Rigorous Numerics In Dynamics

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Rigorous Numerics in Dynamics

Author : Jan Bouwe van den Berg
Publisher : American Mathematical Soc.
ISBN 13 : 1470428148
Total Pages : 226 pages
Book Rating : 4.4/5 (74 download)

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Book Synopsis Rigorous Numerics in Dynamics by : Jan Bouwe van den Berg

Download or read book Rigorous Numerics in Dynamics written by Jan Bouwe van den Berg and published by American Mathematical Soc.. This book was released on 2018-07-12 with total page 226 pages. Available in PDF, EPUB and Kindle. Book excerpt: This volume is based on lectures delivered at the 2016 AMS Short Course “Rigorous Numerics in Dynamics”, held January 4–5, 2016, in Seattle, Washington. Nonlinear dynamics shapes the world around us, from the harmonious movements of celestial bodies, via the swirling motions in fluid flows, to the complicated biochemistry in the living cell. Mathematically these phenomena are modeled by nonlinear dynamical systems, in the form of ODEs, PDEs and delay equations. The presence of nonlinearities complicates the analysis, and the difficulties are even greater for PDEs and delay equations, which are naturally defined on infinite dimensional function spaces. With the availability of powerful computers and sophisticated software, numerical simulations have quickly become the primary tool to study the models. However, while the pace of progress increases, one may ask: just how reliable are our computations? Even for finite dimensional ODEs, this question naturally arises if the system under study is chaotic, as small differences in initial conditions (such as those due to rounding errors in numerical computations) yield wildly diverging outcomes. These issues have motivated the development of the field of rigorous numerics in dynamics, which draws inspiration from ideas in scientific computing, numerical analysis and approximation theory. The articles included in this volume present novel techniques for the rigorous study of the dynamics of maps via the Conley-index theory; periodic orbits of delay differential equations via continuation methods; invariant manifolds and connecting orbits; the dynamics of models with unknown nonlinearities; and bifurcations diagrams.

Rigorous Numerics in Dynamics

Author : Jan Bouwe Van
Publisher :
ISBN 13 : 9781470447298
Total Pages : 226 pages
Book Rating : 4.4/5 (472 download)

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Book Synopsis Rigorous Numerics in Dynamics by : Jan Bouwe Van

Download or read book Rigorous Numerics in Dynamics written by Jan Bouwe Van and published by . This book was released on 2018 with total page 226 pages. Available in PDF, EPUB and Kindle. Book excerpt: This volume is based on lectures delivered at the 2016 AMS Short Course ""Rigorous Numerics in Dynamics"", held January 4-5, 2016, in Seattle, Washington. Nonlinear dynamics shapes the world around us, from the harmonious movements of celestial bodies, via the swirling motions in fluid flows, to the complicated biochemistry in the living cell. Mathematically these phenomena are modeled by nonlinear dynamical systems, in the form of ODEs, PDEs and delay equations. The presence of nonlinearities complicates the analysis, and the difficulties are even greater for PDEs and delay equations, which a.

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.

Dynamics: Numerical Explorations

Author : Helena E. Nusse
Publisher : Springer
ISBN 13 : 9781468402322
Total Pages : 485 pages
Book Rating : 4.4/5 (23 download)

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Book Synopsis Dynamics: Numerical Explorations by : Helena E. Nusse

Download or read book Dynamics: Numerical Explorations written by Helena E. Nusse and published by Springer. This book was released on 2014-03-03 with total page 485 pages. Available in PDF, EPUB and Kindle. Book excerpt: Co-author J.A. Yorke developed an array of tools to help visualize the properties of dynamical systems, while Yorke found it useful to combine these various basic tools into one single package: Dynamics. The program together with this manual provides an introduction to and an overview of fundamental, sophisticated tools and numerical methods together with many simple examples. All numerical methods described in this handbook are implemented in the program, which is capable of, among others: iterating maps and solving differential equations; plotting trajectories; featuring an array of simple commands; printing a created picture in resolution higher than that of the screen. Requires a UNIX workstation running X11 graphics or a PC.

