Text in Applied Mathematics Vol. 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.
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Text in Applied Mathematics Vol. 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.
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.
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 Hopf bifurcation is the subject of the next section, along with a method to determine whether this bifurcation is subcritical or supercritical.
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. However, the book would explode in length if this were to happen. 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. Last Updated: 6th August
Differential Equations and Dynamical Systems
The final prices may differ from the prices shown due to specifics of VAT rules About this Textbook This textbook presents a systematic study of the qualitative and geometric theory of nonlinear differential equations and dynamical systems. Although the main topic of the book is the local and global behavior of nonlinear systems and their bifurcations, a thorough treatment of linear systems is given at the beginning of the text. All the material necessary for a clear understanding of the qualitative behavior of dynamical systems is contained in this textbook, including an outline of the proof and examples illustrating the proof of the Hartman-Grobman theorem, the use of the Poincare map in the theory of limit cycles, the theory of rotated vector fields and its use in the study of limit cycles and homoclinic loops, and a description of the behavior and termination of one-parameter families of limit cycles. The text succeeds admiraby Examples abound, figures are used to advantage, and a reasonable balance is maintained between what is proved in detail and what is asserted with supporting references Each section closes with a set of problems, many of which are quite interesting and round out the text material
Differential Equations and Dynamical Systems