Numerical Continuation Methods For Dynamical Systems

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Numerical Continuation Methods for Dynamical Systems

Author : Bernd Krauskopf
Publisher : Springer
ISBN 13 : 1402063563
Total Pages : 399 pages
Book Rating : 4.4/5 (2 download)

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Book Synopsis Numerical Continuation Methods for Dynamical Systems by : Bernd Krauskopf

Download or read book Numerical Continuation Methods for Dynamical Systems written by Bernd Krauskopf and published by Springer. This book was released on 2007-11-06 with total page 399 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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 for Dynamical Systems

Author : Bernd Krauskopf
Publisher : Springer
ISBN 13 : 9781402063558
Total Pages : 399 pages
Book Rating : 4.0/5 (635 download)

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Book Synopsis Numerical Continuation Methods for Dynamical Systems by : Bernd Krauskopf

Download or read book Numerical Continuation Methods for Dynamical Systems written by Bernd Krauskopf and published by Springer. This book was released on 2007-07-26 with total page 399 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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 Methods for Bifurcations of Dynamical Equilibria

Author : Willy J. F. Govaerts
Publisher : SIAM
ISBN 13 : 9780898719543
Total Pages : 384 pages
Book Rating : 4.7/5 (195 download)

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Book Synopsis Numerical Methods for Bifurcations of Dynamical Equilibria by : Willy J. F. Govaerts

Download or read book Numerical Methods for Bifurcations of Dynamical Equilibria written by Willy J. F. Govaerts and published by SIAM. This book was released on 2000-01-01 with total page 384 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.

Numerical Continuation Methods

Author : Eugene L. Allgower
Publisher : Springer Science & Business Media
ISBN 13 : 3642612571
Total Pages : 402 pages
Book Rating : 4.6/5 (426 download)

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Book Synopsis Numerical Continuation Methods by : Eugene L. Allgower

Download or read book Numerical Continuation Methods written by Eugene L. Allgower and published by Springer Science & Business Media. This book was released on 2012-12-06 with total page 402 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.

Numerical Continuation and Bifurcation in Nonlinear PDEs

Author : Hannes Uecker
Publisher : SIAM
ISBN 13 : 1611976618
Total Pages : 380 pages
Book Rating : 4.6/5 (119 download)

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Book Synopsis Numerical Continuation and Bifurcation in Nonlinear PDEs by : Hannes Uecker

Download or read book Numerical Continuation and Bifurcation in Nonlinear PDEs written by Hannes Uecker and published by SIAM. This book was released on 2021-08-19 with total page 380 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.

Dynamical Systems and Numerical Analysis

Author : Andrew Stuart
Publisher : Cambridge University Press
ISBN 13 : 9780521645638
Total Pages : 708 pages
Book Rating : 4.6/5 (456 download)

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Book Synopsis Dynamical Systems and Numerical Analysis by : Andrew Stuart

Download or read book Dynamical Systems and Numerical Analysis written by Andrew Stuart and published by Cambridge University Press. This book was released on 1998-11-28 with total page 708 pages. Available in PDF, EPUB and Kindle. Book excerpt: The first three chapters contain the elements of the theory of dynamical systems and the numerical solution of initial-value problems. In the remaining chapters, numerical methods are formulated as dynamical systems and the convergence and stability properties of the methods are examined.

Handbook of Dynamical Systems

Author : B. Fiedler
Publisher : Gulf Professional Publishing
ISBN 13 : 0080532845
Total Pages : 1099 pages
Book Rating : 4.0/5 (85 download)

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Book Synopsis Handbook of Dynamical Systems by : B. Fiedler

