zbMATH — the first resource for mathematics

Methods and rules for the qualitative study of dynamical systems on the plane. (Metody i priemy kachestvennogo issledovaniya dinamicheskikh sistem na ploskosti). 2nd ed. (Metody i priemy kachestvennogo issledovaniya dinamicheskikh sistem na ploskosti.) (Russian. English summary) Zbl 0785.34004
Spravochnaya Matematicheskaya Biblioteka. 11. Moskva: Nauka. 488 p. (1990).
Review to the first edition (1976):
The book consists of four parts. In the first part (Chapters 1-6) the authors deal with second order autonomous systems with analytic right- hand sides. The first chapter contains the general theory of dynamical systems in the plane. The authors give theorems for the existence and uniqueness of a solution and for its dependence on initial data. They also give geometric interpretations and a number of examples. The second chapter is devoted to exposition of the Poincaré-Bendixson theory. The next two chapters discuss the qualitative structure of the neighborhood of a singular point. The last two chapters of the first part contain the discussion of the qualitative structure of limit cycles. In the second part (Chapters 7-13), the authors consider the theory of bifurcations. Chapter 7 contains properties of conservative systems, the dependence of solutions on the right-hand sides of the dynamical system. Chapter 8 is devoted to structurally stable systems. The characteristic properties of such systems are given. In the next chapter the authors consider some simple systems which are not structurally stable. Chapter 10 contains a discussion of bifurcations arising by varying the right-hand sides. In Chapter 11 the authors consider dynamical systems depending on parameters. In Chapter 12 they give an exposition of systems with a cylindrical phase space. The last chapter of the second part goes into a discussion of the adequacy of nonlinear physical processes and the qualitative theory and bifurcation theory of dynamical systems. In the third part (Chapters 14-16) the authors consider concrete dynamical systems with analytic right-hand sides. All three chapters of this part are devoted to exposition of examples. Electric circuits with a tunnel diode, a two-dimensional model of a laser, the flight of an aircraft in a vertical plane and some other problems are discussed. The fourth part (Chapters 17-20) contains the theory of “piecewise-patched” systems. Chapter 17 is devoted to the general theory of such systems. The following two chapters contain a number of examples in which physical and engineering processes are described by piecewise-patched systems. In the last chapter, the authors consider some approximations by piecewise smooth functions and give three examples of dynamical systems describing engineering processes. As the book is a reference book no proofs are given but the great number of pictures should enable the reader to understand the text.
The second edition contains slight changes, mainly in the section dealing with limit cycles of quadratic equations. An appendix concerning dynamical systems on surfaces and a supplementary list of references have been added.

34-02 Research exposition (monographs, survey articles) pertaining to ordinary differential equations
34Cxx Qualitative theory for ordinary differential equations