Numerical Continuation And Bifurcation In Nonlinear Pdes

Author : Hannes Uecker
Publisher : SIAM
Page : 380 pages
File Size : 51,6 Mb
Release : 2021-08-19
Category : Mathematics
ISBN : 9781611976618

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Numerical Continuation and Bifurcation in Nonlinear PDEs by Hannes Uecker Pdf

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.

Author : Eugene L. Allgower,Kurt Georg
Publisher : SIAM
Page : 409 pages
File Size : 50,5 Mb
Release : 2003-01-01
Category : Mathematics
ISBN : 9780898715446

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Introduction to Numerical Continuation Methods by Eugene L. Allgower,Kurt Georg Pdf

Numerical continuation methods have provided important contributions toward the numerical solution of nonlinear systems of equations for many years. The methods may be used not only to compute solutions, which might otherwise be hard to obtain, but also to gain insight into qualitative properties of the solutions. Introduction to Numerical Continuation Methods, originally published in 1979, was the first book to provide easy access to the numerical aspects of predictor corrector continuation and piecewise linear continuation methods. Not only do these seemingly distinct methods share many common features and general principles, they can be numerically implemented in similar ways. Introduction to Numerical Continuation Methods also features the piecewise linear approximation of implicitly defined surfaces, the algorithms of which are frequently used in computer graphics, mesh generation, and the evaluation of surface integrals.

Author : Dirk Roose,Bart De Dier,Alastair Spence
Publisher : Springer Science & Business Media
Page : 415 pages
File Size : 47,8 Mb
Release : 2012-12-06
Category : Mathematics
ISBN : 9789400906594

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Continuation and Bifurcations: Numerical Techniques and Applications by Dirk Roose,Bart De Dier,Alastair Spence Pdf

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

Author : Bernd Krauskopf,Hinke M. Osinga,Jorge Galan-Vioque
Publisher : Springer
Page : 399 pages
File Size : 50,6 Mb
Release : 2007-11-06
Category : Science
ISBN : 9781402063565

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Numerical Continuation Methods for Dynamical Systems by Bernd Krauskopf,Hinke M. Osinga,Jorge Galan-Vioque Pdf

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.

Author : Eugene L. Allgower,Kurt Georg
Publisher : Springer Science & Business Media
Page : 402 pages
File Size : 53,5 Mb
Release : 2012-12-06
Category : Mathematics
ISBN : 9783642612572

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Numerical Continuation Methods by Eugene L. Allgower,Kurt Georg Pdf

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.

Author : MITTELMANN,FISCHER
Publisher : Birkhäuser
Page : 218 pages
File Size : 55,7 Mb
Release : 2013-11-21
Category : Science
ISBN : 9783034856812

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Continuation Techniques and Bifurcation Problems by MITTELMANN,FISCHER Pdf

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.

Author : M. Kubicek,M. Marek
Publisher : Springer Science & Business Media
Page : 253 pages
File Size : 41,5 Mb
Release : 2012-12-06
Category : Science
ISBN : 9783642859571

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Computational Methods in Bifurcation Theory and Dissipative Structures by M. Kubicek,M. Marek Pdf

"Dissipative structures" is a concept which has recently been used in physics to discuss the formation of structures organized in space and/or time at the expense of the energy flowing into the system from the outside. The space-time structural organization of biological systems starting from the subcellular level up to the level of ecological systems, coherent structures in laser and of elastic stability in mechanics, instability in hydro plasma physics, problems dynamics leading to the development of turbulence, behavior of electrical networks and chemical reactors form just a short list of problems treated in this framework. Mathematical models constructed to describe these systems are usually nonlinear, often formed by complicated systems of algebraic, ordinary differ ential, or partial differential equations and include a number of character istic parameters. In problems of theoretical interest as well as engineering practice, we are concerned with the dependence of solutions on parameters and particularly with the values of parameters where qualitatively new types of solutions, e.g., oscillatory solutions, new stationary states, and chaotic attractors, appear (bifurcate). Numerical techniques to determine both bifurcation points and the depen dence of steady-state and oscillatory solutions on parameters are developed and discussed in detail in this text. The text is intended to serve as a working manual not only for students and research workers who are interested in dissipative structures, but also for practicing engineers who deal with the problems of constructing models and solving complicated nonlinear systems.

Author : Bernd Krauskopf,Hinke M. Osinga,Jorge Galan-Vioque
Publisher : Springer
Page : 399 pages
File Size : 43,5 Mb
Release : 2007-07-26
Category : Science
ISBN : 1402063555

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Numerical Continuation Methods for Dynamical Systems by Bernd Krauskopf,Hinke M. Osinga,Jorge Galan-Vioque Pdf

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.

Author : Henk A. Dijkstra,Fred W. Wubs
Publisher : Cambridge University Press
Page : 343 pages
File Size : 44,8 Mb
Release : 2023-06-30
Category : Science
ISBN : 9781108852524

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Bifurcation Analysis of Fluid Flows by Henk A. Dijkstra,Fred W. Wubs Pdf

A better understanding of the mechanisms leading a fluid system to exhibit turbulent behavior is one of the grand challenges of the physical and mathematical sciences. Over the last few decades, numerical bifurcation methods have been extended and applied to a number of flow problems to identify critical conditions for fluid instabilities to occur. This book provides a state-of-the-art account of these numerical methods, with much attention to modern linear systems solvers and generalized eigenvalue solvers. These methods also have a broad applicability in industrial, environmental and astrophysical flows. The book is a must-have reference for anyone working in scientific fields where fluid flow instabilities play a role. Exercises at the end of each chapter and Python code for the bifurcation analysis of canonical fluid flow problems provide practice material to get to grips with the methods and concepts presented in the book.

