Differential Dynamical Systems, Revised Edition

Author: James D. Meiss

Publisher: SIAM

ISBN: 161197464X

Category: Mathematics

Page: 392

View: 779

Differential equations are the basis for models of any physical systems that exhibit smooth change. This book combines much of the material found in a traditional course on ordinary differential equations with an introduction to the more modern theory of dynamical systems. Applications of this theory to physics, biology, chemistry, and engineering are shown through examples in such areas as population modeling, fluid dynamics, electronics, and mechanics.÷ Differential Dynamical Systems begins with coverage of linear systems, including matrix algebra; the focus then shifts to foundational material on nonlinear differential equations, making heavy use of the contraction-mapping theorem. Subsequent chapters deal specifically with dynamical systems concepts?flow, stability, invariant manifolds, the phase plane, bifurcation, chaos, and Hamiltonian dynamics. This new edition contains several important updates and revisions throughout the book. Throughout the book, the author includes exercises to help students develop an analytical and geometrical understanding of dynamics. Many of the exercises and examples are based on applications and some involve computation; an appendix offers simple codes written in Maple?, Mathematica?, and MATLAB? software to give students practice with computation applied to dynamical systems problems.

Nonlinear Ordinary Differential Equations

An Introduction for Scientists and Engineers

Author: Dominic Jordan,Peter Smith

Publisher: Oxford University Press on Demand

ISBN: 0199208247

Category: Mathematics

Page: 531

View: 3159

Thoroughly updated and expanded 4th edition of the classic text, including numerous worked examples, diagrams and exercises. An ideal resource for students and lecturers in engineering, mathematics and the sciences it is published alongside a separate Problems and Solutions Sourcebook containing over 500 problems and fully-worked solutions.

Differential Equations, Dynamical Systems, and an Introduction to Chaos

Author: Morris W. Hirsch,Stephen Smale,Robert L. Devaney

Publisher: Academic Press

ISBN: 0123820111

Category: Mathematics

Page: 432

View: 2011

Hirsch, Devaney, and Smale’s classic Differential Equations, Dynamical Systems, and an Introduction to Chaos has been used by professors as the primary text for undergraduate and graduate level courses covering differential equations. It provides a theoretical approach to dynamical systems and chaos written for a diverse student population among the fields of mathematics, science, and engineering. Prominent experts provide everything students need to know about dynamical systems as students seek to develop sufficient mathematical skills to analyze the types of differential equations that arise in their area of study. The authors provide rigorous exercises and examples clearly and easily by slowly introducing linear systems of differential equations. Calculus is required as specialized advanced topics not usually found in elementary differential equations courses are included, such as exploring the world of discrete dynamical systems and describing chaotic systems. Classic text by three of the world’s most prominent mathematicians Continues the tradition of expository excellence Contains updated material and expanded applications for use in applied studies

Gewöhnliche Differentialgleichungen

Author: Vladimir I. Arnold

Publisher: Springer-Verlag

ISBN: 3642564801

Category: Mathematics

Page: 344

View: 9619

nen (die fast unverändert in moderne Lehrbücher der Analysis übernommen wurde) ermöglichten ihm nach seinen eigenen Worten, "in einer halben Vier telstunde" die Flächen beliebiger Figuren zu vergleichen. Newton zeigte, daß die Koeffizienten seiner Reihen proportional zu den sukzessiven Ableitungen der Funktion sind, doch ging er darauf nicht weiter ein, da er zu Recht meinte, daß die Rechnungen in der Analysis bequemer auszuführen sind, wenn man nicht mit höheren Ableitungen arbeitet, sondern die ersten Glieder der Reihenentwicklung ausrechnet. Für Newton diente der Zusammenhang zwischen den Koeffizienten der Reihe und den Ableitungen eher dazu, die Ableitungen zu berechnen als die Reihe aufzustellen. Eine von Newtons wichtigsten Leistungen war seine Theorie des Sonnensy stems, die in den "Mathematischen Prinzipien der Naturlehre" ("Principia") ohne Verwendung der mathematischen Analysis dargestellt ist. Allgemein wird angenommen, daß Newton das allgemeine Gravitationsgesetz mit Hilfe seiner Analysis entdeckt habe. Tatsächlich hat Newton (1680) lediglich be wiesen, daß die Bahnkurven in einem Anziehungsfeld Ellipsen sind, wenn die Anziehungskraft invers proportional zum Abstandsquadrat ist: Auf das Ge setz selbst wurde Newton von Hooke (1635-1703) hingewiesen (vgl. § 8) und es scheint, daß es noch von weiteren Forschern vermutet wurde.

