Author: Jan Bouwe van den BergPublish On: 2018-07-12

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.

Author: Jan Bouwe van den Berg

Publisher: American Mathematical Soc.

ISBN: 9781470428143

Category: Nonlinear mechanics

Page: 224

View: 179

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.

Sander, E., Wanner, T.: Validated saddle-node bifurcations and applications to
lattice dynamical systems. ... 2(1), 53–117 (2002) van den Berg, J.B.: Introduction
to rigorous numerics in dynamics: general functional analytic setup and an ...

Author: Mitsuhiro T. Nakao

Publisher: Springer Nature

ISBN: 9789811376696

Category: Mathematics

Page: 467

View: 656

In the last decades, various mathematical problems have been solved by computer-assisted proofs, among them the Kepler conjecture, the existence of chaos, the existence of the Lorenz attractor, the famous four-color problem, and more. In many cases, computer-assisted proofs have the remarkable advantage (compared with a “theoretical” proof) of additionally providing accurate quantitative information. The authors have been working more than a quarter century to establish methods for the verified computation of solutions for partial differential equations, mainly for nonlinear elliptic problems of the form -∆u=f(x,u,∇u) with Dirichlet boundary conditions. Here, by “verified computation” is meant a computer-assisted numerical approach for proving the existence of a solution in a close and explicit neighborhood of an approximate solution. The quantitative information provided by these techniques is also significant from the viewpoint of a posteriori error estimates for approximate solutions of the concerned partial differential equations in a mathematically rigorous sense. In this monograph, the authors give a detailed description of the verified computations and computer-assisted proofs for partial differential equations that they developed. In Part I, the methods mainly studied by the authors Nakao and Watanabe are presented. These methods are based on a finite dimensional projection and constructive a priori error estimates for finite element approximations of the Poisson equation. In Part II, the computer-assisted approaches via eigenvalue bounds developed by the author Plum are explained in detail. The main task of this method consists of establishing eigenvalue bounds for the linearization of the corresponding nonlinear problem at the computed approximate solution. Some brief remarks on other approaches are also given in Part III. Each method in Parts I and II is accompanied by appropriate numerical examples that confirm the actual usefulness of the authors’ methods. Also in some examples practical computer algorithms are supplied so that readers can easily implement the verification programs by themselves.

Author: Cornell University. Dept. of MathematicsPublish On: 2000

Towards a rigorous numerical study of the Kot - Schaffer The Kneser - Poulsen
conjecture for spherical polytopes ( with model , Dynamic Systems and
Applications 12 no . 1-2 K. Bezdek ) , submitted . ( 2003 ) , 87–98 . Straightening
polygonal ...

Another advantage is that one does not need to investigate the dynamics in the
central direction , which from the point of view of rigorous numerics is very
awkward to handle . The paper is organized as follows . In the second section we
give ...

... nonlinear particle dynamics via the Taylor series representation of maps .
Historically , the treatment of functions in numerics has been done based on the
treatment of numbers ; and as a result , virtually all classical numerical algorithms
are ...

This handbook is volume II in a series collecting mathematical state-of-the-art surveys in the field of dynamical systems.

Author: B. Fiedler

Publisher: Gulf Professional Publishing

ISBN: 0080532845

Category: Science

Page: 1098

View: 224

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 name just a few, are ubiquitous dynamical concepts throughout the articles.

A rigorous numerical method for the global analysis of infinite dimensional
discrete dynamical systems , In progress . ( 2 ] J . K . Hale and G . Raugel .
Regularity , determining modes and Galerkin methods . CDSNS archive , 2002 . (
3 ) Mark ...

Author: Society for Industrial and Applied MathematicsPublish On: 2003-10

S.-N. CHOW AND K. J. PALMER ( 1991 ) , On the numerical computation of orbits
of dynamical systems : The one ... 35-43 . B. A. COOMES , H. KOÇAK , AND K. J.
PALMER ( 1995a ) , Rigorous computational shadowing of orbits of ordinary ...

Author: Society for Industrial and Applied Mathematics

Nonlinear Dynamics and CFD When we try to use numerical methods to gain
insight into the fluid physics ... of realistic flows and configurations further ,
dependence on the numerics takes over even though rigorous analysis lags
behind .

This book summarizes and highlights progress in Dynamical Systems achieved during six years of the German Priority Research Program "Ergotic Theory, Analysis, and Efficient Simulation of Dynamical Systems", funded by the Deutsche ...

Author: Collectif

Publisher: Springer Science & Business Media

ISBN: 3540412905

Category: Mathematics

Page: 820

View: 578

This book summarizes and highlights progress in Dynamical Systems achieved during six years of the German Priority Research Program "Ergotic Theory, Analysis, and Efficient Simulation of Dynamical Systems", funded by the Deutsche Forschungsgemeinschaft (DFG). The three fundamental topics of large time behavior, dimension, and measure are tackled with by a rich circle of uncompromisingly rigorous mathematical concepts. The range of applied issues comprises such diverse areas as crystallization and dendrite growth, the dynamo effect, efficient simulation of biomolecules, fluid dynamics and reacting flows, mechanical problems involving friction, population biology, the spread of infectious diseases, and quantum chaos. The surveys in the book are addressed to experts and non-experts in the mathematical community alike. In addition they intend to convey the significance of the results for applications fair into the neighboring disciplines of Science.

behavior , and a more rigorous , numeric solution of Fresnel ' s equations was
actually used for data analysis20 . However , equation ( 3 . 3 ) provides us with
the general dependence of the SPR data on various system parameters and will
be ...

However, in the case that the precondition can't be satisfied, numerical
computing becomes an important approach to research such a system, thus an
accurate description of the dynamic system is mainly dependent on a rigorous numerical ...