Numerical Methods in Fluid Dynamics

Author : Gary A. Sod
Publisher : Cambridge University Press
ISBN 13 : 9780521259248
Total Pages : 464 pages
Book Rating : 4.2/5 (592 download)

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Book Synopsis Numerical Methods in Fluid Dynamics by : Gary A. Sod

Download or read book Numerical Methods in Fluid Dynamics written by Gary A. Sod and published by Cambridge University Press. This book was released on 1985-10-31 with total page 464 pages. Available in PDF, EPUB and Kindle. Book excerpt: Here is an introduction to numerical methods for partial differential equations with particular reference to those that are of importance in fluid dynamics. The author gives a thorough and rigorous treatment of the techniques, beginning with the classical methods and leading to a discussion of modern developments. For easier reading and use, many of the purely technical results and theorems are given separately from the main body of the text. The presentation is intended for graduate students in applied mathematics, engineering and physical sciences who have a basic knowledge of partial differential equations.

Numerical Methods for Nonsmooth Dynamical Systems

Author : Vincent Acary
Publisher : Springer Science & Business Media
ISBN 13 : 3540753923
Total Pages : 529 pages
Book Rating : 4.5/5 (47 download)

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Book Synopsis Numerical Methods for Nonsmooth Dynamical Systems by : Vincent Acary

Download or read book Numerical Methods for Nonsmooth Dynamical Systems written by Vincent Acary and published by Springer Science & Business Media. This book was released on 2008-01-30 with total page 529 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book concerns the numerical simulation of dynamical systems whose trajec- ries may not be differentiable everywhere. They are named nonsmooth dynamical systems. They make an important class of systems, rst because of the many app- cations in which nonsmooth models are useful, secondly because they give rise to new problems in various elds of science. Usually nonsmooth dynamical systems are represented as differential inclusions, complementarity systems, evolution va- ational inequalities, each of these classes itself being split into several subclasses. The book is divided into four parts, the rst three parts being sketched in Fig. 0. 1. The aim of the rst part is to present the main tools from mechanics and applied mathematics which are necessary to understand how nonsmooth dynamical systems may be numerically simulated in a reliable way. Many examples illustrate the th- retical results, and an emphasis is put on mechanical systems, as well as on electrical circuits (the so-called Filippov’s systems are also examined in some detail, due to their importance in control applications). The second and third parts are dedicated to a detailed presentation of the numerical schemes. A fourth part is devoted to the presentation of the software platform Siconos. This book is not a textbook on - merical analysis of nonsmooth systems, in the sense that despite the main results of numerical analysis (convergence, order of consistency, etc. ) being presented, their proofs are not provided.

Dynamical Systems Method and Applications

Author : Alexander G. Ramm
Publisher : John Wiley & Sons
ISBN 13 : 111819960X
Total Pages : 522 pages
Book Rating : 4.1/5 (181 download)

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Book Synopsis Dynamical Systems Method and Applications by : Alexander G. Ramm

Download or read book Dynamical Systems Method and Applications written by Alexander G. Ramm and published by John Wiley & Sons. This book was released on 2013-06-07 with total page 522 pages. Available in PDF, EPUB and Kindle. Book excerpt: Demonstrates the application of DSM to solve a broad range of operator equations The dynamical systems method (DSM) is a powerful computational method for solving operator equations. With this book as their guide, readers will master the application of DSM to solve a variety of linear and nonlinear problems as well as ill-posed and well-posed problems. The authors offer a clear, step-by-step, systematic development of DSM that enables readers to grasp the method's underlying logic and its numerous applications. Dynamical Systems Method and Applications begins with a general introduction and then sets forth the scope of DSM in Part One. Part Two introduces the discrepancy principle, and Part Three offers examples of numerical applications of DSM to solve a broad range of problems in science and engineering. Additional featured topics include: General nonlinear operator equations Operators satisfying a spectral assumption Newton-type methods without inversion of the derivative Numerical problems arising in applications Stable numerical differentiation Stable solution to ill-conditioned linear algebraic systems Throughout the chapters, the authors employ the use of figures and tables to help readers grasp and apply new concepts. Numerical examples offer original theoretical results based on the solution of practical problems involving ill-conditioned linear algebraic systems, and stable differentiation of noisy data. Written by internationally recognized authorities on the topic, Dynamical Systems Method and Applications is an excellent book for courses on numerical analysis, dynamical systems, operator theory, and applied mathematics at the graduate level. The book also serves as a valuable resource for professionals in the fields of mathematics, physics, and engineering.

