Differential equations and dynamical systems. (English) Zbl 0717.34001

Perko, Lawrence

Texts in Applied Mathematics, 7. New York etc.: Springer-Verlag. xii, 403 p. DM 78.00 (1991).

This book is an introduction to several fundamental and basic topics in the qualitative theory of ordinary differential equations. As the author points out, it is written for upper divisions or first-year graduate students. The book provides a clear exposition of geometrical aspects of nonlinear systems and stresses ideas as well as illustrative examples and exercises. It is divided into 4 chapters. The first one contains notions concerning linear systems of ODE. Chapter 2 discusses the local theory of dynamical systems and includes Harman-Grobman Theorem and Stable Manifold Theorem. The Chapter 3 treats global theory and includes, among others, the study of limit sets (attractors), limit cycles, Stable Manifold Theorem for period orbits etc. and ends with the global phase portrait of a 2-dimensional system. The last chapter is devoted to the study of structural stability and bifurcation theory. Some important concepts and phenomena of the local and global theory are discussed, including Hopf bifurcation, homoclinic bifurcation, chaotic dynamics and Melnikov’s method.

Reviewer: M.A.Teixeira

MSC:

34-01	Introductory exposition (textbooks, tutorial papers, etc.) pertaining to ordinary differential equations
34C23	Bifurcation theory for ordinary differential equations
34A34	Nonlinear ordinary differential equations and systems, general theory
37G99	Local and nonlocal bifurcation theory for dynamical systems
37G15	Bifurcations of limit cycles and periodic orbits in dynamical systems
34A30	Linear ordinary differential equations and systems, general
34C05	Topological structure of integral curves, singular points, limit cycles of ordinary differential equations
34D30	Structural stability and analogous concepts of solutions to ordinary differential equations
34C30	Manifolds of solutions of ODE (MSC2000)

Keywords:

qualitative theory of ordinary differential equations; Harman-Grobman Theorem; Stable Manifold Theorem; limit sets; attractors; limit cycles; period orbits; structural stability; bifurcation theory; Hopf bifurcation; homoclinic bifurcation; chaotic dynamics; Melnikov’s method