Author: Prasad Yarlagadda

Publisher: Trans Tech Publications Ltd

ISBN: 9783038260752

Category: Technology & Engineering

Page: 3300

View: 450

This volume records the accepted papers of 2013 International Conference on Mechatronics and Industrial Informatics (ICMII 2013) which took place in Guangzhou, China between 30-31 March 2013. Volume is indexed by Thomson Reuters CPCI-S (WoS). The papers are grouped as follows: Chapter 1: Theory of Mechanisms and Mechanical Engineering, Dynamics of System Applications; Chapter 2: Materials Research, Manufacturing Technologies in Materials; Chapter 3: Electronics and Microelectronics Technology; Chapter 4: Optoelectronic Devices and Technology; Chapter 5: Sensors and Information Fusion Technology; Chapter 6: Measurement Technology and Instruments; Chapter 7: Modeling and Simulation Technology of Systems; Chapter 8: Voice, Image and Video Processing; Chapter 9: Signal Processing Systems Design and Implementation; Chapter 10: Power Engineering and Automation; Chapter 11: Industrial Robotics and Automation; Chapter 12: Vehicle Control Systems; Chapter 13: Design and Control in Modern System Engineering and Mechatronics; Chapter 14: Intelligent Control, Structural Engineering Analysis, CAD Optimized Design; Chapter 15: Artificial Intelligence Techniques; Chapter 16: Intelligent Optimization Algorithms and Applications; Chapter 17: Computer Information Processing Technology; Chapter 18: Industrial Informatics and Applications; Chapter 19: Database System; Chapter 20: Information Security; Chapter 21: Computer Networks and Communication; Chapter 22: Software Engineering; Chapter 23: E-Commerce/E-Government; Chapter 24: Engineering Management and Engineering Education

Rigorous Numerics of Chaotic Dynamical Systems * Marian Mrozek Instytut
Informatyki , Uniwersytet Jagielloński , PL ... 1 Introduction The importance of
computers in the present day research on dynamical systems cannot be
questioned .

Author: Piotr Garbaczewski

Publisher: Springer

ISBN: UCAL:B4423451

Category: Computers

Page: 571

View: 300

The study of chaotic behaviour of dynamical systems has triggered new efforts to reconcile deterministic and stochastic processes as well as classical and quantum physics. New efforts are made to understand complex and unpredictable behaviour. The papers collected in this volume give a broad overview of these activities. Readers will get a glimpse of the growing importance of Lévy processes for physics. They will find new views on fundamental concepts of quantum physics and will see many applications of chaotic and essentially random phenomena to a number of physical problems.

This book provides algorithms of computation and some practical details of their implementation.

Author: Àlex Haro

Publisher: Springer

ISBN: 9783319296623

Category: Mathematics

Page: 267

View: 938

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.

Recently , Stoffer and Palmer ( 15 ) gave a computer - assisted proof of the
existence of the horseshoe dynamics for the 25th iterate ... The number of cycles
with small period [ 3,4 ] and non - rigorous numerics [ 13 ] suggest that H ( h ) =
0.465 .

Based on the Proceedings of a Conference on Dynamics of Numerics and Numerics of Dynamics Organized by the ... IMA Journal of Numerical Analysis will
be devoted to submitted papers , that have undergone the usual rigorous
refereeing ...

Author: David S. Broomhead

Publisher: Oxford University Press, USA

ISBN: UOM:39015025259113

Category: Social Science

Page: 258

View: 284

This collection of conference papers presents the applications of dynamical systems in numerical analysis and of numerical problems and techniques in dynamical systems.

In a strict sense , neither functions ( for example , C® ) nor numbers ( for example
, the reals R ) can be treated on a ... one floating point number , but rather by an
interval of floating point numbers providing a rigorous upper and lower bound .

Author: Parsa

Publisher: Amer Inst of Physics

ISBN: STANFORD:36105020820952

Category: Science

Page: 237

View: 929

The book reports on the third of three symposia hosted by the Institute for Theoretical Physics and supported by its sponsor, the National Science Foundation. The work deals with some of the fundamental theoretical problems of accelerator physics as discussed by leaders from accelerator and mathematics communities, together with those from other fields of physics. The focus was on nonlinear dynamics and beam stability. This volume begins with some defining talks on relevant mathematical topics such as single-particle Hamiltonian dynamics, chaos, and new ideas in symplectic integrators. The physics topics included single-particle and many-particle dynamics as they relate to circular accelerators in which particles circulate for a very large number of turns. These concepts were also applied to linear accelerators, where space charge and wakefields induced in accelerating cavities play a strong role.

It provides a geometrically natural algorithm that converges regardless of the
restricted dynamics . ... MR2275297 ( 2008h : 37086 ) 37M20 35B32 35K35
35K55 35Q55 37B30 65F35 Miller , Ulrich Rigorous numerics using Conley
index theory ...

Shub , M . ( 1987 ) : Global Stability of Dynamical Systems . Springer , Berlin ...
Random Comput . Dynamics ( submitted ) THE CONLEY INDEX AND RIGOROUS NUMERICS M . Mrozek Jagiellonian Hyperbolic Structures in ODE '
s 173.

Author: F. Zanolin

Publisher: Springer

ISBN: UOM:39015049315784

Category: Mathematics

Page: 209

View: 928

The area covered by this volume represents a broad choice of some interesting research topics in the field of dynamical systems and applications of nonlinear analysis to ordinary and partial differential equations. The contributed papers, written by well known specialists, make this volume a useful tool both for the experts (who can find recent and new results) and for those who are interested in starting a research work in one of these topics (who can find some updated and carefully presented papers on the state of the art of the corresponding subject).