Download or read book Topics in Stability and Bifurcation Theory written by David H. Sattinger and published by Springer. This book was released on 2006-11-15 with total page 197 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book Nonlinear Stability and Bifurcation Theory written by Hans Troger and published by Springer Science & Business Media. This book was released on 2012-12-06 with total page 419 pages. Available in PDF, EPUB and Kindle. Book excerpt: Every student in engineering or in other fields of the applied sciences who has passed through his curriculum knows that the treatment of nonlin ear problems has been either avoided completely or is confined to special courses where a great number of different ad-hoc methods are presented. The wide-spread believe that no straightforward solution procedures for nonlinear problems are available prevails even today in engineering cir cles. Though in some courses it is indicated that in principle nonlinear problems are solveable by numerical methods the treatment of nonlinear problems, more or less, is considered to be an art or an intellectual game. A good example for this statement was the search for Ljapunov functions for nonlinear stability problems in the seventies. However things have changed. At the beginning of the seventies, start ing with the work of V.1. Arnold, R. Thom and many others, new ideas which, however, have their origin in the work of H. Poincare and A. A. Andronov, in the treatment of nonlinear problems appeared. These ideas gave birth to the term Bifurcation Theory. Bifurcation theory allows to solve a great class of nonlinear problems under variation of parameters in a straightforward manner.
Download or read book Topics in Bifurcation Theory and Applications written by Grard Iooss and published by World Scientific. This book was released on 1998 with total page 204 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
Download or read book Elementary Stability and Bifurcation Theory written by Gerard Iooss and published by Springer Science & Business Media. This book was released on 2012-12-06 with total page 347 pages. Available in PDF, EPUB and Kindle. Book excerpt: This substantially revised second edition teaches the bifurcation of asymptotic solutions to evolution problems governed by nonlinear differential equations. Written not just for mathematicians, it appeals to the widest audience of learners, including engineers, biologists, chemists, physicists and economists. For this reason, it uses only well-known methods of classical analysis at foundation level, while the applications and examples are specially chosen to be as varied as possible.
Download or read book Stability and Bifurcation of Structures written by Angelo Luongo and published by Springer Nature. This book was released on 2023-06-27 with total page 712 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book overcomes the separation existing in literature between the static and the dynamic bifurcation worlds. It brings together buckling and post-buckling problems with nonlinear dynamics, the bridge being represented by the perturbation method, i.e., a mathematical tool that allows for solving static and dynamic problems virtually in the same way. The book is organized as follows: Chapter one gives an overview; Chapter two illustrates phenomenological aspect of static and dynamic bifurcations; Chapter three deals with linear stability analysis of dynamical systems; Chapter four and five discuss the general theory and present examples of buckling and post-buckling of elastic structures; Chapter six describes a linearized approach to buckling, usually adopted in the technical literature, in which pre-critical deformations are neglected; Chapters seven to ten, analyze elastic and elasto-plastic buckling of planar systems of beams, thin-walled beams and plate assemblies, respectively; Chapters eleven to thirteen, illustrate dynamic instability phenomena, such as flutter induced by follower forces, aeroelastic bifurcations caused by wind flow, and parametric excitation triggered by pulsating loads. Finally, Chapter fourteen discusses a large gallery of solved problems, concerning topics covered in the book. An Appendix presents the Vlasov theory of open thin-walled beams. The book is devoted to advanced undergraduate and graduate students, as well as engineers and practitioners. The methods illustrated here are immediately applicable to model real problems. The Book Introduces, in a simple way, complex concepts of bifurcation theory, by making use of elementary mathematics Gives a comprehensive overview of bifurcation of linear and nonlinear structures, in static and dynamic fields Contains a chapter in which many problems are solved, either analytically or numerically, and results commented
Download or read book Practical Bifurcation and Stability Analysis written by Rüdiger Seydel and published by Springer Science & Business Media. This book was released on 2009-12-14 with total page 493 pages. Available in PDF, EPUB and Kindle. Book excerpt: Probably the first book to describe computational methods for numerically computing steady state and Hopf bifurcations. Requiring only a basic knowledge of calculus, and using detailed examples, problems, and figures, this is an ideal textbook for graduate students.
