Planar dynamical systems. Selected classical problems (Q2877537)

This book presents in an elementary way the recent significant developments in the qualitative theory of planar autonomous systems. The subjects covered are as follows: center-focus and isochronous center problems, multiple Hopf bifurcations, local and global bifurcations of equivariant planar vector fields related to Hilbert's 16th problem. The materials of this book are based mainly on the published papers of the authors.NEWLINENEWLINEThe book is divided into ten chapters. The first one provides some basic results of the theory of complex analytic autonomous systems including normal forms, integrability and linearization in a neighborhood of an elementary singular point. In addition, there is a study of the quasi-algebraic integrals for some polynomial systems. In the second chapter, the authors consider a class of real planar systems with analytic right hand sides in a neighborhood of the origin under the assumption that the origin is a focus or a center. The elementary theory to solve the center-focus problem is introduced. In Chapter 3 the bifurcations of limit cycles from the origin of the mentioned class of the systems are discussed. In Chapter 4 the concepts of the period constant and the isochronous center in real systems are extended to complex systems. Chapter 5 deals with the study of the center-focus problem at infinity and it concerns the bifurcation of limit cycles from infinity for a class of systems. Chapter 6 represents an introduction to the theory of center-focus problem for a class of multiple singular points. Chapter 7 contains the complete solution of the center-focus and isochronous center problems as well as the bifurcation of limit cycles for a class of nonanalytic systems which is called ''quasi-analytic systems''. The aim of Chapter 8 is to study the second part of Hilbert's 16th problem for \(Z_q\)-equivariant perturbed planar Hamiltonian vector fields by using the Poincaré-Pontrjagin-Andronov theorem and Melnikov's result. Chapter 9 presents the complete solution of the center problem for a class of \(Z_2\)-symmetric cubic systems and the proof that this class of cubic systems has at least \(13\) limit cycles. In the final Chapter 10 the authors study the center-focus problem and bifurcations of limit cycles from nilpotent singular points of multiplicity three.NEWLINENEWLINEThe book is intended for graduate students, post-doctors and researchers in the field of planar dynamical systems. It will be useful also for engineers and other scientists who are interested in the applications of dynamical systems. It requires a minimum background of a one-year course on nonlinear differential equations.