# Differential Equations and Dynamical Systems

## By Lawrence Perko

### Reviewed by Jane Sexton

Springer-Verlag 2001
Text in Applied Mathematics Vol. 7
Pages: 554
ISBN: 0-387-95116-4 (Hardcover)
£44.50

Normally when I pick up a book on differential equations and dynamical systems, my heart beat picks up a few notches as I glance through the contents and flick to some favourite section of the book. This time, that didn't happen. However, to be fair, the intention of this book is not to stimulate the interest of an undergraduate, but to act as a classical reference text for a graduate.

The preface gives the outline to the book with the theory of linear systems dealt with first, followed by the local and global theory of nonlinear systems, and concluding with bifurcation theory. Without further ado, Perko launches immediately into the theory of linear systems. If the book is to be used as a reference text, then obviously the reader can skip to the chapter of interest. However, if the reader is not familiar with nonlinear systems, then it is advised to first read the chapter on the theory of linear systems. To begin to understand nonlinear systems, the local theory must be understood and this is described in chapter 2. Using the techniques shown in chapter 1, the stability and qualitative behaviour in a neighbourhood of a hyperbolic critical point of a nonlinear system can be solved by examining the corresponding linearized system as described by the stable manifold theorem and Hartman-Grobman theorem. For undergraduates studying linear algebra and ordinary differential equations, chapters one and two would prove most useful as most critical results are presented with proofs.

The exciting world of nonlinear systems is introduced in chapter 3 where the techniques needed to construct a global portrait of a nonlinear system are described. As Perko states, "This qualitative information combined with the quantitative information about individual trajectories that can be obtained on a computer is generally as close as we can come to solving a nonlinear system of differential equations; but, in a sense, this information is better than obtaining a formula for the solution since it geometrically describes the behavior of every solution for all time." This is an important statement, since a computer can really only give a local solution and therefore a global understanding of the problem is imperative before this step is taken. (Even when a utility such as AUTO (a continuation package) is being utilised, a thorough understanding of the global portrait should be sought first before using a package such as this, as important features may not be detected.) In this chapter, Perko describes limit sets, attractors, limit cycles and heteroclinic and homoclinic orbits. The theory associated with periodic orbits is given its due attention, including the Poincare map and theorem, and the stable manifold and centre manifold theorems for periodic orbits. The Poincare-Bendixson theorem and Lienard's theorem are described which determine the exact number of limit cycles for certain planar systems. This then leads to the next problem of determining when a system contains no limit cycles, which is achieved using Bendixson's and Dulac's criteria.

Of particular interest to me is chapter 4 which describes bifurcation theory and comprises a significant portion of the book. This theory is invaluable when determining how a system is altered as a system parameter changes and can yield some very powerful results. To begin this chapter, Perko presents the concept of structural stability when equilibrium points are hyperbolic. The remaining part of the chapter consists of bifurcations occurring at nonhyperbolic equilibrium points and periodic orbits as well as bifurcations of periodic orbits from equilibrium points and homoclinic loops which can depend on one or more parameters. These include the saddle-node, transcritical, pitchfork, cusp, and a new one to me, the swallow-tail bifurcation. The concept of codimension and universal unfolding is described which is instrumental in determining the behaviour of a nonlinear system at its 'minimum' level. The Hopf bifurcation is the subject of the next section, along with a method to determine whether this bifurcation is subcritical or supercritical. Melnikov's method for studying homoclinic loop bifurcations and establishing the existence of transverse homoclinic orbits for perturbed dynamical systems is included, and a new addition to this edition is Francoise's algorithm for determining the first nonzero Melnikov function for a perturbed Hamiltonian system. Next, the Taken-Bogdanov bifurcation is described, followed by the insoluble Coppel problem (also a new addition to this edition), which is given a significant amount of attention.

This text is a high quality source for all the theoretical background necessary to launch into the exciting field of dynamical systems. Problem sets are given at the end of each section and I was pleased to see that a solution manual is available from the author, (at no additional cost), which is invaluable if the solutions are well worked. I would have liked to have seen some practical examples for three reasons; interspersing a text with examples gives the reader a break from all the theory, it can demonstrate the application of the theory, and by showing some real problem, then the theory can then illustrate how useful it actually is. So many times we see texts where examples have been conceived so a neat solution is found, and hence can be rarely applied to other problems. My view of applied mathematics is about using techniques to describe the physical world, and to ask why and how about those processes. So as you read this book, you may wonder what these techniques can be used for. There are numerous fields where the examples can be found, and they don't necessarily have to be sourced from the physical sciences. However, the book would explode in length if this were to happen. If this book is to serve as a reference text, then those of us who are not theoretically oriented (and I'll admit that there were sections of this book with which I struggled) would probably balk at it. There is a great opportunity to project this knowledge into the domain of engineers, chemists, biologists, economists and so on, where the theory of dynamical systems would enhance the understanding of their systems.

In summary, this book is a great reference text for those interested in differential equations and dynamical systems as the local and global nonlinear theory is fully presented leading to the description of important elements of bifurcation theory. With this book in hand, one would be confident in making a full and thorough description of any desired dynamical system.

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

UK Nonlinear News thanks Springer-Verlag for providing a review copy of this book.

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Last Updated: 6th August 2001.
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