##
**An introduction to dynamical systems. Continuous and discrete.**
*(English)*
Zbl 1073.37001

Upper Saddle River, NJ: Pearson/Prentice Hall (ISBN 0-13-143140-4/hbk). xiii, 652 p. (2004).

This excellent book is intended for an advanced undergraduate course in nonlinear ordinary differential equations and discrete dynamical systems. The book can serve as several different types of courses on the dynamical systems, according to the aims and student’s audience.

The book is divided into two parts: Part 1 – Systems of nonlinear differential equations, consists of the following 7 chapters: Chapter 1 – Geometric approach to differential equations; Chapter 2 – Linear systems; Chapter 3 – The Flow: Solutions of nonlinear equations; Chapter 4 – Phase portraits with emphasis on fixed-points; Chapter 5 – Phase portraits using energy and other test functions; Chapter 6 – Periodic orbits; Chapter 7 – Chaotic attractors.

Part 2 – Iteration of functions – also consists of 7 chapters: Chapter 8 – Iteration of functions as dynamics; Chapter 9 – Periodic points of one-dimensional maps; Chapter 10 – Itineraries for one-dimensional maps; Chapter 11 – Invariant sets for one-dimensional maps; Chapter 12 – Periodic points of higher-dimensional maps; Chapter 13 – Invariant sets for higher-dimensional maps; Chapter 14 – Fractals.

Almost every chapter has the same structure. The main concepts are presented in the first sections of each chapters. These sections are followed by a section that presents some applications and then by a section that contains proofs of the more difficult results and more theoretical material. The presentation of the applications is valuable, because these applications provide motivation and illustrate the usefulness of the theory of dynamical systems. Also, all chapters, apart from Chapters 1 and 8, are provided by exercises.

The book is well written and organized and can be recommended to all ones, which are interested in dynamical systems.

The book is divided into two parts: Part 1 – Systems of nonlinear differential equations, consists of the following 7 chapters: Chapter 1 – Geometric approach to differential equations; Chapter 2 – Linear systems; Chapter 3 – The Flow: Solutions of nonlinear equations; Chapter 4 – Phase portraits with emphasis on fixed-points; Chapter 5 – Phase portraits using energy and other test functions; Chapter 6 – Periodic orbits; Chapter 7 – Chaotic attractors.

Part 2 – Iteration of functions – also consists of 7 chapters: Chapter 8 – Iteration of functions as dynamics; Chapter 9 – Periodic points of one-dimensional maps; Chapter 10 – Itineraries for one-dimensional maps; Chapter 11 – Invariant sets for one-dimensional maps; Chapter 12 – Periodic points of higher-dimensional maps; Chapter 13 – Invariant sets for higher-dimensional maps; Chapter 14 – Fractals.

Almost every chapter has the same structure. The main concepts are presented in the first sections of each chapters. These sections are followed by a section that presents some applications and then by a section that contains proofs of the more difficult results and more theoretical material. The presentation of the applications is valuable, because these applications provide motivation and illustrate the usefulness of the theory of dynamical systems. Also, all chapters, apart from Chapters 1 and 8, are provided by exercises.

The book is well written and organized and can be recommended to all ones, which are interested in dynamical systems.

Reviewer: Alois Klíč (Praha)

### MSC:

37-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to dynamical systems and ergodic theory |

39-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to difference and functional equations |

34-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to ordinary differential equations |

00-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to mathematics in general |