Newton s Method and Dynamical Systems

Newton   s Method and Dynamical Systems

Newton's method has recently become one of the paradigms in the revival of Julia set theory and complex dynamical systems. This paper, to a large extent experimental in nature, investigates Newton's method for some particular model ...

Author: H.-O. Peitgen

Publisher: Springer Science & Business Media

ISBN: 9789400922815

Category: Science

Page: 226

View: 370

Categories: Science

Numerical Continuation Methods for Dynamical Systems

Numerical Continuation Methods for Dynamical Systems

Then the matrix (o) (; ;) in Newton's method is a “bordered tridiagonal' matrix of the form We now show how to solve such linear systems efficiently. 1.2.4 The Bordering Algorithm The linear systems in Newton's method for ...

Author: Bernd Krauskopf

Publisher: Springer

ISBN: 9781402063565

Category: Science

Page: 399

View: 219

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.
Categories: Science

A First Course In Chaotic Dynamical Systems

A First Course In Chaotic Dynamical Systems

Recall from Chapter 13 that Newton's method is an iterative procedure to solve certain equations. Given a function F, to find the solutions of F(x) = 0, we simply choose an initial guess x0 and then compute the orbit of x0 ...

Author: Robert L. L. Devaney

Publisher: CRC Press

ISBN: 9781000065657

Category: Mathematics

Page: 318

View: 365

A First Course in Chaotic Dynamical Systems: Theory and Experiment, Second Edition The long-anticipated revision of this well-liked textbook offers many new additions. In the twenty-five years since the original version of this book was published, much has happened in dynamical systems. Mandelbrot and Julia sets were barely ten years old when the first edition appeared, and most of the research involving these objects then centered around iterations of quadratic functions. This research has expanded to include all sorts of different types of functions, including higher-degree polynomials, rational maps, exponential and trigonometric functions, and many others. Several new sections in this edition are devoted to these topics. The area of dynamical systems covered in A First Course in Chaotic Dynamical Systems: Theory and Experiment, Second Edition is quite accessible to students and also offers a wide variety of interesting open questions for students at the undergraduate level to pursue. The only prerequisite for students is a one-year calculus course (no differential equations required); students will easily be exposed to many interesting areas of current research. This course can also serve as a bridge between the low-level, often non-rigorous calculus courses, and the more demanding higher-level mathematics courses. Features More extensive coverage of fractals, including objects like the Sierpinski carpet and others that appear as Julia sets in the later sections on complex dynamics, as well as an actual chaos "game." More detailed coverage of complex dynamical systems like the quadratic family and the exponential maps. New sections on other complex dynamical systems like rational maps. A number of new and expanded computer experiments for students to perform. About the Author Robert L. Devaney is currently professor of mathematics at Boston University. He received his PhD from the University of California at Berkeley under the direction of Stephen Smale. He taught at Northwestern University and Tufts University before coming to Boston University in 1980. His main area of research is dynamical systems, primarily complex analytic dynamics, but also including more general ideas about chaotic dynamical systems. Lately, he has become intrigued with the incredibly rich topological aspects of dynamics, including such things as indecomposable continua, Sierpinski curves, and Cantor bouquets.
Categories: Mathematics

MacMath 9 2

MacMath 9  2

We have developed this text and software for a junior-senior level course in Applicable Mathematics at Cornell University, in order to take advantage of the new qualitative and geometric insights made possible by the advent of excellent and ...

Author: John H. Hubbard

Publisher: Springer-Verlag

ISBN: 9783662253687

Category: Mathematics

Page: 177

View: 284

MacMath is a scientific toolkit for the Macintosh TM computer developed by John H. Hubbard and Beverly H. West, consisting of twelve graphics programs. It supports mathematical computation and experimentation in dynamical systems, both for differential equations and for iteration. The MacMath package was designed to accompany the textbook Differential Equations: A Dynamical Sys tems Approach, also by J. H. Hubbard and B. H. West (Part I, One Dimensional Equations, 1990; Part II, Higher Dimensional Systems, 1991; Springer-Verlag). We have developed this text and software for a junior-senior level course in Applicable Mathematics at Cornell University, in order to take advantage of the new qualitative and geometric insights made possible by the advent of excellent and easily accessible graphics. Our primary reasons are two: 1. A picture is worth a thousand words. Graphics are far more than just a luxury - the human brain is made to process visual information; more information can be assimilated in a few seconds of looking at a graphics output than in months of analyzing a tabulated computer printout, perhaps a centimeter thick, carrying the same numerical information. 2. From qualitative analysis we can obtain excellent quantitative information.
Categories: Mathematics

Newton s Method as a Dynamical System Global Convergence and Predictability

Newton s Method as a Dynamical System  Global Convergence and Predictability

Newton's method as an iterative scheme to compute both unstable and stable fixed points of a discrete dynamical system is considered. It is shown for Newton iterations that the basins of attraction are intertwined in a complicated manner.

