Nonautonomous Bifurcation Theory
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Nonautonomous Bifurcation Theory
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Author : Vasso Anagnostopoulou
language : en
Publisher: Springer Nature
Release Date : 2023-05-31
Nonautonomous Bifurcation Theory written by Vasso Anagnostopoulou and has been published by Springer Nature this book supported file pdf, txt, epub, kindle and other format this book has been release on 2023-05-31 with Mathematics categories.
Bifurcation theory is a major topic in dynamical systems theory with profound applications. However, in contrast to autonomous dynamical systems, it is not clear what a bifurcation of a nonautonomous dynamical system actually is, and so far, various different approaches to describe qualitative changes have been suggested in the literature. The aim of this book is to provide a concise survey of the area and equip the reader with suitable tools to tackle nonautonomous problems. A review, discussion and comparison of several concepts of bifurcation is provided, and these are formulated in a unified notation and illustrated by means of comprehensible examples. Additionally, certain relevant tools needed in a corresponding analysis are presented.
Bifurcation In Autonomous And Nonautonomous Differential Equations With Discontinuities
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Author : Marat Akhmet
language : en
Publisher: Springer
Release Date : 2017-01-23
Bifurcation In Autonomous And Nonautonomous Differential Equations With Discontinuities written by Marat Akhmet and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2017-01-23 with Mathematics categories.
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.
Attractivity And Bifurcation For Nonautonomous Dynamical Systems
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Author : Martin Rasmussen
language : en
Publisher: Springer
Release Date : 2007-05-26
Attractivity And Bifurcation For Nonautonomous Dynamical Systems written by Martin Rasmussen and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2007-05-26 with Mathematics categories.
Although, bifurcation theory of equations with autonomous and periodic time dependence is a major object of research in the study of dynamical systems since decades, the notion of a nonautonomous bifurcation is not yet established. In this book, two different approaches are developed which are based on special definitions of local attractivity and repulsivity. It is shown that these notions lead to nonautonomous Morse decompositions.
Stability And Bifurcation Theory For Non Autonomous Differential Equations
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Author : Anna Capietto
language : en
Publisher: Springer
Release Date : 2012-12-14
Stability And Bifurcation Theory For Non Autonomous Differential Equations written by Anna Capietto and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2012-12-14 with Mathematics categories.
This volume contains the notes from five lecture courses devoted to nonautonomous differential systems, in which appropriate topological and dynamical techniques were described and applied to a variety of problems. The courses took place during the C.I.M.E. Session "Stability and Bifurcation Problems for Non-Autonomous Differential Equations," held in Cetraro, Italy, June 19-25 2011. Anna Capietto and Jean Mawhin lectured on nonlinear boundary value problems; they applied the Maslov index and degree-theoretic methods in this context. Rafael Ortega discussed the theory of twist maps with nonperiodic phase and presented applications. Peter Kloeden and Sylvia Novo showed how dynamical methods can be used to study the stability/bifurcation properties of bounded solutions and of attracting sets for nonautonomous differential and functional-differential equations. The volume will be of interest to all researchers working in these and related fields.
Averaging Methods In Nonautonomous Bifurcation Theory
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Author : P. Gross
language : en
Publisher:
Release Date : 1992
Averaging Methods In Nonautonomous Bifurcation Theory written by P. Gross and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 1992 with categories.
An Introduction To Nonautonomous Dynamical Systems And Their Attractors
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Author : Peter Kloeden
language : en
Publisher: World Scientific
Release Date : 2020-11-25
An Introduction To Nonautonomous Dynamical Systems And Their Attractors written by Peter Kloeden and has been published by World Scientific this book supported file pdf, txt, epub, kindle and other format this book has been release on 2020-11-25 with Mathematics categories.
The nature of time in a nonautonomous dynamical system is very different from that in autonomous systems, which depend only on the time that has elapsed since starting rather than on the actual time itself. Consequently, limiting objects may not exist in actual time as in autonomous systems. New concepts of attractors in nonautonomous dynamical system are thus required.In addition, the definition of a dynamical system itself needs to be generalised to the nonautonomous context. Here two possibilities are considered: two-parameter semigroups or processes and the skew product flows. Their attractors are defined in terms of families of sets that are mapped onto each other under the dynamics rather than a single set as in autonomous systems. Two types of attraction are now possible: pullback attraction, which depends on the behaviour from the system in the distant past, and forward attraction, which depends on the behaviour of the system in the distant future. These are generally independent of each other.The component subsets of pullback and forward attractors exist in actual time. The asymptotic behaviour in the future limit is characterised by omega-limit sets, in terms of which form what are called forward attracting sets. They are generally not invariant in the conventional sense, but are asymptotically invariant in general and, if the future dynamics is appropriately uniform, also asymptotically negatively invariant.Much of this book is based on lectures given by the authors in Frankfurt and Wuhan. It was written mainly when the first author held a 'Thousand Expert' Professorship at the Huazhong University of Science and Technology in Wuhan.
