Download or read book Newton s Method and Dynamical Systems written by H.-O. Peitgen and published by Springer Science & Business Media. This book was released on 2012-12-06 with total page 227 pages. Available in PDF, EPUB and Kindle. Book excerpt:

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.

Download or read book Newton s Method as a Dynamical System Global Convergence and Predictability written by R. G. Holt and published by . This book was released on 1985 with total page 16 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.

Download or read book Newton s Method Applied to Two Quadratic Equations in C2 Viewed as a Global Dynamical System written by John Hamal Hubbard and published by . This book was released on 2008 with total page 160 pages. Available in PDF, EPUB and Kindle. Book excerpt: Studies the Newton map $N: \mathbb{C} DEGREES2\rightarrow\mathbb{C} DEGREES2$ associated to two equations in two unknowns, as a dynamical system. This title focuses on the first non-trivial case: two simultaneous quadratics, to intersect two conics. It proves among other things: the Russakovksi-Shiffman measure does not change the points of

Download or read book Newton s Method Applied to Two Quadratic Equations in mathbb C 2 Viewed as a Global Dynamical System written by John H. Hubbard and published by American Mathematical Soc.. This book was released on 2008 with total page 160 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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: The Russakovksi-Shiffman measure does not change the points of indeterminancy. The lines joining pairs of roots are invariant, and the Julia set of the restriction of $N$ to such a line has under appropriate circumstances an invariant manifold, which shares features of a stable manifold and a center manifold. The main part of the article concerns the behavior of $N$ at infinity. To compactify $\mathbb{C}^2$ in such a way that $N$ extends to the compactification, the authors must take the projective limit of an infinite sequence of blow-ups. The simultaneous presence of points of indeterminancy and of critical curves forces the authors to define a new kind of blow-up: the Farey blow-up. This construction is studied in its own right in chapter 4, where they show among others that the real oriented blow-up of the Farey blow-up has a topological structure reminiscent of the invariant tori of the KAM theorem. They also show that the cohomology, completed under the intersection inner product, is naturally isomorphic to the classical Sobolev space of functions with square-integrable derivatives. In chapter 5 the authors apply these results to the mapping $N$ in a particular case, which they generalize in chapter 6 to the intersection of any two conics.

Download or read book Solving Nonlinear Equations with Newton s Method written by C. T. Kelley and published by SIAM. This book was released on 2003-01-01 with total page 117 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book on Newton's method is a user-oriented guide to algorithms and implementation. In just over 100 pages, it shows, via algorithms in pseudocode, in MATLAB, and with several examples, how one can choose an appropriate Newton-type method for a given problem, diagnose problems, and write an efficient solver or apply one written by others. It contains trouble-shooting guides to the major algorithms, their most common failure modes, and the likely causes of failure. It also includes many worked-out examples (available on the SIAM website) in pseudocode and a collection of MATLAB codes, allowing readers to experiment with the algorithms easily and implement them in other languages.

Download or read book Newton s Method as a Dynamical System written by Johannes Rückert and published by . This book was released on 2006 with total page 112 pages. Available in PDF, EPUB and Kindle. Book excerpt: We study transcendental and rational mappings that arise as Newton maps of entire functions. Our first result is that "in between" any two accesses to infinity of an immediate basin, a Newton map exhibits either another immediate basin, a virtual immediate basin or infinitely many preimages of some point. This result is joint work with Dierk Schleicher and allows to locate virtual immediate basins. An important corollary is a proof of the folklore result that for Newton maps of polynomials, every complementary component of an immediate basin contains another immediate basin. Our second main result, which is joint work with Xavier Buff, shows an interesting connection between virtual immediate basins of the Newton map N_f and asymptotic values of f, answering a 2003 question of Douady: in many cases, 0 is an asymptotic value of f if N_f has a virtual immediate basin. Conversely, if f has an asymptotic value of logarithmic type at 0, then N_f has a virtual immediate basin. We show by way of counterexamples that this is not true for other types of asymptotic values. Our third main result gives a combinatorial classification of a class of Newton maps of polynomials. Let N be the Newton map of a polynomial such that all critical points of N land on a fixed point after finitely many iterations. In this case, we construct a graph that characterizes N uniquely up to Möbius conjugation. Conversely, we show that every graph with an associated map that satisfies several natural conditions is realized by a unique Newton map. In an appendix, we introduce a class of bounded type transcendental entire functions with the property that its set of escaping points is organized in the form of unbounded rays. This fourth main result is joint work with Günter Rottenfußer, Lasse Rempe and Dierk Schleicher, and is part of an answer to a long-standing conjecture of Fatou and Eremenko.

Download or read book Newton s Method Applied to Two Quadratic Equations in C2 Viewed as a Global Dynamical System written by John H. Hubbard and published by American Mathematical Soc.. This book was released on 2008 with total page 162 pages. Available in PDF, EPUB and Kindle. Book excerpt: Introduction Fundamental properties of Newton maps Invariant 3-manifolds associated to invariant circles The behavior at infinity when $a=b=0$ The Farey blow-up The compactification when $a=b=0$ The case where $a$ and $b$ are arbitrary Bibliography

Download or read book A First Course In Chaotic Dynamical Systems written by Robert L. Devaney and published by CRC Press. This book was released on 2018-05-04 with total page 386 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.

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.