Author: R. G. Holt

Publisher:

ISBN: OCLC:227647148

Category:

Page: 16

View: 113

Newton's method as an iterative scheme to compute both unstable and stable fixed points of a discrete dynamical system is considered. It is shown for Newton iterations that the basins of attraction are intertwined in a complicated manner. This complex structure appears to be fractal, and its dimension is estimated. Consequences of predictability for the final state are given in terms of imprecision in the initial data. Keywords include: Newton's method, Predictability, Basin boundaries, Fractal, Nonlinear dynamic.
Categories:

A First Course In Chaotic Dynamical Systems

A First Course In Chaotic Dynamical Systems

A First Course in Chaotic Dynamical Systems: Theory and Experiment is the first book to introduce modern topics in dynamical systems at the undergraduate level.

Author: Robert L. Devaney

Publisher: CRC Press

ISBN: 9780429983115

Category: Mathematics

Page: 340

View: 809

A First Course in Chaotic Dynamical Systems: Theory and Experiment is the first book to introduce modern topics in dynamical systems at the undergraduate level. Accessible to readers with only a background in calculus, the book integrates both theory and computer experiments into its coverage of contemporary ideas in dynamics. It is designed as a gradual introduction to the basic mathematical ideas behind such topics as chaos, fractals, Newton's method, symbolic dynamics, the Julia set, and the Mandelbrot set, and includes biographies of some of the leading researchers in the field of dynamical systems. Mathematical and computer experiments are integrated throughout the text to help illustrate the meaning of the theorems presented. Chaotic Dynamical Systems Software, Labs 1-6 is a supplementary labouratory software package, available separately, that allows a more intuitive understanding of the mathematics behind dynamical systems theory. Combined with A First Course in Chaotic Dynamical Systems , it leads to a rich understanding of this emerging field.
Categories: Mathematics

Simulating Analyzing and Animating Dynamical Systems

Simulating  Analyzing  and Animating Dynamical Systems

The way to solve these is to use Newton's method which we write as an iteration: An+1 || | | \n 3x3 – 3y” —6xy —1 x * – 3xy” – 1 | |||—||—|| 6xy to | 3x”y – y” | Here (xn, yn) is the nth approximation to the root.

Author: Bard Ermentrout

Publisher: SIAM

ISBN: 9780898715064

Category: Mathematics

Page: 290

View: 584

Simulating, Analyzing, and Animating Dynamical Systems: A Guide to XPPAUT for Researchers and Students provides sophisticated numerical methods for the fast and accurate solution of a variety of equations, including ordinary differential equations, delay equations, integral equations, functional equations, and some partial differential equations, as well as boundary value problems. It introduces many modeling techniques and methods for analyzing the resulting equations. Instructors, students, and researchers will all benefit from this book, which demonstrates how to use software tools to simulate and study sets of equations that arise in a variety of applications. Instructors will learn how to use computer software in their differential equations and modeling classes, while students will learn how to create animations of their equations that can be displayed on the World Wide Web. Researchers will be introduced to useful tricks that will allow them to take full advantage of XPPAUT's capabilities.
Categories: Mathematics

Numerical Methods for Nonsmooth Dynamical Systems

Numerical Methods for Nonsmooth Dynamical Systems

This book concerns the numerical simulation of dynamical systems whose trajec- ries may not be differentiable everywhere.

Author: Vincent Acary

Publisher: Springer Science & Business Media

ISBN: 3540753923

Category: Technology & Engineering

Page: 525

View: 180

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.
Categories: Technology & Engineering

Numerical Methods for Nonlinear Estimating Equations

Numerical Methods for Nonlinear Estimating Equations

In this section we study the theoretical aspects of these algorithms from the viewpoint of dynamical systems . We begin with a brief review of Newton's method by regarding this method as a discrete time dynamical system .

Author: Christopher G. Small

Publisher: Oxford University Press on Demand

ISBN: 0198506880

Category: Mathematics

Page: 309

View: 123

This book provides a comprehensive study of nonlinear estimating equations and artificial likelihoods for statistical inference. It includes a variety of examples from practical applications and is ideal for research statisticians and advanced graduate students.
Categories: Mathematics

Newton s Method Applied to Two Quadratic Equations in C2 Viewed as a Global Dynamical System

Newton s Method Applied to Two Quadratic Equations in C2 Viewed as a Global Dynamical System

The authors study the Newton map $N:\mathbb{C}^2\rightarrow\mathbb{C}^2$ associated to two equations in two unknowns, as a dynamical system.