Download or read book Handbook of Dynamical Systems written by B. Fiedler and published by Gulf Professional Publishing. This book was released on 2002-02-21 with total page 1099 pages. Available in PDF, EPUB and Kindle. Book excerpt: This handbook is volume II in a series collecting mathematical state-of-the-art surveys in the field of dynamical systems. Much of this field has developed from interactions with other areas of science, and this volume shows how concepts of dynamical systems further the understanding of mathematical issues that arise in applications. Although modeling issues are addressed, the central theme is the mathematically rigorous investigation of the resulting differential equations and their dynamic behavior. However, the authors and editors have made an effort to ensure readability on a non-technical level for mathematicians from other fields and for other scientists and engineers. The eighteen surveys collected here do not aspire to encyclopedic completeness, but present selected paradigms. The surveys are grouped into those emphasizing finite-dimensional methods, numerics, topological methods, and partial differential equations. Application areas include the dynamics of neural networks, fluid flows, nonlinear optics, and many others. While the survey articles can be read independently, they deeply share recurrent themes from dynamical systems. Attractors, bifurcations, center manifolds, dimension reduction, ergodicity, homoclinicity, hyperbolicity, invariant and inertial manifolds, normal forms, recurrence, shift dynamics, stability, to namejust a few, are ubiquitous dynamical concepts throughout the articles.

Numerical Methods for Bifurcations of Dynamical Equilibria

Author : Willy J. F. Govaerts
Publisher : SIAM
ISBN 13 : 0898714427
Total Pages : 376 pages
Book Rating : 4.8/5 (987 download)

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Book Synopsis Numerical Methods for Bifurcations of Dynamical Equilibria by : Willy J. F. Govaerts

Numerical Methods for Bifurcation Problems and Large-Scale Dynamical Systems

Author : Eusebius Doedel
Publisher : Springer Science & Business Media
ISBN 13 : 1461212081
Total Pages : 482 pages
Book Rating : 4.4/5 (612 download)

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Book Synopsis Numerical Methods for Bifurcation Problems and Large-Scale Dynamical Systems by : Eusebius Doedel

Download or read book Numerical Methods for Bifurcation Problems and Large-Scale Dynamical Systems written by Eusebius Doedel and published by Springer Science & Business Media. This book was released on 2012-12-06 with total page 482 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.

Recipes for Continuation

Author : Harry Dankowicz
Publisher : SIAM
ISBN 13 : 1611972566
Total Pages : 585 pages
Book Rating : 4.6/5 (119 download)

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Book Synopsis Recipes for Continuation by : Harry Dankowicz

Download or read book Recipes for Continuation written by Harry Dankowicz and published by SIAM. This book was released on 2013-08-08 with total page 585 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book provides a comprehensive introduction to the mathematical methodology of parameter continuation. It develops a systematic formalism for constructing and implementing abstract representations of continuation problems with equal emphasis on theoretical rigor, algorithm development and software engineering. The book demonstrates the use of fully developed toolbox templates for boundary-value problems to the analysis of periodic orbits, quasi-periodic invariant tori, and connecting orbits between equilibria and/or periodic orbits. The book contains extensive and fully-worked examples that illustrate the application of the MATLAB-based Computational Continuation Core (COCO) to cutting-edge research in applied dynamical systems. Many exercises and open-ended projects on both theoretical and algorithmic aspects of the material are provided, suitable for self-study and course assignments. It is intended for students and teachers of nonlinear dynamics and engineering at the advanced undergraduate or first-year graduate level, as well as practitioners engaged in modeling dynamical systems or software development.

Elements of Applied Bifurcation Theory

Author : Yuri Kuznetsov
Publisher : Springer Science & Business Media
ISBN 13 : 1475739788
Total Pages : 648 pages
Book Rating : 4.4/5 (757 download)

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Book Synopsis Elements of Applied Bifurcation Theory by : Yuri Kuznetsov

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.