Download or read book Elements of Applied Bifurcation Theory written by Yuri Kuznetsov and published by Springer Science & Business Media. This book was released on 2013-03-09 with total page 648 pages. Available in PDF, EPUB and Kindle. Book excerpt: Providing readers with a solid basis in dynamical systems theory, as well as explicit procedures for application of general mathematical results to particular problems, the focus here is on efficient numerical implementations of the developed techniques. The book is designed for advanced undergraduates or graduates in applied mathematics, as well as for Ph.D. students and researchers in physics, biology, engineering, and economics who use dynamical systems as model tools in their studies. A moderate mathematical background is assumed, and, whenever possible, only elementary mathematical tools are used. This new edition preserves the structure of the first while updating the context to incorporate recent theoretical developments, in particular new and improved numerical methods for bifurcation analysis.
Download or read book Topics in Dynamic Bifurcation Theory written by Jack K. Hale and published by American Mathematical Soc.. This book was released on 1981-12-31 with total page 90 pages. Available in PDF, EPUB and Kindle. Book excerpt: Presents the general theory of first order bifurcation that occur for vector fields in finite dimensional space. This book includes formulation of structural stability and bifurcation in infinite dimensions.
Download or read book Topics in Bifurcation Theory and Applications written by Gérard Iooss and published by World Scientific Publishing Company. This book was released on 1999-01-22 with total page 196 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
Download or read book Elementary Stability and Bifurcation Theory written by G. Iooss and published by Springer Science & Business Media. This book was released on 2013-03-09 with total page 300 pages. Available in PDF, EPUB and Kindle. Book excerpt: In its most general form bifurcation theory is a theory of equilibrium solutions of nonlinear equations. By equilibrium solutions we mean, for example, steady solutions, time-periodic solutions, and quasi-periodic solutions. The purpose of this book is to teach the theory of bifurcation of equilibrium solutions of evolution problems governed by nonlinear differential equations. We have written this book for the broaqest audience of potentially interested learners: engineers, biologists, chemists, physicists, mathematicians, econom ists, and others whose work involves understanding equilibrium solutions of nonlinear differential equations. To accomplish our aims, we have thought it necessary to make the analysis 1. general enough to apply to the huge variety of applications which arise in science and technology, and 2. simple enough so that it can be understood by persons whose mathe matical training does not extend beyond the classical methods of analysis which were popular in the 19th Century. Of course, it is not possible to achieve generality and simplicity in a perfect union but, in fact, the general theory is simpler than the detailed theory required for particular applications. The general theory abstracts from the detailed problems only the essential features and provides the student with the skeleton on which detailed structures of the applications must rest. It is generally believed that the mathematical theory of bifurcation requires some functional analysis and some of the methods of topology and dynamics.
Download or read book Dynamical Systems V written by V.I. Arnold and published by Springer Science & Business Media. This book was released on 2013-12-01 with total page 279 pages. Available in PDF, EPUB and Kindle. Book excerpt: Bifurcation theory and catastrophe theory are two well-known areas within the field of dynamical systems. Both are studies of smooth systems, focusing on properties that seem to be manifestly non-smooth. Bifurcation theory is concerned with the sudden changes that occur in a system when one or more parameters are varied. Examples of such are familiar to students of differential equations, from phase portraits. Understanding the bifurcations of the differential equations that describe real physical systems provides important information about the behavior of the systems. Catastrophe theory became quite famous during the 1970's, mostly because of the sensation caused by the usually less than rigorous applications of its principal ideas to "hot topics", such as the characterization of personalities and the difference between a "genius" and a "maniac". Catastrophe theory is accurately described as singularity theory and its (genuine) applications. The authors of this book, previously published as Volume 5 of the Encyclopaedia, have given a masterly exposition of these two theories, with penetrating insight.
