Integrability of dynamical systems: algebra and analysis
type: Item
instance of: scholarly article
title: Integrability of dynamical systems: algebra and analysis
zbMATH Open document ID: 1373.37053
DOI: 10.1007/978-981-10-4226-3
author: Xiang Zhang
published in: Developments in Mathematics
publication date: 2017-03-20T00:00:00Z
review text: The book is devoted to the theory of dynamical and integrable systems, focusing primarily on recent developments and modern techniques, relying especially on the theory of algebraic curves. This allows essential advances and generalizations of the classical results on the subject, which in many cases have been proven by the author and his collaborators. This is not, however, a mere collection of technical results based on the author's published papers. Each topic is carefully introduced and explained from basic concepts while quickly progressing to quite advanced topics. This defines the main readership of the book, which will be mainly useful for advanced undergraduate and beginning graduate students interested in the modern theory of dynamical and integrable systems, and who want to quickly acquire the necessary knowledge in basics and be brought up to speed with the latest research. The prerequisites are quite modest -- standard undergraduate courses on differential equations, differential geometry, measure theory and algebraic curves should suffice to read the book. When needed the author introduces the new concepts right in the text, together with numerous references, although it would have been more convenient to relegate some of these introductory notions to appendices. Almost each theorem is proven in generous details, and numerous remarks clarifying, explaining and supplementing the statements of the theorem are given, together with many mostly elementary (but very useful) examples for the reader to be able to apply the statements from general theorems in practice. Some of the presented results are very recent, proved only in 2016, and very often the author also poses open problems, which can serve as concrete project for graduate students. Even though there are no exercises in the book, a number of routine proofs of some theorems are left as ones. In the first chapter, the author introduces the main concepts of dynamical systems and discusses the main theo…
reviewed by: A. Melikyan
Mathematics Subject Classification ID: 37Cxx
Mathematics Subject Classification ID: 34C05
Mathematics Subject Classification ID: 34C07
Mathematics Subject Classification ID: 34C08
Mathematics Subject Classification ID: 37J35
Mathematics Subject Classification ID: 14H70
zbMATH DE Number: 6696156
zbMATH Keywords: dynamical system
zbMATH Keywords: Darboux integrability
zbMATH Keywords: Hilbert's 16th problem
zbMATH Keywords: Liouville integrability
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1007/978-981-10-4226-3
OpenAlex ID: W2602429862
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description: scientific article; zbMATH DE number 6696156