Theory and Applications of Hopf Bifurcation
Title | Theory and Applications of Hopf Bifurcation PDF eBook |
Author | B. D. Hassard |
Publisher | CUP Archive |
Total Pages | 324 |
Release | 1981-02-27 |
Genre | Mathematics |
ISBN | 9780521231589 |
This text will be of value to all those interested in and studying the subject in the mathematical, natural and engineering sciences.
Theory and Applications of Hopf Bifurcation
Title | Theory and Applications of Hopf Bifurcation PDF eBook |
Author | B. D. Hassard |
Publisher | |
Total Pages | 0 |
Release | 1981 |
Genre | |
ISBN |
The Hopf Bifurcation and Its Applications
Title | The Hopf Bifurcation and Its Applications PDF eBook |
Author | J. E. Marsden |
Publisher | Springer Science & Business Media |
Total Pages | 420 |
Release | 2012-12-06 |
Genre | Mathematics |
ISBN | 1461263743 |
The goal of these notes is to give a reasonahly com plete, although not exhaustive, discussion of what is commonly referred to as the Hopf bifurcation with applications to spe cific problems, including stability calculations. Historical ly, the subject had its origins in the works of Poincare [1] around 1892 and was extensively discussed by Andronov and Witt [1] and their co-workers starting around 1930. Hopf's basic paper [1] appeared in 1942. Although the term "Poincare Andronov-Hopf bifurcation" is more accurate (sometimes Friedrichs is also included), the name "Hopf Bifurcation" seems more common, so we have used it. Hopf's crucial contribution was the extension from two dimensions to higher dimensions. The principal technique employed in the body of the text is that of invariant manifolds. The method of Ruelle Takens [1] is followed, with details, examples and proofs added. Several parts of the exposition in the main text come from papers of P. Chernoff, J. Dorroh, O. Lanford and F. Weissler to whom we are grateful. The general method of invariant manifolds is common in dynamical systems and in ordinary differential equations: see for example, Hale [1,2] and Hartman [1]. Of course, other methods are also available. In an attempt to keep the picture balanced, we have included samples of alternative approaches. Specifically, we have included a translation (by L. Howard and N. Kopell) of Hopf's original (and generally unavailable) paper.
Bifurcation Theory And Applications
Title | Bifurcation Theory And Applications PDF eBook |
Author | Shouhong Wang |
Publisher | World Scientific |
Total Pages | 391 |
Release | 2005-06-27 |
Genre | Science |
ISBN | 9814480592 |
This book covers comprehensive bifurcation theory and its applications to dynamical systems and partial differential equations (PDEs) from science and engineering, including in particular PDEs from physics, chemistry, biology, and hydrodynamics.The book first introduces bifurcation theories recently developed by the authors, on steady state bifurcation for a class of nonlinear problems with even order nondegenerate nonlinearities, regardless of the multiplicity of the eigenvalues, and on attractor bifurcations for nonlinear evolution equations, a new notion of bifurcation.With this new notion of bifurcation, many longstanding bifurcation problems in science and engineering are becoming accessible, and are treated in the second part of the book. In particular, applications are covered for a variety of PDEs from science and engineering, including the Kuramoto-Sivashinsky equation, the Cahn-Hillard equation, the Ginzburg-Landau equation, reaction-diffusion equations in biology and chemistry, the Benard convection problem, and the Taylor problem. The applications provide, on the one hand, general recipes for other applications of the theory addressed in this book, and on the other, full classifications of the bifurcated attractor and the global attractor as the control parameters cross certain critical values, dictated usually by the eigenvalues of the linearized problems. It is expected that the book will greatly advance the study of nonlinear dynamics for many problems in science and engineering.
Bifurcation Theory
Title | Bifurcation Theory PDF eBook |
Author | Hansjörg Kielhöfer |
Publisher | Springer Science & Business Media |
Total Pages | 355 |
Release | 2006-04-10 |
Genre | Mathematics |
ISBN | 0387216332 |
In the past three decades, bifurcation theory has matured into a well-established and vibrant branch of mathematics. This book gives a unified presentation in an abstract setting of the main theorems in bifurcation theory, as well as more recent and lesser known results. It covers both the local and global theory of one-parameter bifurcations for operators acting in infinite-dimensional Banach spaces, and shows how to apply the theory to problems involving partial differential equations. In addition to existence, qualitative properties such as stability and nodal structure of bifurcating solutions are treated in depth. This volume will serve as an important reference for mathematicians, physicists, and theoretically-inclined engineers working in bifurcation theory and its applications to partial differential equations.
Topics in Bifurcation Theory and Applications
Title | Topics in Bifurcation Theory and Applications PDF eBook |
Author | Grard Iooss |
Publisher | World Scientific |
Total Pages | 204 |
Release | 1998 |
Genre | Technology & Engineering |
ISBN | 9789810237288 |
This textbook presents the most efficient analytical techniques in the local bifurcation theory of vector fields. It is centered on the theory of normal forms and its applications, including interaction with symmetries. The first part of the book reviews the center manifold reduction and introduces normal forms (with complete proofs). Basic bifurcations are studied together with bifurcations in the presence of symmetries. Special attention is given to examples with reversible vector fields, including the physical example given by the water waves. In this second edition, many problems with detailed solutions are added at the end of the first part (some systems being in infinite dimensions). The second part deals with the Couette-Taylor hydrodynamical stability problem, between concentric rotating cylinders. The spatial structure of various steady or unsteady solutions results directly from the analysis of the reduced system on a center manifold. In this part we also study bifurcations (simple here) from group orbits of solutions in an elementary way (avoiding heavy algebra). The third part analyzes bifurcations from time periodic solutions of autonomous vector fields. A normal form theory is developed, covering all cases, and emphasizing a partial Floquet reduction theory, which is applicable in infinite dimensions. Studies of period doubling as well as Arnold's resonance tongues are included in this part.
Bifurcation Theory of Functional Differential Equations
Title | Bifurcation Theory of Functional Differential Equations PDF eBook |
Author | Shangjiang Guo |
Publisher | Springer Science & Business Media |
Total Pages | 295 |
Release | 2013-07-30 |
Genre | Mathematics |
ISBN | 1461469929 |
This book provides a crash course on various methods from the bifurcation theory of Functional Differential Equations (FDEs). FDEs arise very naturally in economics, life sciences and engineering and the study of FDEs has been a major source of inspiration for advancement in nonlinear analysis and infinite dimensional dynamical systems. The book summarizes some practical and general approaches and frameworks for the investigation of bifurcation phenomena of FDEs depending on parameters with chap. This well illustrated book aims to be self contained so the readers will find in this book all relevant materials in bifurcation, dynamical systems with symmetry, functional differential equations, normal forms and center manifold reduction. This material was used in graduate courses on functional differential equations at Hunan University (China) and York University (Canada).