Download or read book Methods of Bifurcation Theory written by S.-N. Chow and published by Springer Science & Business Media. This book was released on 2012-12-06 with total page 529 pages. Available in PDF, EPUB and Kindle. Book excerpt: An alternative title for this book would perhaps be Nonlinear Analysis, Bifurcation Theory and Differential Equations. Our primary objective is to discuss those aspects of bifurcation theory which are particularly meaningful to differential equations. To accomplish this objective and to make the book accessible to a wider we have presented in detail much of the relevant background audience, material from nonlinear functional analysis and the qualitative theory of differential equations. Since there is no good reference for some of the mate rial, its inclusion seemed necessary. Two distinct aspects of bifurcation theory are discussed-static and dynamic. Static bifurcation theory is concerned with the changes that occur in the structure of the set of zeros of a function as parameters in the function are varied. If the function is a gradient, then variational techniques play an important role and can be employed effectively even for global problems. If the function is not a gradient or if more detailed information is desired, the general theory is usually local. At the same time, the theory is constructive and valid when several independent parameters appear in the function. In differential equations, the equilibrium solutions are the zeros of the vector field. Therefore, methods in static bifurcation theory are directly applicable.
Download or read book Bifurcation Theory And Applications written by Shouhong Wang and published by World Scientific. This book was released on 2005-06-27 with total page 391 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
Download or read book Bifurcation Analysis Algorithms Applications written by KÜPPER and published by Birkhäuser. This book was released on 2013-03-08 with total page 368 pages. Available in PDF, EPUB and Kindle. Book excerpt: The conference on BIFURCATIONS: ANALYSIS, ALGORITHMS, APPLICATIONS took place in Dortmund in August 18 - 22, 1986. More then 150 Scientists from 16 countries participated in the meeting, among them mathematicians, engi neers, and physicists. A broad spectrum of new results on bifurcation was covered by 49 talks. The diversity of the range of treated topics and of involved fields inspired fruitful discussions. 36 refereed papers are contained in these proceedings. The subjects covered treat bifurcation problems, ranging from theoretical investigations to numerical results, with emphasis placed upon applications. The more theoreti cal papers include the topics symmetry breaking, delay differential equations, Cornu spirals, homoclinic orbits, and selfsimilarity. Different kinds of bifurcations are treated: Hopf bifurcation, bifurcation from continuous spec trum, complex bifurcation, and bifurcation near tori. Several numerical as pects are discussed, among them continuation, block elimination, and spectral methods. Algorithms are proposed for approximating manifolds, calculating pe riodic solutions and handling multi-parameter problems. Ample space is devoted to· applications. Classical phenomena from fluid mechanics (such as convection rolls and th~ Taylor vortex problem), buckling, and reaction-diffusion pro blems are considered. Other applications of bifurcations include railway vehicle dynamics, computer graphics, semiconductors, drilling processes, simu lation of oil reservoirs, and rotor dynamics. The proceedings reflect current research in bifurcation. They are an attempt to bring together researchers from differ~nt disciplines to stimulate common effort towards a better understanding and handling of bifurcation pro blems.
Download or read book Nonlinear Differential Equations written by Piero de Mottoni and published by Academic Press. This book was released on 2014-05-10 with total page 370 pages. Available in PDF, EPUB and Kindle. Book excerpt: Nonlinear Differential Equations: Invariance, Stability, and Bifurcation presents the developments in the qualitative theory of nonlinear differential equations. This book discusses the exchange of mathematical ideas in stability and bifurcation theory. Organized into 26 chapters, this book begins with an overview of the initial value problem for a nonlinear wave equation. This text then focuses on the interplay between stability exchange for a stationary solution and the appearance of bifurcating periodic orbits. Other chapters consider the development of methods for ascertaining stability and boundedness and explore the development of bifurcation and stability analysis in nonlinear models of applied sciences. This book discusses as well nonlinear hyperbolic equations in further contributions, featuring stability properties of periodic and almost periodic solutions. The reader is also introduced to the stability problem of the equilibrium of a chemical network. The final chapter deals with suitable spaces for studying functional equations. This book is a valuable resource for mathematicians.
Download or read book Bifurcation Theory written by Hansjörg Kielhöfer and published by Springer Science & Business Media. This book was released on 2006-04-10 with total page 355 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
Download or read book Studies in Non linear Stability and Bifurcation Theory written by Jan Sijbrand and published by . This book was released on 1981 with total page 280 pages. Available in PDF, EPUB and Kindle. Book excerpt:
