商品描述
This textbook offers a rigorous yet accessible introduction to the qualitative theory of dynamical systems, focusing on both discrete- and continuous-time systems--those defined by iterated maps and differential equations. With clarity and precision, it provides a conceptual framework and the essential tools needed to describe, analyze, and understand the behavior of real-world systems across the sciences and engineering. Designed for advanced undergraduates and early graduate students, the book assumes only a foundational background in analysis, linear algebra, and differential equations. It bridges the gap between introductory courses and more advanced treatments by offering a self-contained and balanced approach--one that integrates geometric intuition with analytical rigor. Key features include:
A carefully curated selection of topics essential for applied contexts Full, detailed proofs of cornerstone results, including the Poincaré-Bendixson theorem, Lyapunov's stability criteria, Grobman-Hartman theorem, Center Manifold theorem A unified treatment of discrete- and continuous-time systems, with discrete methods often paving the way for their continuous counterparts Employing modern functional analytic techniques to streamline and clarify complex arguments Special attention to invariant manifolds, symbolic dynamics, and topological normal forms for codimension-one bifurcations
Whether for students planning further study in pure or applied mathematics, or for those in disciplines such as physics, biology, or engineering seeking to apply dynamical systems theory in practice, this book offers a concise yet comprehensive entry point. Instructors will appreciate its modular structure and completeness, while students will benefit from its clarity, rigor, and insightful presentation.
商品描述(中文翻譯)
這本教科書提供了一個嚴謹但易於理解的動態系統質性理論入門，專注於離散時間和連續時間系統——即由迭代映射和微分方程定義的系統。它以清晰和精確的方式提供了一個概念框架以及描述、分析和理解現實世界系統行為所需的基本工具，涵蓋科學和工程領域。
本書為高年級本科生和早期研究生設計，僅假設讀者具備分析、線性代數和微分方程的基礎背景。它通過提供一個自成體系且平衡的方法，彌補了入門課程與更高級處理之間的差距——這種方法將幾何直覺與分析嚴謹性相結合。
主要特點包括：
- 精心策劃的主題選擇，對於應用情境至關重要
- 重要結果的完整詳細證明，包括Poincaré-Bendixson定理、Lyapunov穩定性標準、Grobman-Hartman定理、中心流形定理
- 對離散時間和連續時間系統的統一處理，離散方法通常為其連續對應物鋪平道路
- 採用現代函數分析技術來簡化和澄清複雜的論證
- 特別關注不變流形、符號動力學和對於一維分岔的拓撲標準型
無論是計劃在純數學或應用數學進一步學習的學生，還是希望在實踐中應用動態系統理論的物理學、生物學或工程學等學科的學生，本書都提供了一個簡明而全面的入門點。教師將欣賞其模組化結構和完整性，而學生則將受益於其清晰性、嚴謹性和深刻的呈現。
作者簡介
Yuri Aleksandrovich Kuznetsov (b. in Alma-Ata, USSR, September 8, 1957) received graduate degree in Theoretical Physics from the Rostov-on-Don State University, USSR, in 1979, and the Ph.D. in Physics and Mathematics from the Institute of Biophysics, USSR Academy of Sciences, Pushchino, Moscow Region, in 1986. From 1979 until 2007, he was affiliated with the Research Computing Center (later: Institute of Mathematical Problems of Biology), USSR /Russian Academy of Sciences, Pushchino, where he held different positions, including that of Senior Researcher. During 1991-1998, he visited as Invited Professor/Researcher IIASA (Laxenburg, Austria), Politecnico di Milano (Italy), CWI (Amsterdam, The Netherlands), and ENS (Paris/Lion, France). Since 1999 he is affiliated with Utrecht University (The Netherlands), being Associate Professor at the Department of Mathematics. In 2011, he was appointed as Professor of Numerical Bifurcation Methods at the University of Twente (Enschede, The Netherlands). Wolf-Jürgen Beyn, born on April 6, 1949 in Hamburg (Germany), studied Mathematics, Physics, and Computer Science at the Universities of Hamburg and Münster (Germany). He graduated in Mathematics at the University of Münster in 1973 and received his PhD there in 1975. He worked as a Scientific Assistant at the Mathematics Institutes of the Universities of Münster (1976 - 1979) and Konstanz (1979 -1981). In 1981 he received his habilitation in Mathematics at the University of Konstanz, and he has been a lecturer ('Privatdozent') there till 1990. Research stays as visiting professor led him to the University of Calgary (Canada) in 1979, to the California Institute of Technology (Pasadena) in 1986, and to the Universities of New Mexico (Albuquerque) and Missouri (Columbia) during the period 1990-2011. In 1990 he was appointed Associate Professor of Mathematics at Bielefeld University (Germany), where he was promoted to full in 2002. Since then he has been the head of the Numerical Analysis group at the Faculty of Mathematics of Bielefeld University. Odo Diekmann was born on April 14, 1948, in Diepenveen (the Netherlands). He studied mathematics and physics at the University of Amsterdam and received his PhD from that same university in 1978, with a thesis devoted to mathematical epidemiology of infectious diseases and nonlinear integral equations. He worked at the Centre for Mathematics and Computer Science CWI in Amsterdam from 1974 to 1995. From 1986 to 1995 this was combined with a part-time professorship (joint with J.A.J. Metz) at the Biology Department of Leiden University. In 1995 he became Professor in Applied Mathematics at Utrecht University (The Netherlands).
作者簡介(中文翻譯)
尤里·亞歷山德羅維奇·庫茲涅佐夫（Yuri Aleksandrovich Kuznetsov，1957年9月8日出生於蘇聯阿爾馬阿塔）於1979年在蘇聯羅斯托夫州立大學獲得理論物理碩士學位，並於1986年在蘇聯科學院生物物理研究所（位於莫斯科地區的普希金）獲得物理與數學博士學位。從1979年到2007年，他在蘇聯/俄羅斯科學院普希金的研究計算中心（後來稱為生物學數學問題研究所）工作，擔任不同職位，包括高級研究員。在1991年至1998年間，他曾作為受邀教授/研究員訪問IIASA（奧地利拉克森堡）、米蘭理工大學（意大利）、CWI（荷蘭阿姆斯特丹）和ENS（法國巴黎/里昂）。自1999年以來，他在荷蘭烏特勒支大學任職，擔任數學系的副教授。2011年，他被任命為荷蘭特溫特大學數值分岔方法教授。
沃夫-尤爾根·貝因（Wolf-Jürgen Beyn）於1949年4月6日出生於德國漢堡，他在漢堡和明斯特大學學習數學、物理和計算機科學。1973年，他在明斯特大學獲得數學學位，並於1975年獲得博士學位。他曾在明斯特大學（1976-1979）和康斯坦茨大學（1979-1981）的數學研究所擔任科學助理。1981年，他在康斯坦茨大學獲得數學的資格認證，並在那裡擔任講師（'Privatdozent'）直到1990年。作為訪問教授的研究期間，他曾於1979年到卡爾加里大學（加拿大）、1986年到加州理工學院（帕薩迪納）以及1990年至2011年間到新墨西哥大學（阿爾伯克基）和密蘇里大學（哥倫比亞）進行研究。1990年，他被任命為德國比勒費爾德大學的數學副教授，並於2002年晉升為正教授。自那時以來，他一直是比勒費爾德大學數學系數值分析小組的負責人。
奧多·迪克曼（Odo Diekmann）於1948年4月14日出生於荷蘭迪彭芬。他在阿姆斯特丹大學學習數學和物理，並於1978年在該大學獲得博士學位，論文專注於傳染病的數學流行病學和非線性積分方程。他於1974年至1995年在阿姆斯特丹的數學與計算機科學中心CWI工作。1986年至1995年期間，他還兼任萊頓大學生物系的兼職教授（與J.A.J. Metz共同）。1995年，他成為荷蘭烏特勒支大學的應用數學教授。