Download or read book Dynamical Systems and Fractals written by Karl-Heinz Becker and published by Cambridge University Press. This book was released on 1989-10-26 with total page 420 pages. Available in PDF, EPUB and Kindle. Book excerpt: This 1989 book is about chaos, fractals and complex dynamics.

Download or read book An Introduction To Chaotic Dynamical Systems written by Robert Devaney and published by CRC Press. This book was released on 2018-03-09 with total page 360 pages. Available in PDF, EPUB and Kindle. Book excerpt: The study of nonlinear dynamical systems has exploded in the past 25 years, and Robert L. Devaney has made these advanced research developments accessible to undergraduate and graduate mathematics students as well as researchers in other disciplines with the introduction of this widely praised book. In this second edition of his best-selling text, Devaney includes new material on the orbit diagram fro maps of the interval and the Mandelbrot set, as well as striking color photos illustrating both Julia and Mandelbrot sets. This book assumes no prior acquaintance with advanced mathematical topics such as measure theory, topology, and differential geometry. Assuming only a knowledge of calculus, Devaney introduces many of the basic concepts of modern dynamical systems theory and leads the reader to the point of current research in several areas.

Download or read book Complex Dynamical Systems written by Robert L. Devaney and published by American Mathematical Soc.. This book was released on 1994-12-20 with total page 232 pages. Available in PDF, EPUB and Kindle. Book excerpt: In the last fifteen years, the Mandelbrot set has emerged as one of the most recognizable objects in mathematics. While there is no question of its beauty, relatively few people appreciate the fact that the mathematics behind such images is equally beautiful. This book presents lectures delivered during the AMS Short Course entitled ``Complex Dynamical Systems: The Mathematics Behind the Mandelbrot and Julia Sets'', held at the Joint Mathematics Meetings in Cincinnati in January 1994. The lectures cover a wide range of topics, including the classical work of Julia and Fatou on local dynamics of analytic maps as well as recent work on the dynamics of quadratic and cubic polynomials, the geometry of Julia sets, and the structure of various parameter spaces. Among the other topics are recent results on Yoccoz puzzles and tableaux, limiting dynamics near parabolic points, the spider algorithm, extensions of the theory to rational maps, Newton's method, and entire transcendental functions. Much of the book is accessible to anyone with a background in the basics of dynamical systems and complex analysis.

Download or read book An Introduction To Chaotic Dynamical Systems written by Robert L. Devaney and published by CRC Press. This book was released on 2021-11-28 with total page 434 pages. Available in PDF, EPUB and Kindle. Book excerpt: There is an explosion of interest in dynamical systems in the mathematical community as well as in many areas of science. The results have been truly exciting: systems which once seemed completely intractable from an analytic point of view can now be understood in a geometric or qualitative sense rather easily. Scientists and engineers realize the power and the beauty of the geometric and qualitative techniques. These techniques apply to a number of important nonlinear problems ranging from physics and chemistry to ecology and economics. Computer graphics have allowed us to view the dynamical behavior geometrically. The appearance of incredibly beautiful and intricate objects such as the Mandelbrot set, the Julia set, and other fractals have really piqued interest in the field. This is text is aimed primarily at advanced undergraduate and beginning graduate students. Throughout, the author emphasizes the mathematical aspects of the theory of discrete dynamical systems, not the many and diverse applications of this theory. The field of dynamical systems and especially the study of chaotic systems has been hailed as one of the important breakthroughs in science in the past century and its importance continues to expand. There is no question that the field is becoming more and more important in a variety of scientific disciplines. New to this edition: •Greatly expanded coverage complex dynamics now in Chapter 2 •The third chapter is now devoted to higher dimensional dynamical systems. •Chapters 2 and 3 are independent of one another. •New exercises have been added throughout.

Download or read book Differential Equations A Dynamical Systems Approach written by John H. Hubbard and published by Springer Science & Business Media. This book was released on 1997-10-17 with total page 374 pages. Available in PDF, EPUB and Kindle. Book excerpt: This corrected third printing retains the authors'main emphasis on ordinary differential equations. It is most appropriate for upper level undergraduate and graduate students in the fields of mathematics, engineering, and applied mathematics, as well as the life sciences, physics and economics. The authors have taken the view that a differential equations theory defines functions; the object of the theory is to understand the behaviour of these functions. The tools the authors use include qualitative and numerical methods besides the traditional analytic methods, and the companion software, MacMath, is designed to bring these notions to life.

Download or read book Dynamical Systems written by Shlomo Sternberg and published by Courier Corporation. This book was released on 2014-06-10 with total page 276 pages. Available in PDF, EPUB and Kindle. Book excerpt: A pioneer in the field of dynamical systems discusses one-dimensional dynamics, differential equations, random walks, iterated function systems, symbolic dynamics, and Markov chains. Supplementary materials include PowerPoint slides and MATLAB exercises. 2010 edition.

Download or read book Convergence and Applications of Newton type Iterations written by Ioannis K. Argyros and published by Springer Science & Business Media. This book was released on 2008-06-12 with total page 513 pages. Available in PDF, EPUB and Kindle. Book excerpt: This monograph is devoted to a comprehensive treatment of iterative methods for solving nonlinear equations with particular emphasis on semi-local convergence analysis. Theoretical results are applied to engineering, dynamic economic systems, input-output systems, nonlinear and linear differential equations, and optimization problems. Accompanied by many exercises, some with solutions, the book may be used as a supplementary text in the classroom for an advanced course on numerical functional analysis.