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Stability theory of dynamical systems. Reprint of the 1970 edition. (English) Zbl 0993.37001
Classics in Mathematics. Berlin: Springer. xi, 225 p. (2002).
For a review of the original (1970) see Zbl 0213.10904.
The book presents a systematic treatment of the theory of dynamical systems and their stability written at the graduate and advanced undergraduate level. The text consists of two parts.
The first part (Chapters 1-7) contains the basic theory of dynamical systems in metric spaces and the second part (Chapters 8 and 9) contains applications and extensions of the stability theory to dynamical systems defined by ordinary differential equations.
Specifically, Chapter 1 contains the definition of a dynamical system and some examples to indicate various fields of application. Chapter 2 contains elementary notions which remain invariant under certain topological transformations of dynamical systems. Chapter 3 deals mainly with minimal sets and their structure. Chapter 4 is devoted to the study of dispersive and parallelizable dynamical systems and concludes the part of the book devoted to the basic theory. Chapter 5 develops the main theme of the book, i.e., the stability and attraction theory. The theory presented here differs rather strongly from the one developed by Zubov, being essentially based on the concept of the weak attraction (absent in Zubov’s work).
Chapter 6 is devoted to a more specific problem: the classification of the flows near a compact invariant set. Some results are given, but many problems in this are still open. Chapter 7 contains the theory of higher prolongations originated by T. Ura with applications to absolute stability and generalized recurrence.
Chapter 8 deals with the geometrical theory of stability for ordinary autonomous differential equations including various extensions of Lyapunov’s direct method. Chapter 9 is again devoted to a more specific problem of characterizing stability and attraction concepts via non-continuous Lyapunov functions.
The book is well written and contains a number of examples and exercises.

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
37C75 Stability theory for smooth dynamical systems
34Dxx Stability theory for ordinary differential equations