Author : Eusebius Doedel,Laurette S. Tuckerman
Publisher : Springer Science & Business Media
Page : 482 pages
File Size : 41,6 Mb
Release : 2012-12-06
Category : Mathematics
ISBN : 9781461212089

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Numerical Methods for Bifurcation Problems and Large-Scale Dynamical Systems by Eusebius Doedel,Laurette S. Tuckerman Pdf

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.

Author : Klaus Schmitt
Publisher : Unknown
Page : 182 pages
File Size : 45,6 Mb
Release : 1982
Category : Bifurcation theory
ISBN : UOM:39015015721098

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A Study of Eigenvalue and Bifurcation Problems for Nonlinear Elliptic Partial Differential Equations Via Topological Continuation Methods by Klaus Schmitt Pdf

Author : Willy J. F. Govaerts
Publisher : SIAM
Page : 384 pages
File Size : 52,8 Mb
Release : 2000-01-01
Category : Mathematics
ISBN : 0898719542

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Numerical Methods for Bifurcations of Dynamical Equilibria by Willy J. F. Govaerts Pdf

Dynamical systems arise in all fields of applied mathematics. The author focuses on the description of numerical methods for the detection, computation, and continuation of equilibria and bifurcation points of equilibria of dynamical systems. This subfield has the particular attraction of having links with the geometric theory of differential equations, numerical analysis, and linear algebra.

Author : H.W. Broer,I. Hoveijn,F. Takens,S.A. van Gils
Publisher : Birkhäuser
Page : 464 pages
File Size : 50,7 Mb
Release : 2013-11-11
Category : Mathematics
ISBN : 9783034875189

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Nonlinear Dynamical Systems and Chaos by H.W. Broer,I. Hoveijn,F. Takens,S.A. van Gils Pdf

Symmetries in dynamical systems, "KAM theory and other perturbation theories", "Infinite dimensional systems", "Time series analysis" and "Numerical continuation and bifurcation analysis" were the main topics of the December 1995 Dynamical Systems Conference held in Groningen in honour of Johann Bernoulli. They now form the core of this work which seeks to present the state of the art in various branches of the theory of dynamical systems. A number of articles have a survey character whereas others deal with recent results in current research. It contains interesting material for all members of the dynamical systems community, ranging from geometric and analytic aspects from a mathematical point of view to applications in various sciences.

Author : Jose L Huertas,Wai-Kai Chen,Rabinder N Madan
Publisher : World Scientific
Page : 872 pages
File Size : 55,5 Mb
Release : 1999-07-03
Category : Science
ISBN : 9789814496629

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Visions of Nonlinear Science in the 21st Century by Jose L Huertas,Wai-Kai Chen,Rabinder N Madan Pdf

Authoritative and visionary, this festschrift features 12 highly readable expositions of virtually all currently active aspects of nonlinear science. It has been painstakingly researched and written by leading scientists and eminent expositors, including L Shilnikov, R Seydel, I Prigogine, W Porod, C Mira, M Lakshmanan, W Lauterborn, A Holden, H Haken, C Grebogi, E Doedel and L Chua; each chapter addresses a current and intensively researched area of nonlinear science and chaos, including nonlinear dynamics, mathematics, numerics and technology. Handsomely produced with high resolution color graphics for enhanced readability, this book has been carefully written at a high level of exposition and is somewhat self-contained. Each chapter includes a tutorial and background information, as well as a survey of each area's main results and state of the art. Of special interest to both beginners and seasoned researchers is the identification of future trends and challenging yet tractable problems that are likely to be solved before the end of the 21st century. The visionary and provocative nature of this book makes it a valuable and lasting reference. Contents:Chua's Circuit and the Qualitative Theory of Dynamical Systems (C Mira)Nonlinear Science and the Laws of Nature (I Prigogine)Visions of Synergetics (H Haken)Mathematical Problems of Nonlinear Dynamics: A Tutorial (L Shilnikov)Experimental Nonlinear Physics (W Lauterborn et al.)Nonlinear Physics: Integrability, Chaos and Beyond (M Lakshmanan)Nonlinear Science: The Impact of Biology (A V Holden)Nonlinear Computation (R Seydel)Nonlinear Numerics (E Doedel)Some Historical Aspects of Nonlinear Dynamics: Possible Trends for the Future (C Mira)Control and Applications of Chaos (C Grebogi et al.)Quantum Dot Devices and Quantum-Dot Cellular Automata (W Porod)CNN: A Paradigm for Complexity (L O Chua) Readership: Nonlinear scientists. Keywords:Chua's Circuit;Qualitative Theory;Dynamical Systems;Nonlinear Science;Laws of Nature;Visions of Synergetics;Experimental Nonlinear Physics;Nonlinear Dynamics;Nonlinear Physics;Integrability;Chaos;Nonlinear Computation;Nonlinear Numerics;Control of Chaos;Applications of Chaos;Quantum Dot Devices;Quantum-Dot Cellular Automata;CNN;Cellular Neural Networks

Author : Panayotis G. Kevrekidis
Publisher : Springer Nature
Page : 337 pages
File Size : 42,7 Mb
Release : 2024-06-16
Category : Electronic
ISBN : 9783031549786

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Fractional Dispersive Models and Applications by Panayotis G. Kevrekidis Pdf