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Continuation Techniques And Bifurcation Problems

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Numerical Continuation Methods For Dynamical Systems

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Author : Bernd Krauskopf
language : en
Publisher: Springer
Release Date : 2007-11-06

Numerical Continuation Methods For Dynamical Systems written by Bernd Krauskopf and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2007-11-06 with Science categories.

Path following in combination with boundary value problem solvers has emerged as a continuing and strong influence in the development of dynamical systems theory and its application. It is widely acknowledged that the software package AUTO - developed by Eusebius J. Doedel about thirty years ago and further expanded and developed ever since - plays a central role in the brief history of numerical continuation. This book has been compiled on the occasion of Sebius Doedel's 60th birthday. Bringing together for the first time a large amount of material in a single, accessible source, it is hoped that the book will become the natural entry point for researchers in diverse disciplines who wish to learn what numerical continuation techniques can achieve. The book opens with a foreword by Herbert B. Keller and lecture notes by Sebius Doedel himself that introduce the basic concepts of numerical bifurcation analysis. The other chapters by leading experts discuss continuation for various types of systems and objects and showcase examples of how numerical bifurcation analysis can be used in concrete applications. Topics that are treated include: interactive continuation tools, higher-dimensional continuation, the computation of invariant manifolds, and continuation techniques for slow-fast systems, for symmetric Hamiltonian systems, for spatially extended systems and for systems with delay. Three chapters review physical applications: the dynamics of a SQUID, global bifurcations in laser systems, and dynamics and bifurcations in electronic circuits.

Numerical Continuation Methods

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Author : Eugene L. Allgower
language : en
Publisher: Springer Science & Business Media
Release Date : 2012-12-06

Numerical Continuation Methods written by Eugene L. Allgower 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 2012-12-06 with Mathematics categories.

Over the past fifteen years two new techniques have yielded extremely important contributions toward the numerical solution of nonlinear systems of equations. This book provides an introduction to and an up-to-date survey of numerical continuation methods (tracing of implicitly defined curves) of both predictor-corrector and piecewise-linear types. It presents and analyzes implementations aimed at applications to the computation of zero points, fixed points, nonlinear eigenvalue problems, bifurcation and turning points, and economic equilibria. Many algorithms are presented in a pseudo code format. An appendix supplies five sample FORTRAN programs with numerical examples, which readers can adapt to fit their purposes, and a description of the program package SCOUT for analyzing nonlinear problems via piecewise-linear methods. An extensive up-to-date bibliography spanning 46 pages is included. The material in this book has been presented to students of mathematics, engineering and sciences with great success, and will also serve as a valuable tool for researchers in the field.

Continuation Techniques And Bifurcation Problems

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Author : MITTELMANN
language : en
Publisher: Birkhäuser
Release Date : 2014-10-05

Continuation Techniques And Bifurcation Problems written by MITTELMANN and has been published by Birkhäuser this book supported file pdf, txt, epub, kindle and other format this book has been release on 2014-10-05 with Science categories.

The analysis of parameter-dependent nonlinear has received much attention in recent years. Numerical continuation techniques allow the efficient computation of solution branches in a one-parameter problem. In many cases continuation procedures are used as part of a more complete analysis of a nonlinear problem, based on bifurcation theory and singularity theory. These theories contribute to the understanding of many nonlinear phenomena in nature and they form the basis for various analytical and numerical tools, which provide qualitative and quantitative results about nonlinear systems. In this issue we have collected a number of papers dealing with continuation techniques and bifurcation problems. Readers familiar with the notions of continuation and bifurcation will find recent research results addressing a variety of aspects in this issue. Those who intend to learn about the field or a specific topic in it may find it useful to first consult earlier literature on the numerical treatment of these problems together with some theoretical background. The papers in this issue fall naturally into different groups.

Numerical Continuation And Bifurcation In Nonlinear Pdes

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Author : Hannes Uecker
language : en
Publisher: SIAM
Release Date : 2021-08-19

Numerical Continuation And Bifurcation In Nonlinear Pdes written by Hannes Uecker and has been published by SIAM this book supported file pdf, txt, epub, kindle and other format this book has been release on 2021-08-19 with Mathematics categories.