Dynamical Systems

Author: George David Birkhoff

Publisher: American Mathematical Soc.

ISBN: 082181009X

Category: Mathematics

Page: 305

View: 7880

His research in dynamics constitutes the middle period of Birkhoff's scientific career, that of maturity and greatest power. --Yearbook of the American Philosophical Society The author's great book ... is well known to all, and the diverse active modern developments in mathematics which have been inspired by this volume bear the most eloquent testimony to its quality and influence. --Zentralblatt MATH In 1927, G. D. Birkhoff wrote a remarkable treatise on the theory of dynamical systems that would inspire many later mathematicians to do great work. To a large extent, Birkhoff was writing about his own work on the subject, which was itself strongly influenced by Poincare's approach to dynamical systems. With this book, Birkhoff also demonstrated that the subject was a beautiful theory, much more than a compendium of individual results. The influence of this work can be found in many fields, including differential equations, mathematical physics, and even what is now known as Morse theory. The present volume is the revised 1966 reprinting of the book, including a new addendum, some footnotes, references added by Jurgen Moser, and a special preface by Marston Morse. Although dynamical systems has thrived in the decades since Birkhoff's book was published, this treatise continues to offer insight and inspiration for still more generations of mathematicians.

Averaging Methods in Nonlinear Dynamical Systems

Author: Jan A. Sanders,Ferdinand Verhulst,James Murdock

Publisher: Springer Science & Business Media

ISBN: 0387489185

Category: Mathematics

Page: 434

View: 2304

Perturbation theory and in particular normal form theory has shown strong growth in recent decades. This book is a drastic revision of the first edition of the averaging book. The updated chapters represent new insights in averaging, in particular its relation with dynamical systems and the theory of normal forms. Also new are survey appendices on invariant manifolds. One of the most striking features of the book is the collection of examples, which range from the very simple to some that are elaborate, realistic, and of considerable practical importance. Most of them are presented in careful detail and are illustrated with illuminating diagrams.

Mathematics for Dynamic Modeling

Author: Edward Beltrami,Edward J. Beltrami

Publisher: Academic Press

ISBN: 9780120855667

Category: Mathematics

Page: 219

View: 8141

This new edition of Mathematics for Dynamic Modeling updates a widely used and highly-respected textbook. The text is appropriate for upper-level undergraduate and graduate level courses in modeling, dynamical systems, differential equations, and linear multivariable systems offered in a variety of departments including mathematics, engineering, computer science, and economics. The text features many different realistic applications from a wide variety of disciplines. The book covers important tools such as linearization, feedback concepts, the use of Liapunov functions, and optimal control. This new edition is a valuable tool for understanding and teaching a rapidly growing field. Practitioners and researchers may also find this book of interest. Contains a new chapter on stability of dynamic models Covers many realistic applications from a wide variety of fields in an accessible manner Provides a broad introduction to the full scope of dynamical systems Incorporates new developments such as new models for chemical reactions and autocatalysis Integrates MATLAB throughout the text in both examples and illustrations Includes a new introduction to nonlinear differential equations

Introduction to the Modern Theory of Dynamical Systems

Author: Anatole Katok,Boris Hasselblatt

Publisher: Cambridge University Press

ISBN: 9780521575577

Category: Mathematics

Page: 802

View: 9941

This book provides a self-contained comprehensive exposition of the theory of dynamical systems. The book begins with a discussion of several elementary but crucial examples. These are used to formulate a program for the general study of asymptotic properties and to introduce the principal theoretical concepts and methods. The main theme of the second part of the book is the interplay between local analysis near individual orbits and the global complexity of the orbit structure. The third and fourth parts develop the theories of low-dimensional dynamical systems and hyperbolic dynamical systems in depth. The book is aimed at students and researchers in mathematics at all levels from advanced undergraduate and up.

