Bifurcation dynamics in polynomial discrete systems. With 61 figures
In this book, the author presented the bifurcation dynamics of one-dimensional polynomial nonlinear discrete systems and discusses bifurcation trees caused by period doubling and monotone saddle-node bifurcations for forward and inverse polynomial discrete systems. The mechanism of formation of bifurcation trees caused by monotonic saddle-node bifurcations is determined. Arising and switching bifurcations of prime and higher fixed points of period-1 are also discussed here. The book consists of a Preface, five chapters and an Index. References are provided at the end of each chapter. Chapter 1 presents the global bifurcation theory for polynomial discrete systems of the fourth degree, and bifurcation trees by means of period doubling and monotone saddle-node bifurcations are discussed for direct and inverse quadratic systems. Chapter 2 discusses the global bifurcation theory for cubic polynomial discrete systems, and also presents bifurcation trees through period doubling and monotone saddle-node bifurcation, which is different from a quadratic discrete system. In Chapter 3, the global bifurcation theory for polynomial discrete systems of the fourth degree is presented for extension to polynomial discrete systems of $(2m)$th degree. The bifurcation and stability of polynomial discrete systems $(2m)$ and $(2m+1)$ degrees are presented in Chapters 4 and 5 as a general theory of stability and bifurcations for polynomial nonlinear discrete systems. The monograph is written on a completely new topic: the dynamics of nonlinear discrete polynomial systems. For the classes of polynomial equations under consideration, the author carried out a very detailed study of stability conditions and bifurcations and, in fact, opened a new direction of research in nonlinear dynamics. The author does not give examples of the application of the developed theory to specific problems of science and technology. The book is of great interest for specialists working with polynomial discrete systems.
scientific article; zbMATH DE number 7224521