Numerical Bifurcation Analysis of Maps

Author : Yuri A. Kuznetsov
Publisher : Cambridge University Press
ISBN 13 : 1108695140
Total Pages : 424 pages
Book Rating : 4.1/5 (86 download)

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Book Synopsis Numerical Bifurcation Analysis of Maps by : Yuri A. Kuznetsov

Download or read book Numerical Bifurcation Analysis of Maps written by Yuri A. Kuznetsov and published by Cambridge University Press. This book was released on 2019-03-28 with total page 424 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book combines a comprehensive state-of-the-art analysis of bifurcations of discrete-time dynamical systems with concrete instruction on implementations (and example applications) in the free MATLAB® software MatContM developed by the authors. While self-contained and suitable for independent study, the book is also written with users in mind and is an invaluable reference for practitioners. Part I focuses on theory, providing a systematic presentation of bifurcations of fixed points and cycles of finite-dimensional maps, up to and including cases with two control parameters. Several complementary methods, including Lyapunov exponents, invariant manifolds and homoclinic structures, and parts of chaos theory, are presented. Part II introduces MatContM through step-by-step tutorials on how to use the general numerical methods described in Part I for simple dynamical models defined by one- and two-dimensional maps. Further examples in Part III show how MatContM can be used to analyze more complicated models from modern engineering, ecology, and economics.

Mathematics of Complexity and Dynamical Systems

Author : Robert A. Meyers
Publisher : Springer Science & Business Media
ISBN 13 : 1461418054
Total Pages : 1885 pages
Book Rating : 4.4/5 (614 download)

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Book Synopsis Mathematics of Complexity and Dynamical Systems by : Robert A. Meyers

Download or read book Mathematics of Complexity and Dynamical Systems written by Robert A. Meyers and published by Springer Science & Business Media. This book was released on 2011-10-05 with total page 1885 pages. Available in PDF, EPUB and Kindle. Book excerpt: Mathematics of Complexity and Dynamical Systems is an authoritative reference to the basic tools and concepts of complexity, systems theory, and dynamical systems from the perspective of pure and applied mathematics. Complex systems are systems that comprise many interacting parts with the ability to generate a new quality of collective behavior through self-organization, e.g. the spontaneous formation of temporal, spatial or functional structures. These systems are often characterized by extreme sensitivity to initial conditions as well as emergent behavior that are not readily predictable or even completely deterministic. The more than 100 entries in this wide-ranging, single source work provide a comprehensive explication of the theory and applications of mathematical complexity, covering ergodic theory, fractals and multifractals, dynamical systems, perturbation theory, solitons, systems and control theory, and related topics. Mathematics of Complexity and Dynamical Systems is an essential reference for all those interested in mathematical complexity, from undergraduate and graduate students up through professional researchers.

Knowledge Discovery in Spatial Data

Author : Yee Leung
Publisher : Springer Science & Business Media
ISBN 13 : 3642026648
Total Pages : 381 pages
Book Rating : 4.6/5 (42 download)

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Book Synopsis Knowledge Discovery in Spatial Data by : Yee Leung

Download or read book Knowledge Discovery in Spatial Data written by Yee Leung and published by Springer Science & Business Media. This book was released on 2010-03-14 with total page 381 pages. Available in PDF, EPUB and Kindle. Book excerpt: When I ?rst came across the term data mining and knowledge discovery in databases, I was excited and curious to ?nd out what it was all about. I was excited because the term tends to convey a new ?eld that is in the making. I was curious because I wondered what it was doing that the other ?elds of research, such as statistics and the broad ?eld of arti?cial intelligence, were not doing. After reading up on the literature, I have come to realize that it is not much different from conventional data analysis. The commonly used de?nition of knowledge discovery in databases: “the non-trivial process of identifying valid, novel, potentially useful, and ultimately understandable patterns in data” is actually in line with the core mission of conventional data analysis. The process employed by conventional data analysis is by no means trivial, and the patterns in data to be unraveled have, of course, to be valid, novel, useful and understandable. Therefore, what is the commotion all about? Careful scrutiny of the main lines of research in data mining and knowledge discovery again told me that they are not much different from that of conventional data analysis. Putting aside data warehousing and database m- agement aspects, again a main area of research in conventional database research, the rest of the tasks in data mining are largely the main concerns of conventional data analysis.

