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Introduction to Applied Nonlinear Dynamical Systems and Chaos de Stephen Wiggins

Introduction to Applied Nonlinear Dynamical Systems and Chaos: Texts in Applied Mathematics, cartea 2

Autor Stephen Wiggins
en Limba Engleză Hardback – oct 2003

Subliniem faptul că Introduction to Applied Nonlinear Dynamical Systems and Chaos reprezintă un pilon central în seria Texts in Applied Mathematics, fiind conceput special pentru a pregăti studenții în fața complexității cercetării moderne. Structura materialului urmărește o progresie riguroasă, pornind de la conceptele fundamentale de stabilitate liniară și funcții Liapunov, trecând prin teoria varietăților invariante și ajungând la fenomene complexe precum bifurcațiile și dinamica simbolică. Metodologia autorului pune accent pe oferirea unui „set de unelte” matematic extins, necesar pentru a naviga explozia de date și modele din științele fizice și biologice ale ultimelor decenii.

Această a doua ediție revizuită și adăugită restructurează prezentarea temelor clasice și introduce secțiuni vitale despre formele normale și sistemele reversibile. Credem că volumul reușește să mențină un echilibru între rigoarea analitică și aplicabilitatea practică, un aspect vizibil în capitolele dedicate condițiilor Conley-Moser sau interpretării diagramelor de bifurcație. Putem afirma că lucrarea completează perspectiva oferită de Differential Equations and Dynamical Systems de Lawrence Perko, adăugând o profunzime sporită în zona metodelor analitice pentru haos și bifurcații globale, acolo unde textul lui Perko rămâne focusat pe sistemele diferențiale fundamentale.

În contextul operei sale, această carte rafinează conceptele explorate de Stephen Wiggins în Global Bifurcations and Chaos, oferind un cadru pedagogic mai larg. Dacă în ORDINARY DIFFERENTIAL EQUATIONS autorul stabilea bazele curriculare, aici el extinde analiza către regimurile neliniare. Cuprinsul detaliat indică o acoperire enciclopedică, de la teorema Poincaré-Bendixson până la dinamica din proximitatea punctelor homoclinice, făcând din acest volum de 864 de pagini o resursă esențială pentru cercetarea în dinamica neliniară.

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Specificații

ISBN-13: 9780387001777
ISBN-10: 0387001778
Pagini: 868
Ilustrații: XXXVIII, 844 p.
Dimensiuni: 160 x 241 x 52 mm
Greutate: 1.45 kg
Ediția:Second Edition 2003
Editura: Springer
Colecția Texts in Applied Mathematics
Seria Texts in Applied Mathematics

Locul publicării:New York, NY, United States

Public țintă

Research

De ce să citești această carte

Această ediție este indispensabilă pentru studenții de master și doctorat care doresc să stăpânească instrumentele analitice ale haosului și sistemelor neliniare. Cititorul câștigă o înțelegere profundă a stabilității și bifurcațiilor, beneficiind de expertiza unui lider în domeniu. Este recomandată celor care au parcurs deja bazele ecuațiilor diferențiale și caută rigoarea necesară pentru a trece la cercetarea aplicată în matematică sau fizică teoretică.

Despre autor

Stephen Wiggins este un renumit profesor de matematică aplicată și șeful Școlii de Matematică de la Universitatea din Bristol. Cu o carieră dedicată studiului sistemelor dinamice, Wiggins este recunoscut pentru capacitatea de a sintetiza teorii matematice abstracte în tratate fundamentale pentru cercetarea contemporană. Opera sa vastă include manuale de referință despre ecuații diferențiale și mecanică clasică, fiind unul dintre editorii și autorii cheie care au definit direcția seriei Texts in Applied Mathematics de la Springer.

Descriere scurtă

Mathematics is playing an ever more important role in the physical and biological sciences, provoking a blurring of boundaries between scientific disciplines and a resurgence of interest in the modern as well as the classical techniques of applied mathematics. This renewal of interest, both in - search and teaching, has led to the establishment of the series Texts in Applied Mathematics (TAM). The development of new courses is a natural consequence of a high level of excitement on the research frontier as newer techniques, such as nume- cal and symbolic computer systems, dynamical systems, and chaos, mix with and reinforce the traditional methods of applied mathematics. Thus, the purpose of this textbook series is to meet the current and future needs of these advances and to encourage the teaching of new courses. TAM will publish textbooks suitable for use in advanced undergraduate and beginning graduate courses, and will complement the Applied Mat- matical Sciences (AMS) series, whichwill focus on advanced textbooks and research-level monographs. Pasadena, California J.E. Marsden Providence, Rhode Island L. Sirovich College Park, Maryland S.S. Antman Preface to the Second Edition This edition contains a signi?cant amount of new material. The main r- son for this is that the subject of applied dynamical systems theory has seen explosive growth and expansion throughout the 1990s. Consequently, a student needs a much larger toolbox today in order to begin research on signi?cant problems.

