Ordinary differential equations. Qualitative theory. Transl. from the Portuguese by the authors (Q2885321)

This book contains a wide range of material on ODEs from the introductory to the advanced level. As the authors explain in the preface, several combinations of topics are admissible, possibly in correspondence with different steps of training, in order to design a course based on the text.NEWLINENEWLINEAt the beginning, the book deals with the basic theory of existence and uniqueness for first-order systems of differential equations from a moderately advanced point of view. The smooth dependence on the initial conditions is presented via the use of ``fiber contractions'' in order to accommodate parameters in a convenient way. The initial chapter includes pays attention to phase diagrams, equations with ``conservation of energy'' and gives a quick introduction to differential equations on manifolds.NEWLINENEWLINEChapter 2 begins with the theory of linear systems, including a detailed study of plane linear systems and Floquet's theory for periodic coefficients. The final sections of the chapter introduce the notions of conjugacy (linear, topological) and the criterion of topological conjugacy for hyperbolic matrices in terms of the count of eigenvalues with positive real part.NEWLINENEWLINEChapter 3 begins with the description of stability for linear systems and the principle of linearized (exponential) stability. The perturbation of nonautonomous linear systems is considered too. Lyapunov functions are introduced and used to yield results on stability and asymptotic stability.NEWLINENEWLINEChapter 4 gives an account of hyperbolic critical points and includes the proof of the Grobman-Hartman theorem. Hölder continuity of the local conjugacy for perturbations with sufficiently small Lipschitz constant is addressed.NEWLINENEWLINEChapter 5 is devoted to invariant manifolds. First, Lipschitz invariant manifolds are constructed at a hyperbolic critical point, and, as a second step, the Hadamard-Perron theorem for the class \(C^1\) is proved.NEWLINENEWLINEChapter 6 gives an overview of the notions of index of a vector field concerning a closed path and the index of an isolated critical point.NEWLINENEWLINEChapter 7 contains the statements and proofs for the Poincaré-Bendixson theory. As in most other texts, pictures are used to abbreviate the proof. It may be interesting to recall the ``full detailed'' approach in the book of \textit{J. Cronin} [Differential equations. Introduction and qualitative theory. New York: Marcel Dekker (1994; Zbl 0798.34001)] (see also the paper by \textit{E. Serra} and \textit{M. Tarallo} [Riv. Mat. Pura Appl. 7, 81--86 (1990; Zbl 0723.34031)]).NEWLINENEWLINEThe subject of the final chapters, 8--9, is usually less likely to be found in textbooks similar to this one. In Chapter 8, the center manifold at a critical point is constructed, and followed by applications to the study of stability and bifurcation. Moreover, an introduction to normal forms is given there, with deep results on the reduction of some systems by convenient changes of variables. Chapter 9 is a brief exposition on Hamiltonian systems, focusing on stability and integrability, including the statement of the Liouville-Arnold theorem and the KAM theorem.NEWLINENEWLINETogether with the elegant presentation of the material of chapters 4--5, these two final chapters may be seen as the distinctive feature of the book, and evidence of the experience and expertise of the authors. Combining mainstream material with special features, this book will range among similar works written by prestigious authors. Rigor pervades the presentation even if the expository style does not look heavy. A wealth of interesting and instructive examples and exercises (with answers) complements and illustrates the material.