Numerical Methods For Bifurcation Problems
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Numerical Methods For Bifurcation Problems And Large Scale Dynamical Systems
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Author : Eusebius Doedel
language : en
Publisher: Springer Science & Business Media
Release Date : 2012-12-06
Numerical Methods For Bifurcation Problems And Large Scale Dynamical Systems written by Eusebius Doedel 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.
The Institute for Mathematics and its Applications (IMA) devoted its 1997-1998 program to Emerging Applications of Dynamical Systems. Dynamical systems theory and related numerical algorithms provide powerful tools for studying the solution behavior of differential equations and mappings. In the past 25 years computational methods have been developed for calculating fixed points, limit cycles, and bifurcation points. A remaining challenge is to develop robust methods for calculating more complicated objects, such as higher- codimension bifurcations of fixed points, periodic orbits, and connecting orbits, as well as the calcuation of invariant manifolds. Another challenge is to extend the applicability of algorithms to the very large systems that result from discretizing partial differential equations. Even the calculation of steady states and their linear stability can be prohibitively expensive for large systems (e.g. 10_3- -10_6 equations) if attempted by simple direct methods. Several of the papers in this volume treat computational methods for low and high dimensional systems and, in some cases, their incorporation into software packages. A few papers treat fundamental theoretical problems, including smooth factorization of matrices, self -organized criticality, and unfolding of singular heteroclinic cycles. Other papers treat applications of dynamical systems computations in various scientific fields, such as biology, chemical engineering, fluid mechanics, and mechanical engineering.
Numerical Methods For Bifurcation Problems And Large Scale Dynamical Systems
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Author : Eusebius Doedel
language : en
Publisher:
Release Date : 2000-03-17
Numerical Methods For Bifurcation Problems And Large Scale Dynamical Systems written by Eusebius Doedel and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 2000-03-17 with categories.
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 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.
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.
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.
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.
Numerical Methods For Bifurcation Problems
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Author : T. Kupper
language : en
Publisher:
Release Date : 1984-07-01
Numerical Methods For Bifurcation Problems written by T. Kupper and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 1984-07-01 with categories.
Nonlinear Stability And Bifurcation Theory
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Author : Hans Troger
language : en
Publisher: Springer Science & Business Media
Release Date : 2012-12-06
Nonlinear Stability And Bifurcation Theory written by Hans Troger 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 Science categories.
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.
Continuation Techniques And Bifurcation Problems
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Author : MITTELMANN
language : en
Publisher: Birkhäuser
Release Date : 2013-11-21
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 2013-11-21 with Social 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.