Cuprins

Equilibrium Solutions, Stability, and Linearized Stability.- Liapunov Functions.- Invariant Manifolds: Linear and Nonlinear Systems.- Periodic Orbits.- Vector Fields Possessing an Integral.- Index Theory.- Some General Properties of Vector Fields: Existence, Uniqueness, Differentiability, and Flows.- Asymptotic Behavior.- The Poincaré-Bendixson Theorem.- Poincaré Maps.- Conjugacies of Maps, and Varying the Cross-Section.- Structural Stability, Genericity, and Transversality.- Lagrange’s Equations.- Hamiltonian Vector Fields.- Gradient Vector Fields.- Reversible Dynamical Systems.- Asymptotically Autonomous Vector Fields.- Center Manifolds.- Normal Forms.- Bifurcation of Fixed Points of Vector Fields.- Bifurcations of Fixed Points of Maps.- On the Interpretation and Application of Bifurcation Diagrams: A Word of Caution.- The Smale Horseshoe.- Symbolic Dynamics.- The Conley-Moser Conditions, or “How to Prove That a Dynamical System is Chaotic”.- Dynamics Near Homoclinic Points of Two-Dimensional Maps.- Orbits Homoclinic to Hyperbolic Fixed Points in Three-Dimensional Autonomous Vector Fields.- Melnikov–s Method for Homoclinic Orbits in Two-Dimensional, Time-Periodic Vector Fields.- Liapunov Exponents.- Chaos and Strange Attractors.- Hyperbolic Invariant Sets: A Chaotic Saddle.- Long Period Sinks in Dissipative Systems and Elliptic Islands in Conservative Systems.- Global Bifurcations Arising from Local Codimension—Two Bifurcations.- Glossary of Frequently Used Terms.

Recenzii

From the reviews of the second edition:
"This is a very substantial revision of the author’s original textbook published in 1990. It does not only contain much new material, for instance on invariant manifold theory and normal forms, it has also been restructured. … The presentation is intended for advanced undergraduates … . This second edition … will serve as one of the most eminent introductions to the geometric theory of dynamical systems." (R. Bürger, Monatshefte für Mathematik, Vol. 145 (4), 2005)
"This is an extensively rewritten version of the first edition which appeared in 1990, taking into account the many changes in the subject during the intervening time period. … The book is suitable for use as a textbook for graduate courses in applied mathematics or cognate fields. It is written in a readable style, with considerable motivation and many insightful examples. … Overall, the book provides a very accessible, up-to-date and comprehensive introduction to applied dynamical systems." (P.E. Kloeden, ZAMM-Zeitschrift für Angewandte Mathematik und Mechanik, Vol. 85 (1), 2005)
"The second edition of this popular text … is an encyclopedic introduction to dynamical systems theory and applications that includes substantial revisions and new material. It should be on the reading list of every student of the subject … . Also, the new organization makes the book more suitable as a textbook that can be used in graduate courses. This book will also be a useful reference for applied scientists … as well as a guide to the literature." (Carmen Chicone, Mathematical Reviews, 2004h)
"This volume includes a significant amount of new material. … Each chapter starts with a narrative … and ends with a large collection of excellent exercises. … An extensive bibliography … provide a useful guide for future study. … This is a highly recommended book for advanced undergraduate and first-year graduate students. It contains most of the necessary mathematical tools … to apply the results of the subject to problems in the physical and engineering sciences." (Tibor Krisztin, Acta Scientiarum Mathematicarum, Vol. 75, 2009)
“It is certainly one of the most complete introductory textbooks about dynamical systems, though no single book can be really complete. … Some chapters can certainly be used as a course text for a master’s course, but the whole book is to thick for a single course. … a suitable first text for Ph.D. students who want to do research in dynamical systems, and a useful reference work for more experienced people. I definitely enjoyed reading this book and can only recommend it.” (Kurt Lust, Bulletin of the Belgian Mathematical Society, Vol. 15 (1), 2008)

Textul de pe ultima copertă

This volume is intended for advanced undergraduate or first-year graduate students as an introduction to applied nonlinear dynamics and chaos. The author has placed emphasis on teaching the techniques and ideas that will enable students to take specific dynamical systems and obtain some quantitative information about the behavior of these systems. He has included the basic core material that is necessary for higher levels of study and research. Thus, people who do not necessarily have an extensive mathematical background, such as students in engineering, physics, chemistry, and biology, will find this text as useful as students of mathematics.
This new edition contains extensive new material on invariant manifold theory and normal forms (in particular, Hamiltonian normal forms and the role of symmetry). Lagrangian, Hamiltonian, gradient, and reversible dynamical systems are also discussed. Elementary Hamiltonian bifurcations are covered, as well as the basic propertiesof circle maps. The book contains an extensive bibliography as well as a detailed glossary of terms, making it a comprehensive book on applied nonlinear dynamical systems from a geometrical and analytical point of view.