Ordinary Differential Equations. Introduction and Qualitative Theory
This is the third edition of the classical textbook on qualitative theory of ordinary differential equations (the first edition was published by M. Dekker in 1980). Chapters 1-5 cover all the basic facts: the existence and uniqueness of solutions and their dependence on initial conditions; linear systems with constant, periodic and continuous coefficients; autonomous systems and, for planar systems, their behaviour near an isolated equilibrium and the Poincaré-Bendixson theory; and stability and instability results via linearisation and Lyapunov functions. Chapters 6-8 deal with more advanced topics on the existence of periodic solutions. Topological methods and Poincaré's perturbation method (for nonautonomous as well as autonomous equations) and Hopf's bifurcation theorem are discussed in detail. The last chapter is devoted to a brief discussion of the averaging method. For this third edition the sections on local behaviour near a singular equilibrium and on periodic solutions in small parameter problems have been extended and/or unified. The book is written for advanced undergraduate students with a good knowledge of calculus and linear algebra. Several appendices are included for understanding of the more advanced chapters. The largest chapter discusses degree theory. There are many examples throughout the text serving either as motivations or as illustrations of general results. A lot of them come from biological, chemical and, of course, mechanical models.