Author: John H. Hubbard

Publisher: American Mathematical Soc.

ISBN: 9780821840566

Category: Mathematics

Page: 146

View: 680

The authors study the Newton map $N:\mathbb{C 2\rightarrow\mathbb{C 2$ associated to two equations in two unknowns, as a dynamical system. They focus on the first non-trivial case: two simultaneous quadratics, to intersect two conics. In the first two chapters, the authors prove among other things:
Categories: Mathematics

Dynamical Systems

Dynamical Systems

We solve the nonlinear system of equations using Newton's method . Each Newton step results in a linear system of equations to be solved . Because we typically consider L with many points , indirect methods will be used to solve the ...

Author:

Publisher: Springer Science & Business Media

ISBN: 3540407863

Category:

Page:

View: 675

Categories:

A First Course in Discrete Dynamical Systems

A First Course in Discrete Dynamical Systems

Can you determine limn-2 h"(.2345)? Describe Newton's method for approximating the zeros of a function as a dynamical system. (You should be able to find Newton's method in your calculus textbook.) What is the significance of limn-co ...

Author: Richard A. Holmgren

Publisher: Springer Science & Business Media

ISBN: 9781441987327

Category: Mathematics

Page: 223

View: 402

Given the ease with which computers can do iteration it is now possible for almost anyone to generate beautiful images whose roots lie in discrete dynamical systems. Images of Mandelbrot and Julia sets abound in publications both mathematical and not. The mathematics behind the pictures are beautiful in their own right and are the subject of this text. Mathematica programs that illustrate the dynamics are included in an appendix.
Categories: Mathematics

MacMath 9 2

MacMath 9 2

Qualitative analysis of the picture leads to quantitative results and even to new mathematics. This new edition includes the latest version of the Mac Math diskette, 9.2.

Author: John H. Hubbard

Publisher: Springer

ISBN: 9781461383789

Category: Science

Page: 162

View: 613

MacMath is a scientific toolkit for the Macintosh computer consisting of twelve graphics programs. It supports mathematical computation and experimentation in dynamical systems, both for differential equations and for iteration. The MacMath package was designed to accompany the textbooks Differential Equations: A Dynamical Systems Approach Part I & II. The text and software was developed for a junior-senior level course in applicable mathematics at Cornell University, in order to take advantage of excellent and easily accessible graphics. MacMath addresses differential equations and iteration such as: analyzer, diffeq, phase plane, diffeq 3D views, numerical methods, periodic differential equations, cascade, 2D iteration, eigenfinder, jacobidraw, fourier, planets. These versatile programs greatly enhance the understanding of the mathematics in these topics. Qualitative analysis of the picture leads to quantitative results and even to new mathematics. This new edition includes the latest version of the Mac Math diskette, 9.2.
Categories: Science

Chaos and Fractals

Chaos and Fractals

5A recent collection of papers discussing Newton's method as dynamical systems is in Newton's Method and Dynamical Systems, H.-O. Peitgen (ed.), Kluver Academic Publishers, Dordrecht, 1989. See also H.-O. Peitgen P. H. Richter.

Author: Heinz-Otto Peitgen

Publisher: Springer Science & Business Media

ISBN: 9780387202297

Category: Mathematics

Page: 864

View: 829

The fourteen chapters of this book cover the central ideas and concepts of chaos and fractals as well as many related topics including: the Mandelbrot set, Julia sets, cellular automata, L-systems, percolation and strange attractors. This new edition has been thoroughly revised throughout. The appendices of the original edition were taken out since more recent publications cover this material in more depth. Instead of the focussed computer programs in BASIC, the authors provide 10 interactive JAVA-applets for this second edition.
Categories: Mathematics

Modelling and Parameter Estimation of Dynamic Systems

Modelling and Parameter Estimation of Dynamic Systems

... Riccati equation 71,322 maximum likelihood estimation for dynamic system 42–5 efficiency 42 optimisation methods ... 3 modified Gauss-Newton optimisation 106 modified Newton-Raphson method see Gauss-Newton method Monte-Carlo method ...

Author: J.R. Raol

Publisher: IET

ISBN: 9780863413636

Category: Mathematics

Page: 388

View: 419

This book presents a detailed examination of the estimation techniques and problems in dynamic systems. Containing several illustrations and computer programs, the book promotes a better understanding of system modelling and parameter estimation. Parameter estimation involves observation of a dynamic system to develop mathematical models that represent the system dynamics. With the increasing use of high speed digital computers, elegant and innovative techniques like filter error method, H° and artificial neural networks are finding more and more use in parameter estimation problems. The material is presented in an accessible manner and enables the user to implement and execute the programs and, therefore, gain first-hand experience of the estimation progress.
Categories: Mathematics