Dynamical Systems

An Introduction with Applications in Economics and Biology

Author: Pierre N.V. Tu

Publisher: Springer Science & Business Media

ISBN: 3642787932

Category: Science

Page: 314

View: 8727

The favourable reception of the first edition and the encouragement received from many readers have prompted the author to bring out this new edition. This provides the opportunity for correcting a number of errors, typographical and others, contained in the first edition and making further improvements. This second edition has a new chapter on simplifying Dynamical Systems covering Poincare map, Floquet theory, Centre Manifold Theorems, normal forms of dynamical systems, elimination of passive coordinates and Liapunov-Schmidt reduction theory. It would provide a gradual transition to the study of Bifurcation, Chaos and Catastrophe in Chapter 10. Apart from this, most others - in fact all except the first three and last chapters - have been revised and enlarged to bring in some new materials, elaborate some others, especially those sections which many readers felt were rather too concise in the first edition, by providing more explana tion, examples and applications. Chapter 11 provides some good examples of this. Another example may be found in Chapter 4 where the review of Linear Algebra has been enlarged to incorporate further materials needed in this edition, for example the last section on idempotent matrices and projection would prove very useful to follow Liapunov-Schmidt reduction theory presented in Chapter 9.

Partielle Differentialgleichungen

Eine Einführung

Author: Walter A. Strauss

Publisher: Springer-Verlag

ISBN: 366312486X

Category: Mathematics

Page: 458

View: 7424

Dieses Buch ist eine umfassende Einführung in die klassischen Lösungsmethoden partieller Differentialgleichungen. Es wendet sich an Leser mit Kenntnissen aus einem viersemestrigen Grundstudium der Mathematik (und Physik) und legt seinen Schwerpunkt auf die explizite Darstellung der Lösungen. Es ist deshalb besonders auch für Anwender (Physiker, Ingenieure) sowie für Nichtspezialisten, die die Methoden der mathematischen Physik kennenlernen wollen, interessant. Durch die große Anzahl von Beispielen und Übungsaufgaben eignet es sich gut zum Gebrauch neben Vorlesungen sowie zum Selbststudium.

Angewandte Mathematik: Body and Soul

Band 2: Integrale und Geometrie in IRn

Author: Kenneth Eriksson,Donald Estep,Claes Johnson

Publisher: Springer-Verlag

ISBN: 3540269509

Category: Mathematics

Page: 362

View: 9488

"Angewandte Mathematik: Body & Soul" ist ein neuer Grundkurs in der Mathematikausbildung für Studienanfänger in den Naturwissenschaften, der Technik, und der Mathematik, der an der Chalmers Tekniska Högskola in Göteborg entwickelt wurde. Er besteht aus drei Bänden sowie Computer-Software. Das Projekt ist begründet in der Computerrevolution, die ihrerseits völlig neue Möglichkeiten des wissenschaftlichen Rechnens in der Mathematik, den Naturwissenschaften und im Ingenieurwesen eröffnet hat. Es besteht aus einer Synthese der mathematischen Analysis (Soul) mit der numerischen Berechnung (Body) sowie den Anwendungen. Die Bände I-III geben eine moderne Version der Analysis und der linearen Algebra wieder, einschließlich konstruktiver numerischer Techniken und Anwendungen, zugeschnitten auf Anfängerprogramme im Maschinenbau und den Naturwissenschaften. Weitere Bände behandeln Themen wie z.B. dynamische Systeme, Strömungsdynamik, Festkörpermechanik und Elektromagnetismus. Dieser Band entwickelt das Riemann-Integral, um eine Funktion zu einer gegebenen Ableitung zu bestimmen. Darauf aufbauend werden Differentialgleichungen und Anfangswertprobleme mit einer Vielzahl anschaulicher Anwendungen behandelt. Die lineare Algebra wird auf n-dimensionale Räume verallgemeinert, wobei wiederum dem praktischen Umgang und numerischen Lösungstechniken besonderer Platz eingeräumt wird. Die Autoren sind führende Experten im Gebiet des wissenschaftlichen Rechnens und haben schon mehrere erfolgreiche Bücher geschrieben. "[......] Oh, by the way, I suggest immediate purchase of all three volumes!" The Mathematical Association of America Online, 7.7.04