Nonlinear Dynamics in Complex Systems

Author : Armin Fuchs
Publisher : Springer Science & Business Media
ISBN 13 : 3642335527
Total Pages : 237 pages
Book Rating : 4.6/5 (423 download)

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Book Synopsis Nonlinear Dynamics in Complex Systems by : Armin Fuchs

Download or read book Nonlinear Dynamics in Complex Systems written by Armin Fuchs and published by Springer Science & Business Media. This book was released on 2012-09-22 with total page 237 pages. Available in PDF, EPUB and Kindle. Book excerpt: With many areas of science reaching across their boundaries and becoming more and more interdisciplinary, students and researchers in these fields are confronted with techniques and tools not covered by their particular education. Especially in the life- and neurosciences quantitative models based on nonlinear dynamics and complex systems are becoming as frequently implemented as traditional statistical analysis. Unfamiliarity with the terminology and rigorous mathematics may discourage many scientists to adopt these methods for their own work, even though such reluctance in most cases is not justified. This book bridges this gap by introducing the procedures and methods used for analyzing nonlinear dynamical systems. In Part I, the concepts of fixed points, phase space, stability and transitions, among others, are discussed in great detail and implemented on the basis of example elementary systems. Part II is devoted to specific, non-trivial applications: coordination of human limb movement (Haken-Kelso-Bunz model), self-organization and pattern formation in complex systems (Synergetics), and models of dynamical properties of neurons (Hodgkin-Huxley, Fitzhugh-Nagumo and Hindmarsh-Rose). Part III may serve as a refresher and companion of some mathematical basics that have been forgotten or were not covered in basic math courses. Finally, the appendix contains an explicit derivation and basic numerical methods together with some programming examples as well as solutions to the exercises provided at the end of certain chapters. Throughout this book all derivations are as detailed and explicit as possible, and everybody with some knowledge of calculus should be able to extract meaningful guidance follow and apply the methods of nonlinear dynamics to their own work. “This book is a masterful treatment, one might even say a gift, to the interdisciplinary scientist of the future.” “With the authoritative voice of a genuine practitioner, Fuchs is a master teacher of how to handle complex dynamical systems.” “What I find beautiful in this book is its clarity, the clear definition of terms, every step explained simply and systematically.” (J.A.Scott Kelso, excerpts from the foreword)

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.

Advanced Dynamics

Author : Dan B. Marghitu
Publisher : Springer Science & Business Media
ISBN 13 : 1461434750
Total Pages : 616 pages
Book Rating : 4.4/5 (614 download)

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Book Synopsis Advanced Dynamics by : Dan B. Marghitu

Download or read book Advanced Dynamics written by Dan B. Marghitu and published by Springer Science & Business Media. This book was released on 2012-05-24 with total page 616 pages. Available in PDF, EPUB and Kindle. Book excerpt: Advanced Dynamics: Analytical and Numerical Calculations with MATLAB provides a thorough, rigorous presentation of kinematics and dynamics while using MATLAB as an integrated tool to solve problems. Topics presented are explained thoroughly and directly,allowing fundamental principles to emerge through applications from areas such as multibody systems, robotics, spacecraft and design of complex mechanical devices. This book differs from others in that it uses symbolic MATLAB for both theory and applications. Special attention is given to solutions that are solved analytically and numerically using MATLAB. The illustrations and figures generated with MATLAB reinforce visual learning while an abundance of examples offer additional support.

Energy Methods in Dynamics

Author : Khanh Chau Le
Publisher : Springer Science & Business Media
ISBN 13 : 3642224040
Total Pages : 297 pages
Book Rating : 4.6/5 (422 download)

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Book Synopsis Energy Methods in Dynamics by : Khanh Chau Le