Bifurcation Theory
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Author : Ale Jan Homburg
language : en
Publisher: American Mathematical Society
Release Date : 2024-12-02
Bifurcation Theory written by Ale Jan Homburg and has been published by American Mathematical Society this book supported file pdf, txt, epub, kindle and other format this book has been release on 2024-12-02 with Mathematics categories.
This textbook provides a thorough overview of bifurcation theory. Assuming some familiarity with differential equations and dynamical systems, it is suitable for use on advanced undergraduate and graduate level and can, in particular, be used for a graduate course on bifurcation theory. The book combines a solid theoretical basis with a detailed description of classical bifurcations. It is organized in chapters on local, nonlocal, and global bifurcations; a number of appendices develop the toolbox for the study of bifurcations. The discussed local bifurcations include saddle-node and Hopf bifurcations, as well as the more advanced Bogdanov-Takens and Neimark-Sacker bifurcations. The book also covers nonlocal bifurcations, discussing various homoclinic bifurcations, and it surveys global bifurcations and phenomena, such as intermittency and period-doubling cascades. The book develops a broad range of complementary techniques, both geometric and analytic, for studying bifurcations. Techniques include normal form methods, center manifold reductions, the Lyapunov-Schmidt construction, cross-coordinate constructions, Melnikov's method, and Lin's method. Full proofs of the results are provided, also for the material in the appendices. This includes proofs of the stable manifold theorem, of the center manifold theorem, and of Lin's method for studying homoclinic bifurcations.
Nonautonomous Dynamical Systems In The Life Sciences
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Author : Peter E. Kloeden
language : en
Publisher: Springer
Release Date : 2014-01-22
Nonautonomous Dynamical Systems In The Life Sciences written by Peter E. Kloeden and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2014-01-22 with Mathematics categories.
Nonautonomous dynamics describes the qualitative behavior of evolutionary differential and difference equations, whose right-hand side is explicitly time dependent. Over recent years, the theory of such systems has developed into a highly active field related to, yet recognizably distinct from that of classical autonomous dynamical systems. This development was motivated by problems of applied mathematics, in particular in the life sciences where genuinely nonautonomous systems abound. The purpose of this monograph is to indicate through selected, representative examples how often nonautonomous systems occur in the life sciences and to outline the new concepts and tools from the theory of nonautonomous dynamical systems that are now available for their investigation.
Bifurcation Theory And Methods Of Dynamical Systems
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Author : Dingjun Luo
language : en
Publisher: World Scientific
Release Date : 1997
Bifurcation Theory And Methods Of Dynamical Systems written by Dingjun Luo and has been published by World Scientific this book supported file pdf, txt, epub, kindle and other format this book has been release on 1997 with Science categories.
Dynamical bifurcation theory is concerned with the changes that occur in the global structure of dynamical systems as parameters are varied. This book makes recent research in bifurcation theory of dynamical systems accessible to researchers interested in this subject. In particular, the relevant results obtained by Chinese mathematicians are introduced as well as some of the works of the authors which may not be widely known. The focus is on the analytic approach to the theory and methods of bifurcations. The book prepares graduate students for further study in this area, and it serves as a ready reference for researchers in nonlinear sciences and applied mathematics.
Geometric Theory Of Discrete Nonautonomous Dynamical Systems
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Author : Christian Pötzsche
language : en
Publisher: Springer Science & Business Media
Release Date : 2010-09-17
Geometric Theory Of Discrete Nonautonomous Dynamical Systems written by Christian Pötzsche and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2010-09-17 with Language Arts & Disciplines categories.
The goal of this book is to provide an approach to the corresponding geometric theory of nonautonomous discrete dynamical systems in infinite-dimensional spaces by virtue of 2-parameter semigroups (processes).