is hardly covered in
any other text, especially the beautiful illustrations of the
Shil'nikov case involving two independent homoclinic orbits to a
saddle-node. Further recent results on orbit and inclination-flip
homoclinic bifurcations are tucked away in an appendix to Chapter 10.
Also, the projection method for center manifold reduction
is given explicitly, in preference to
the direct approach involving diagonalisation. This yields
simpler formulae than appear in most books for the leading
coefficients of normal forms for the standard codim 1 and codim 2 local
bifurcations. Particular stress is also given to the difference between a
normal form and a * topological* normal form (or universal
unfolding). In cases where there is no topological normal form
(e.g. in Section 8.4 on the Hopf-Hopf bifurcation),
it is explained which features of the dynamics of the normal form are not
structurally stable and what one should expect under generic perturbation

Clearly the book is not perfect. Some may find the author's systematic style off-putting; rarely does he lead by example. The reader is also warned at the start of Chapter 8 (still only half way through the book!) against trying to follow by hand the normal-form calculations that follow. However, we are instead urged to use a symbolic manipulation package and a Maple routine is provided for this purpose as an extended hint to an exercise in Chapter 8. The book is not without typographical mistakes, but the the general quality of production is more than adequate.

More substantive criticisms would have to be that, despite its length at over 500 pages, there are some obvious omissions (many of which are identified in the Preface). There is very little on chaos itself, the book treats only transitions from regular dynamics to chaotic dynamics. While the treatment of bifurcations in flows is comprehensive, there is virtually nothing on global bifurcations in maps (for example homoclinic tangencies in maps receive only a single paragraph on p.227, whereas homoclinic orbits in flows occupy the whole of Chapter 6 and Section 7.1). The book doesn't really cover Hamiltonian systems. There is also no mention of non-smooth dynamics, very little on the theory of maps of the interval, only a basic introduction to dynamics with symmetry, and not much on Melnikov's method (which, for time-independent perturbations, is given the historically more accurate name of the Pontryagin method). The latter omission is deliberate; instead of methods based on averaging, the approach taken is that of local bifurcation analysis allied to specifically constructed numerical methods. Thus, long expositions of Melnikov analysis, as in Chapter 1(!) of [9] for example, are avoided and the reader has instead a far more practical and powerful tool.

This is not a book that sells out to the popular trend of ``chaos theory''. Instead, it follows the tradition of Poincaré, Andronov, Smale, Arnold, etc. on the geometric theory of differential equations and dynamical systems. It illustrates that there is no royal road to enlightenment in nonlinear dynamics. An understanding of concepts such as structural stability, co-dimension, bifurcation, both local and global, and, even, what exactly is meant by a dynamical system is required before one is able to answer the fundamental question; ``Here is a dynamical system, what happens to its dynamics as I vary parameters?''. Kuznetsov seeks to provide this understanding at an elementary level. But, the book does more than this. The reader is given all the tools (simplified methods for calculating center-manifold and normal-form coefficients, symbolic algebra routines, numerical continuation methods and software) for calculating bifurcation diagrams for themselves.

In summary, this is a book that under-promises and over delivers;
despite the title of the book ``*Elements* of Applied Bifurcation
Theory'' the book has the potential to become a primary source book
for applied dynamical systems research and bifurcation theory in
particular.

**1**- Arrowsmith, D.K. & Place, C.M. (1990)
*An Introduction to Dynamical Systems*. Cambridge University Press, Cambridge. **2**- Glendinning, P. (1994)
*Stability, Instability and Chaos*. Cambridge University Press, Cambridge. **3**- Guckenheimer, J. & Holmes, P. (1983)
*Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields*. Springer-Verlag, New York. **4**- Hale, J & Koçak, H.
*Dynamics and Bifurcations*. Springer-Verlag, New York. **5**- Ott, E. (1993)
*Chaos in Dynamical Systems*. Cambridge University Press, Cambridge. **6**- Moon, F.C. (1992)
*Chaotic and Fractal Dynamics*. John Wiley, New York. **7**- Palis, J. & Takens, F.
*Hyperbolicity and sensitive chaotic dynamics at homoclinic bifurcations*. Cambridge University Press, Cambridge. **8**- Strogatz, S. (1994)
*Nonlinear Dynamics and Chaos.*Addison-Wesley, Reading. **9**- Wiggins, S. (1990)
*Introduction to Applied Nonlinear Dynamical Systems and Chaos*. Springer-Verlag, New York.

A listing of books reviewed in `UK Nonlinear News` is available
at:
http://www.amsta.leeds.ac.uk/Applied/news.dir/uknonl-books.html

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