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**Ordinary differential equations and dynamical systems.**
*(English)*
Zbl 1263.34002

Graduate Studies in Mathematics 140. Providence, RI: American Mathematical Society (AMS) (ISBN 978-0-8218-8328-0/hbk). xi, 356 p. (2012).

This book provides a concise but rigorous and self-contained introduction to the field of ordinary differential equations. It covers a variety of important subjects, including classical theory (Part 1), dynamical systems (Part 2) and chaos (Part 3). The text requires only some basic knowledge of linear algebra, calculus and complex analysis.

The book starts with examples introducing the concept of a differential equation. Chapter 1 presents the classical methods for finding explicit solutions to the simplest cases and then some first steps in the qualitative analysis. The basic existence and uniqueness results for initial value problems are included in Chapter 2 as well as parameters dependence and extensibility topics. Special attention is paid to the systems of linear equations (Chapter 3) and Floquet theory. A brief overview of the Jordan canonical form is developed in an appendix to this chapter. The first part finishes with two additional topics: differential equations in the complex domain and the Frobenius method (Chapter 4) and boundary value problems (Chapter 6) with emphasis on Sturm-Liouville and oscillation theories.

The second part concerns the qualitative analysis of autonomous equations in the settings of dynamical systems theory, introducing the concepts of invariant and limit sets, orbits and stability. To make the theory clear, the subsequent topic, dedicated to the planar systems, begins with a review of the well known examples: Volterra-Lotka system, Lienard’s and van der Pol’s equations. The PoincarĂ©-Bendixson theorem is proved as well. Then higher dimensional dynamical systems are considered, including attractors, Hamiltonian systems and the KAM theorem. The last chapter in Part 2 treats the stability theory near fixed points, the stable/unstable manifold theorem and the Hartman-Grobman theorem. A brief introduction to Volterra’s and Hammerstein’s integral equations theory is given in an appendix to this chapter.

The chaotic dynamical systems are treated in the final Part 3. Following the same style throughout the book, this part starts with an introductory example providing a first glance at the topic – the logistic map. Then discrete dynamical systems theory is developed with a special attention to interval maps. Finally, the text focuses on Melnikov’s method and the Smale-Birkhoff theorem.

It is worth noting that the topics are accompanied by mathematical software code included throughout the text. On the other hand, there are plenty of exercises and examples. In conclusion, the book will be useful for any introductory graduate level course in ordinary differential equations.

The book starts with examples introducing the concept of a differential equation. Chapter 1 presents the classical methods for finding explicit solutions to the simplest cases and then some first steps in the qualitative analysis. The basic existence and uniqueness results for initial value problems are included in Chapter 2 as well as parameters dependence and extensibility topics. Special attention is paid to the systems of linear equations (Chapter 3) and Floquet theory. A brief overview of the Jordan canonical form is developed in an appendix to this chapter. The first part finishes with two additional topics: differential equations in the complex domain and the Frobenius method (Chapter 4) and boundary value problems (Chapter 6) with emphasis on Sturm-Liouville and oscillation theories.

The second part concerns the qualitative analysis of autonomous equations in the settings of dynamical systems theory, introducing the concepts of invariant and limit sets, orbits and stability. To make the theory clear, the subsequent topic, dedicated to the planar systems, begins with a review of the well known examples: Volterra-Lotka system, Lienard’s and van der Pol’s equations. The PoincarĂ©-Bendixson theorem is proved as well. Then higher dimensional dynamical systems are considered, including attractors, Hamiltonian systems and the KAM theorem. The last chapter in Part 2 treats the stability theory near fixed points, the stable/unstable manifold theorem and the Hartman-Grobman theorem. A brief introduction to Volterra’s and Hammerstein’s integral equations theory is given in an appendix to this chapter.

The chaotic dynamical systems are treated in the final Part 3. Following the same style throughout the book, this part starts with an introductory example providing a first glance at the topic – the logistic map. Then discrete dynamical systems theory is developed with a special attention to interval maps. Finally, the text focuses on Melnikov’s method and the Smale-Birkhoff theorem.

It is worth noting that the topics are accompanied by mathematical software code included throughout the text. On the other hand, there are plenty of exercises and examples. In conclusion, the book will be useful for any introductory graduate level course in ordinary differential equations.

Reviewer: Tihomir Gyulov (Ruse)

### MSC:

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

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

34A12 | Initial value problems, existence, uniqueness, continuous dependence and continuation of solutions to ordinary differential equations |

34A30 | Linear ordinary differential equations and systems |

34Bxx | Boundary value problems for ordinary differential equations |

34Cxx | Qualitative theory for ordinary differential equations |

34Mxx | Ordinary differential equations in the complex domain |

37C29 | Homoclinic and heteroclinic orbits for dynamical systems |

37C75 | Stability theory for smooth dynamical systems |

37D45 | Strange attractors, chaotic dynamics of systems with hyperbolic behavior |

37B10 | Symbolic dynamics |