Differential equations and dynamical systems.

*(English)*Zbl 0717.34001
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) |