Download or read book Energy Methods in Dynamics written by Khanh Chau Le and published by Springer Science & Business Media. This book was released on 2011-09-25 with total page 297 pages. Available in PDF, EPUB and Kindle. Book excerpt: The above examples should make clear the necessity of understanding the mechanism of vibrations and waves in order to control them in an optimal way. However vibrations and waves are governed by differential equations which require, as a rule, rather complicated mathematical methods for their analysis. The aim of this textbook is to help students acquire both a good grasp of the first principles from which the governing equations can be derived, and the adequate mathematical methods for their solving. Its distinctive features, as seen from the title, lie in the systematic and intensive use of Hamilton's variational principle and its generalizations for deriving the governing equations of conservative and dissipative mechanical systems, and also in providing the direct variational-asymptotic analysis, whenever available, of the energy and dissipation for the solution of these equations. It will be demonstrated that many well-known methods in dynamics like those of Lindstedt-Poincare, Bogoliubov-Mitropolsky, Kolmogorov-Arnold-Moser (KAM), and Whitham are derivable from this variational-asymptotic analysis. This book grew up from the lectures given by the author in the last decade at the Ruhr University Bochum, Germany. Since vibrations and waves are constituents of various disciplines (physics, mechanics, electrical engineering etc.) and cannot be handled in a single textbook, I have restricted myself mainly to vibrations and waves of mechanical nature. The material of this book can be recommended for a one year course in higher dynamics for graduate students of mechanical and civil engineering. For this circle of readers, the emphasis is made on the constructive methods of solution and not on the rigorous mathematical proofs of convergence. As compensation, various numerical simulations of the exact and approximate solutions are provided which demonstrate vividly the validity of the used methods. To help students become more proficient, each chapter ends with exercises, of which some can be solved effectively by using Mathematica.

Dynamics

Author : Helena E. Nusse
Publisher : Springer Science & Business Media
ISBN 13 : 9780387982649
Total Pages : 654 pages
Book Rating : 4.9/5 (826 download)

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Book Synopsis Dynamics by : Helena E. Nusse

Download or read book Dynamics written by Helena E. Nusse and published by Springer Science & Business Media. This book was released on 1998 with total page 654 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book, together with the accompanying computer program Dynamics 2 (included on a diskette), is suitable for the novice and the expert in dynamical systems. It helps the novice begin immediately exploring dynamical systems with a broad array of interactive techniques. The book explains basic ideas of nonlinear dynamical systems, and Dynamics 2 provides many tools developed by the Maryland Chaos group to visualize dynamical systems. Dynamics 2 can be used by undergraduates, by graduate students, and by researchers in a variety of scientific disciplines.

Numerical Methods in Fluid Dynamics

Author : M. Holt
Publisher : Springer Science & Business Media
ISBN 13 : 3642963706
Total Pages : 262 pages
Book Rating : 4.6/5 (429 download)

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Book Synopsis Numerical Methods in Fluid Dynamics by : M. Holt

Download or read book Numerical Methods in Fluid Dynamics written by M. Holt and published by Springer Science & Business Media. This book was released on 2012-12-06 with total page 262 pages. Available in PDF, EPUB and Kindle. Book excerpt: This monograph is based on a graduate course, Mechanical Engipeering 266, which was developed over a number of years at the University of California-Berkeley. Shorter versions of the course were given at the University of Paris VI in 1969, and at the University of Paris XI in 1972. The course was originally presented as the last of a three quarter sequence on Compressible Flow Theory, with emphasis on the treatment of non-linear problems by numerical techniques. This is reflected in the material of the first half of the book, covering several techniques for handling non-linear wave interaction and other problems in Gas Dynamics. The techniques have their origins in the Method of Characteristics (in both two and three dimensions). Besides reviewing the method itself the more recent techniques derived from it, firstly by Godunov and his group, and secondly by Rusanov and his co-workers, are described. Both these approaches are applicable to steady flows calculated as asymptotic states of unsteady flows and treat elliptic prob lems as limiting forms of unsteady hyperbolic problems. They are there fore applicable to low speed as well a~ to high speed flow problems. The second half of the book covers the treatment of a variety of steady flow problems, including effects of both viscosity and compressibi lity, by the Method of Integral Relations, Telenin's Method, and the Method of Lines.

The Parameterization Method for Invariant Manifolds

Author : Àlex Haro
Publisher : Springer
ISBN 13 : 3319296620
Total Pages : 280 pages
Book Rating : 4.3/5 (192 download)

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Book Synopsis The Parameterization Method for Invariant Manifolds by : Àlex Haro

Download or read book The Parameterization Method for Invariant Manifolds written by Àlex Haro and published by Springer. This book was released on 2016-04-18 with total page 280 pages. Available in PDF, EPUB and Kindle. Book excerpt: This monograph presents some theoretical and computational aspects of the parameterization method for invariant manifolds, focusing on the following contexts: invariant manifolds associated with fixed points, invariant tori in quasi-periodically forced systems, invariant tori in Hamiltonian systems and normally hyperbolic invariant manifolds. This book provides algorithms of computation and some practical details of their implementation. The methodology is illustrated with 12 detailed examples, many of them well known in the literature of numerical computation in dynamical systems. A public version of the software used for some of the examples is available online. The book is aimed at mathematicians, scientists and engineers interested in the theory and applications of computational dynamical systems.

