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Dynamical systems with applications using MATLAB. 2nd ed. (English) Zbl 1319.37002

Cham: Birkhäuser/Springer (ISBN 978-3-319-06819-0/hbk; 978-3-319-06820-6/ebook). xv, 514 p. (2014).
The qualitative theory of ordinary differential equations and the concept of a dynamical system constitute powerful analytical tools in the study of differential equations arising in many areas of science and technology. Although the widespread use of computers and sophisticated numerical integration algorithms allows one to construct in a relatively simple way numerical approximations to exact trajectories for a given set of parameters and initial conditions, it is with the theory of dynamical systems that one is able to describe the qualitative behavior of the solution set of a given differential equation. This includes, of course, the invariant sets (fixed points, periodic orbits, limit cycles) and the asymptotic behavior of the flow defined by the equation. In this way, it is possible to get a global picture about the main features of the system and how it changes depending on the particular values of the parameters. Nowadays, in addition, we have at our disposal powerful software packages that combine both approaches, so that it is relatively simple to get phase portraits of (discrete and continuous) dynamical systems of low dimension and analyze how they change in presence of several types of bifurcations.
The present book constitutes an introduction to the main concepts and techniques of dynamical systems by means of one such package, MATLAB (and the associated Symbolic Math Toolbox, the Image Processing Toolbox and Simulink products). Both discrete and continuous systems are considered, and the study not only covers the basic concepts (fixed points, periodic orbits, period-doubling bifurcations and so on) but also more advanced material, such as recent advances in Hilbert’s sixteenth problem, chaos control procedures and even recent work by the author and his collaborators on binary oscillator computing. It is important to remark that the emphasis is put mainly on examples and applications arising in a variety of disciplines: population dynamics, epidemiology, electric circuits, nonlinear optics, chemical reaction kinetics, and neural networks, just to cite a few.
The reader is advised that this is not (and has not the intention to be) a mathematical treatise on dynamical systems, where the concepts and results follow logically from the preceding ones, with rigorous definitions and self-contained mathematical proofs. If he/she is interested in such an approach, other textbooks and monographs (some of them collected in the preface, other disseminated along the list of references of each chapter) might be more convenient indeed for him/her. This book can be used instead either as a valuable complement of these treatises, since the main results of the theory are clearly illustrated in practice (and in many cases with the corresponding code) or as an introduction by itself for those students and practitioners coming from applied sciences and engineering. The author states that “it is written for both senior undergraduates and graduate students” (I would add “of sciences and/or engineering”) and this aim is clearly visible in the very structure of the book. Each chapter begins with a list of aims and objectives and is closed with some selected MATLAB program files and a list of 10 exercises whose solution is provided at the end of the book. The text is supplemented by a chapter dealing with examination-type questions.
This is the second edition of the book, the first having been published in 2004. This fact speaks by itself about the success of the book and the degree of accomplishment of its intended goals. Several changes have been introduced in this edition. The first is related of course with the software package: MATLAB has evolved from version 6 to version R2014b. The second concerns the structure and contents of the book, which have been expanded and updated, especially in the bibliography.
The present edition consists of 22 chapters (Chapter 23 is nothing but the set of answers to the exercises proposed in the previous chapters). There are three of them specifically dedicated to MATLAB: Chapter 1 is a tutorial introduction to the package, whereas Chapter 7 is devoted to the Image Processing toolbox and Chapter 21 deals with Simulink. Chapters 2–6 are dedicated to discrete dynamical systems (linear and nonlinear recurrences, complex iterative maps, fractals and multifractals). Chapters 8–17 deal with continuous dynamical systems of low dimension. Here planar systems are considered, together with Hamiltonian systems, three-dimensional autonomous systems, the advent of “chaos” and several procedures to detect it in practical situations, Poincaré maps, local and global bifurcations and a very illustrative introduction to the second part of Hilbert’s sixteenth problem. Finally, Chapters 18, 19 and 20 cover more advanced topics that are usually not included in introductory books on dynamical systems: neural networks, control of chaos, chaos synchronization and binary oscillator computing.
Overall this book constitutes a valuable reference to the existing literature on dynamical systems, specially for the remarkable collection of examples and applications selected from very different areas as well as for its treatment with MATLAB of these problems.

MSC:

37-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to dynamical systems and ergodic theory
34-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to ordinary differential equations
37-04 Software, source code, etc. for problems pertaining to dynamical systems and ergodic theory
68N15 Theory of programming languages

Citations:

Zbl 1066.37001

Software:

MuPAD; Matlab; Simulink
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