Bifurcation in Autonomous and Nonautonomous Differential Equations with Discontinuities

This book is devoted to bifurcation theory for autonomous and nonautonomous differential equations with discontinuities of different types. That is, those with jumps present either in the right-hand-side or in trajectories or in the arguments of solutions of equations. The results obtained in this book can be applied to various fields such as neural networks, brain dynamics, mechanical systems, weather phenomena, population dynamics, etc. Without any doubt, bifurcation theory should be further developed to different types of differential equations. In this sense, the present book will be a leading one in this field. The reader will benefit from the recent results of the theory and will learn in the very concrete way how to apply this theory to differential equations with various types of discontinuity. Moreover, the reader will learn new ways to analyze nonautonomous bifurcation scenarios in these equations. The book will be of a big interest both for beginners and experts in the field. For the former group of specialists, that is, undergraduate and graduate students, the book will be useful since it provides a strong impression that bifurcation theory can be developed not only for discrete and continuous systems, but those which combine these systems in very different ways. The latter group of specialists will find in this book several powerful instruments developed for the theory of discontinuous dynamical systems with variable moments of impacts, differential equations with piecewise constant arguments of generalized type and Filippov systems. A significant benefit of the present book is expected to be for those who consider bifurcations in systems with impulses since they are presumably nonautonomous systems.

Auteur

1) Prof. Dr. Marat Akhmet is a member of the Department of Mathematics, Middle East Technical University, Turkey. He is a specialist in dynamical models, bifurcation theory, chaos theory and differential equations. He has spent several years investigating the dynamics of neural networks, economic models and mechanical systems. He has published 4 books on different topics of dynamical systems.

2) Dr. Ardak Kashkynbayev obtained his PhD from the Department of Mathematics, Middle East Technical University, Turkey. His research focuses on differential equations, bifurcation theory, chaos theory and applications to mechanical systems.

Résumé
This book focuses on bifurcation theory for autonomous and nonautonomous differential equations with discontinuities of different types those with jumps present either in the right-hand side, or in trajectories or in the arguments of solutions of equations. The results obtained can be applied to various fields, such as neural networks, brain dynamics, mechanical systems, weather phenomena and population dynamics. Developing bifurcation theory for various types of differential equations, the book is pioneering in the field. It presents the latest results and provides a practical guide to applying the theory to differential equations with various types of discontinuity. Moreover, it offers new ways to analyze nonautonomous bifurcation scenarios in these equations. As such, it shows undergraduate and graduate students how bifurcation theory can be developed not only for discrete and continuous systems, but also for those that combine these systems in very different ways. At the same time, it offers specialists several powerful instruments developed for the theory of discontinuous dynamical systems with variable moments of impact, differential equations with piecewise constant arguments of generalized type and Filippov systems.

Contenu
Introduction.- Hopf Bifurcation in Impulsive Systems.- Hopf Bifurcation in Fillopov Systems.- Nonautonomous Transcritical and Pitchfork Bifurcations in an Impulsive Bernoulli Equations.- Nonautonomous Transcritical and Pitchfork Bifurcations in Scalar Non-solvable Impulsive Differential Equations.- Nonautonomous Transcritical and Pitchfork Bifurcations in Bernoulli Equations with Piecewise Constant Argument of Generalized Type.