Finite-Elemente-Methoden

Author: Klaus-Jürgen Bathe

Publisher: DrMaster Publications

ISBN: 9783540668060

Category: Technology & Engineering

Page: 1253

View: 1583

Dieses Lehr- und Handbuch behandelt sowohl die elementaren Konzepte als auch die fortgeschrittenen und zukunftsweisenden linearen und nichtlinearen FE-Methoden in Statik, Dynamik, Festkörper- und Fluidmechanik. Es wird sowohl der physikalische als auch der mathematische Hintergrund der Prozeduren ausführlich und verständlich beschrieben. Das Werk enthält eine Vielzahl von ausgearbeiteten Beispielen, Rechnerübungen und Programmlisten. Als Übersetzung eines erfolgreichen amerikanischen Lehrbuchs hat es sich in zwei Auflagen auch bei den deutschsprachigen Ingenieuren etabliert. Die umfangreichen Änderungen gegenüber der Vorauflage innerhalb aller Kapitel - vor allem aber der fortgeschrittenen - spiegeln die rasche Entwicklung innerhalb des letzten Jahrzehnts auf diesem Gebiet wieder. TOC:Eine Einführung in den Gebrauch von Finite-Elemente-Verfahren.-Vektoren, Matrizen und Tensoren.-Einige Grundbegriffe ingenieurwissenschaftlicher Berechnungen.-Formulierung der Methode der finiten Elemente.-Formulierung und Berechnung von isoparametrischen Finite-Elemente-Matrizen.-Nichtlineare Finite-Elemente-Berechnungen in der Festkörper- und Strukturmechanik.-Finite-Elemente-Berechnungen von Wärmeübertragungs- und Feldproblemen.-Lösung von Gleichgewichtsbeziehungen in statischen Berechnungen.-Lösung von Bewegungsgleichungen in kinetischen Berechnungen.-Vorbemerkungen zur Lösung von Eigenproblemen.-Lösungsverfahren für Eigenprobleme.-Implementierung der Finite-Elemente-Methode.

Partielle Differentialgleichungen und numerische Methoden

Author: Stig Larsson,Vidar Thomee

Publisher: Springer-Verlag

ISBN: 3540274227

Category: Mathematics

Page: 272

View: 8924

Das Buch ist für Studenten der angewandten Mathematik und der Ingenieurwissenschaften auf Vordiplomniveau geeignet. Der Schwerpunkt liegt auf der Verbindung der Theorie linearer partieller Differentialgleichungen mit der Theorie finiter Differenzenverfahren und der Theorie der Methoden finiter Elemente. Für jede Klasse partieller Differentialgleichungen, d.h. elliptische, parabolische und hyperbolische, enthält der Text jeweils ein Kapitel zur mathematischen Theorie der Differentialgleichung gefolgt von einem Kapitel zu finiten Differenzenverfahren sowie einem zu Methoden der finiten Elemente. Den Kapiteln zu elliptischen Gleichungen geht ein Kapitel zum Zweipunkt-Randwertproblem für gewöhnliche Differentialgleichungen voran. Ebenso ist den Kapiteln zu zeitabhängigen Problemen ein Kapitel zum Anfangswertproblem für gewöhnliche Differentialgleichungen vorangestellt. Zudem gibt es ein Kapitel zum elliptischen Eigenwertproblem und zur Entwicklung nach Eigenfunktionen. Die Darstellung setzt keine tiefer gehenden Kenntnisse in Analysis und Funktionalanalysis voraus. Das erforderliche Grundwissen über lineare Funktionalanalysis und Sobolev-Räume wird im Anhang im Überblick besprochen.