Averaging Methods in Nonlinear Dynamical Systems

Author : Jan A. Sanders
Publisher : Springer Science & Business Media
ISBN 13 : 1475745753
Total Pages : 259 pages
Book Rating : 4.4/5 (757 download)

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Book Synopsis Averaging Methods in Nonlinear Dynamical Systems by : Jan A. Sanders

Download or read book Averaging Methods in Nonlinear Dynamical Systems written by Jan A. Sanders and published by Springer Science & Business Media. This book was released on 2013-04-17 with total page 259 pages. Available in PDF, EPUB and Kindle. Book excerpt: In this book we have developed the asymptotic analysis of nonlinear dynamical systems. We have collected a large number of results, scattered throughout the literature and presented them in a way to illustrate both the underlying common theme, as well as the diversity of problems and solutions. While most of the results are known in the literature, we added new material which we hope will also be of interest to the specialists in this field. The basic theory is discussed in chapters two and three. Improved results are obtained in chapter four in the case of stable limit sets. In chapter five we treat averaging over several angles; here the theory is less standardized, and even in our simplified approach we encounter many open problems. Chapter six deals with the definition of normal form. After making the somewhat philosophical point as to what the right definition should look like, we derive the second order normal form in the Hamiltonian case, using the classical method of generating functions. In chapter seven we treat Hamiltonian systems. The resonances in two degrees of freedom are almost completely analyzed, while we give a survey of results obtained for three degrees of freedom systems. The appendices contain a mix of elementary results, expansions on the theory and research problems.

Numerical Analysis in Modern Scientific Computing

Author : Peter Deuflhard
Publisher : Springer Science & Business Media
ISBN 13 : 0387215840
Total Pages : 350 pages
Book Rating : 4.3/5 (872 download)

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Book Synopsis Numerical Analysis in Modern Scientific Computing by : Peter Deuflhard

Download or read book Numerical Analysis in Modern Scientific Computing written by Peter Deuflhard and published by Springer Science & Business Media. This book was released on 2012-12-06 with total page 350 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book introduces the main topics of modern numerical analysis: sequence of linear equations, error analysis, least squares, nonlinear systems, symmetric eigenvalue problems, three-term recursions, interpolation and approximation, large systems and numerical integrations. The presentation draws on geometrical intuition wherever appropriate and is supported by a large number of illustrations, exercises, and examples.

Optimization and Dynamical Systems

Author : Uwe Helmke
Publisher : Springer Science & Business Media
ISBN 13 : 1447134672
Total Pages : 409 pages
Book Rating : 4.4/5 (471 download)

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Book Synopsis Optimization and Dynamical Systems by : Uwe Helmke

Download or read book Optimization and Dynamical Systems written by Uwe Helmke and published by Springer Science & Business Media. This book was released on 2012-12-06 with total page 409 pages. Available in PDF, EPUB and Kindle. Book excerpt: This work is aimed at mathematics and engineering graduate students and researchers in the areas of optimization, dynamical systems, control sys tems, signal processing, and linear algebra. The motivation for the results developed here arises from advanced engineering applications and the emer gence of highly parallel computing machines for tackling such applications. The problems solved are those of linear algebra and linear systems the ory, and include such topics as diagonalizing a symmetric matrix, singular value decomposition, balanced realizations, linear programming, sensitivity minimization, and eigenvalue assignment by feedback control. The tools are those, not only of linear algebra and systems theory, but also of differential geometry. The problems are solved via dynamical sys tems implementation, either in continuous time or discrete time , which is ideally suited to distributed parallel processing. The problems tackled are indirectly or directly concerned with dynamical systems themselves, so there is feedback in that dynamical systems are used to understand and optimize dynamical systems. One key to the new research results has been the recent discovery of rather deep existence and uniqueness results for the solution of certain matrix least squares optimization problems in geomet ric invariant theory. These problems, as well as many other optimization problems arising in linear algebra and systems theory, do not always admit solutions which can be found by algebraic methods.