This book provides a hands-on approach to numerical continuation and bifurcation for nonlinear PDEs in 1D, 2D, and 3D. Partial differential equations (PDEs) are the main tool to describe spatially and temporally extended systems in nature. PDEs usually come with parameters, and the study of the parameter dependence of their solutions is an important task. Letting one parameter vary typically yields a branch of solutions, and at special parameter values, new branches may bifurcate. After a concise review of some analytical background and numerical methods, the author explains the free MATLAB package pde2path by using a large variety of examples with demo codes that can be easily adapted to the reader's given problem. Numerical Continuation and Bifurcation in Nonlinear PDEs will appeal to applied mathematicians and scientists from physics, chemistry, biology, and economics interested in the numerical solution of nonlinear PDEs, particularly the parameter dependence of solutions. It can be used as a supplemental text in courses on nonlinear PDEs and modeling and bifurcation.

Continuation Techniques And Bifurcation Problems

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Author : H. D. Mittelmann
language : en
Publisher: Birkhauser
Release Date : 1990

Continuation Techniques And Bifurcation Problems written by H. D. Mittelmann and has been published by Birkhauser this book supported file pdf, txt, epub, kindle and other format this book has been release on 1990 with Mathematics categories.

Nonlinear Pdes

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Author : Guido Schneider
language : en
Publisher: American Mathematical Soc.
Release Date : 2017-10-26

Nonlinear Pdes written by Guido Schneider and has been published by American Mathematical Soc. this book supported file pdf, txt, epub, kindle and other format this book has been release on 2017-10-26 with Mathematics categories.

This is an introductory textbook about nonlinear dynamics of PDEs, with a focus on problems over unbounded domains and modulation equations. The presentation is example-oriented, and new mathematical tools are developed step by step, giving insight into some important classes of nonlinear PDEs and nonlinear dynamics phenomena which may occur in PDEs. The book consists of four parts. Parts I and II are introductions to finite- and infinite-dimensional dynamics defined by ODEs and by PDEs over bounded domains, respectively, including the basics of bifurcation and attractor theory. Part III introduces PDEs on the real line, including the Korteweg-de Vries equation, the Nonlinear Schrödinger equation and the Ginzburg-Landau equation. These examples often occur as simplest possible models, namely as amplitude or modulation equations, for some real world phenomena such as nonlinear waves and pattern formation. Part IV explores in more detail the connections between such complicated physical systems and the reduced models. For many models, a mathematically rigorous justification by approximation results is given. The parts of the book are kept as self-contained as possible. The book is suitable for self-study, and there are various possibilities to build one- or two-semester courses from the book.

Recipes For Continuation

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Author : Harry Dankowicz
language : en
Publisher: SIAM
Release Date : 2013-01-01

Recipes For Continuation written by Harry Dankowicz and has been published by SIAM this book supported file pdf, txt, epub, kindle and other format this book has been release on 2013-01-01 with Science categories.

This book provides a comprehensive introduction to the mathematical methodology of parameter continuation, the computational analysis of families of solutions to nonlinear mathematical equations. It develops a systematic formalism for constructing abstract representations of continuation problems and for implementing these in an existing computational platform. Recipes for Continuation lends equal importance to theoretical rigor, algorithm development, and software engineering; demonstrates the use of fully developed toolbox templates for single- and multisegment boundary-value problems to the analysis of periodic orbits in smooth and hybrid dynamical systems, quasi-periodic invariant tori, and homoclinic and heteroclinic connecting orbits between equilibria and/or periodic orbits; shows the use of vectorization for optimal computational efficiency, an object-oriented paradigm for the modular construction of continuation problems, and adaptive discretization algorithms for guaranteed bounds on estimated errors; and contains extensive and fully worked examples that illustrate the application of the MATLAB®-based Computational Continuation Core (COCO) to problems from recent research literature that are relevant to dynamical system models from mechanics, electronics, biology, economics, and neuroscience.

Elements Of Applied Bifurcation Theory

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Author : Yuri Kuznetsov
language : en
Publisher: Springer Science & Business Media
Release Date : 2004-06-29

Elements Of Applied Bifurcation Theory written by Yuri Kuznetsov 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 2004-06-29 with Mathematics categories.

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.

Continuation And Bifurcations Numerical Techniques And Applications

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Author : Dirk Roose
language : en
Publisher: Springer Science & Business Media
Release Date : 1990-08-31

Continuation And Bifurcations Numerical Techniques And Applications written by Dirk Roose 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 1990-08-31 with Mathematics categories.

Proceedings of the NATO Advanced Research Workshop, Leuven, Belgium, September 18-22, 1989

Numerical Methods For Bifurcation Problems

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Author : KÜPPER
language : en
Publisher: Birkhäuser
Release Date : 1984

Numerical Methods For Bifurcation Problems written by KÜPPER and has been published by Birkhäuser this book supported file pdf, txt, epub, kindle and other format this book has been release on 1984 with Juvenile Nonfiction categories.

Continuation Techniques And Bifurcation Problems

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