Dynamical Systems and Numerical Analysis

Author: Andrew Stuart,A. R. Humphries

Publisher: Cambridge University Press

ISBN: 9780521645638

Category: Mathematics

Page: 685

View: 8556

This book unites the study of dynamical systems and numerical solution of differential equations. 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 formulted as dynamical systems and the convergence and stability properties of the methods are examined. Topics studied include the stability of numerical methods for contractive, dissipative, gradient and Hamiltonian systems together with the convergence properties of equilibria, periodic solutions and strage attractors under numerical approximation. This book will be an invaluable tool for graduate students and researchers in the fields of numerical analysis and dynamical systems.

Quantenphysik f?r Dummies

Author: Steven Holzner

Publisher: John Wiley & Sons

ISBN: 3527668004

Category: Science

Page: 316

View: 2685

Die Quantenphysik, auch Quantenmechanik, ist ein zentrales, wenn auch von vielen Sch?lern und Studenten ungeliebtes Thema der Physik. Aber keine Panik! Steven Holzner erkl?rt Ihnen freundlich, verst?ndlich, kompetent, was Sie ?ber Quantenphysik wissen m?ssen. Er erl?utert die Grundlagen von Drehimpuls und Spin, gibt Ihnen Tipps, wie Sie komplexe Gleichungen l?sen und nimmt den klassischen Problemen der Quantenphysik den Schrecken. Dabei arbeitet er mit Beispielen, die er ausf?hrlich erkl?rt und gibt Ihnen so zus?tzliche Sicherheit auf einem vor Unsch?rfen wimmelnden Feld.

Dynamical Systems

Stability, Symbolic Dynamics, and Chaos

Author: Clark Robinson

Publisher: CRC Press

ISBN: 1482227878

Category: Mathematics

Page: 520

View: 2882

Several distinctive aspects make Dynamical Systems unique, including: treating the subject from a mathematical perspective with the proofs of most of the results included providing a careful review of background materials introducing ideas through examples and at a level accessible to a beginning graduate student focusing on multidimensional systems of real variables The book treats the dynamics of both iteration of functions and solutions of ordinary differential equations. Many concepts are first introduced for iteration of functions where the geometry is simpler, but results are interpreted for differential equations. The dynamical systems approach of the book concentrates on properties of the whole system or subsets of the system rather than individual solutions. The more local theory discussed deals with characterizing types of solutions under various hypothesis, and later chapters address more global aspects.

An Exploration of Dynamical Systems and Chaos

Completely Revised and Enlarged Second Edition

Author: John Argyris,Gunter Faust,Maria Haase,Rudolf Friedrich

Publisher: Springer

ISBN: 3662460424

Category: Technology & Engineering

Page: 865

View: 3048

This book is conceived as a comprehensive and detailed text-book on non-linear dynamical systems with particular emphasis on the exploration of chaotic phenomena. The self-contained introductory presentation is addressed both to those who wish to study the physics of chaotic systems and non-linear dynamics intensively as well as those who are curious to learn more about the fascinating world of chaotic phenomena. Basic concepts like Poincaré section, iterated mappings, Hamiltonian chaos and KAM theory, strange attractors, fractal dimensions, Lyapunov exponents, bifurcation theory, self-similarity and renormalisation and transitions to chaos are thoroughly explained. To facilitate comprehension, mathematical concepts and tools are introduced in short sub-sections. The text is supported by numerous computer experiments and a multitude of graphical illustrations and colour plates emphasising the geometrical and topological characteristics of the underlying dynamics. This volume is a completely revised and enlarged second edition which comprises recently obtained research results of topical interest, and has been extended to include a new section on the basic concepts of probability theory. A completely new chapter on fully developed turbulence presents the successes of chaos theory, its limitations as well as future trends in the development of complex spatio-temporal structures. "This book will be of valuable help for my lectures" Hermann Haken, Stuttgart "This text-book should not be missing in any introductory lecture on non-linear systems and deterministic chaos" Wolfgang Kinzel, Würzburg “This well written book represents a comprehensive treatise on dynamical systems. It may serve as reference book for the whole field of nonlinear and chaotic systems and reports in a unique way on scientific developments of recent decades as well as important applications.” Joachim Peinke, Institute of Physics, Carl-von-Ossietzky University Oldenburg, Germany