Dynamics of Complex Interacting Systems

Author : E. Goles
Publisher : Springer Science & Business Media
ISBN 13 : 9401713235
Total Pages : 185 pages
Book Rating : 4.4/5 (17 download)

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Book Synopsis Dynamics of Complex Interacting Systems by : E. Goles

Download or read book Dynamics of Complex Interacting Systems written by E. Goles and published by Springer Science & Business Media. This book was released on 2013-04-17 with total page 185 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book contains the courses given at the Fourth School on Statistical Physics and Cooperative Systems held at Santiago, Chile, from 12th to 16th December 1994. This School brings together scientists working on subjects related to recent trends in complex systems. Some of these subjects deal with dynamical systems, ergodic theory, cellular automata, symbolic and arithmetic dynamics, spatial systems, large deviation theory and neural networks. Scientists working in these subjects come from several aeras: pure and applied mathematics, non linear physics, biology, computer science, electrical engineering and artificial intelligence. Each contribution is devoted to one or more of the previous subjects. In most cases they are structured as surveys, presenting at the same time an original point of view about the topic and showing mostly new results. The expository text of Roberto Livi concerns the study of coupled map lattices (CML) as models of spatially extended dynamical systems. CML is one of the most used tools for the investigation of spatially extended systems. The paper emphasizes rigorous results about the dynamical behavior of one dimensional CML; i.e. a uniform real local function defined in the interval [0,1], interacting with its nearest neighbors in a one dimensional lattice.

Applied Symbolic Dynamics And Chaos

Author : Bailin Hao
Publisher : World Scientific
ISBN 13 : 9814495972
Total Pages : 460 pages
Book Rating : 4.8/5 (144 download)

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Book Synopsis Applied Symbolic Dynamics And Chaos by : Bailin Hao

Download or read book Applied Symbolic Dynamics And Chaos written by Bailin Hao and published by World Scientific. This book was released on 1998-07-04 with total page 460 pages. Available in PDF, EPUB and Kindle. Book excerpt: Latest Edition: Applied Symbolic Dynamics and Chaos (2nd Edition)Symbolic dynamics is a coarse-grained description of dynamics. It provides a rigorous way to understand the global systematics of periodic and chaotic motion in a system. In the last decade it has been applied to nonlinear systems described by one- and two-dimensional maps as well as by ordinary differential equations. This book will help practitioners in nonlinear science and engineering to master that powerful tool.

Dynamical Systems

Author : Luis Barreira
Publisher : Springer Science & Business Media
ISBN 13 : 1447148355
Total Pages : 214 pages
Book Rating : 4.4/5 (471 download)

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Book Synopsis Dynamical Systems by : Luis Barreira

Download or read book Dynamical Systems written by Luis Barreira and published by Springer Science & Business Media. This book was released on 2012-12-02 with total page 214 pages. Available in PDF, EPUB and Kindle. Book excerpt: The theory of dynamical systems is a broad and active research subject with connections to most parts of mathematics. Dynamical Systems: An Introduction undertakes the difficult task to provide a self-contained and compact introduction. Topics covered include topological, low-dimensional, hyperbolic and symbolic dynamics, as well as a brief introduction to ergodic theory. In particular, the authors consider topological recurrence, topological entropy, homeomorphisms and diffeomorphisms of the circle, Sharkovski's ordering, the Poincaré-Bendixson theory, and the construction of stable manifolds, as well as an introduction to geodesic flows and the study of hyperbolicity (the latter is often absent in a first introduction). Moreover, the authors introduce the basics of symbolic dynamics, the construction of symbolic codings, invariant measures, Poincaré's recurrence theorem and Birkhoff's ergodic theorem. The exposition is mathematically rigorous, concise and direct: all statements (except for some results from other areas) are proven. At the same time, the text illustrates the theory with many examples and 140 exercises of variable levels of difficulty. The only prerequisites are a background in linear algebra, analysis and elementary topology. This is a textbook primarily designed for a one-semester or two-semesters course at the advanced undergraduate or beginning graduate levels. It can also be used for self-study and as a starting point for more advanced topics.