The Theory of Chaotic Attractors

ISBN-13: 9780387403496 / Angielski / Twarda / 2004 / 514 str.

Brian R. Hunt; Judy A. Kennedy; Tien-Yien Li

Opis

The editors felt that the time was right for a book on an important topic, the history and development of the notions of chaotic attractors and their "natu ral" invariant measures. We wanted to bring together a coherent collection of readable, interesting, outstanding papers for detailed study and comparison. We hope that this book will allow serious graduate students to hold seminars to study how the research in this field developed. Limitation of space forced us painfully to exclude many excellent, relevant papers, and the resulting choice reflects the interests of the editors. Since James Alan Yorke was born August 3, 1941, we chose to have this book commemorate his sixtieth birthday, honoring his research in this field. The editors are four of his collaborators. We would particularly like to thank Achi Dosanjh (senior editor math ematics), Elizabeth Young (assistant editor mathematics), Joel Ariaratnam (mathematics editorial), and Yong-Soon Hwang (book production editor) from Springer Verlag in New York for their efforts in publishing this book."

Recenzje

From the reviews:

"This book is a collection, from the last 40 years, of the most influential papers in the development of chaos theory. ... The editors are eminent in this field ... . Their selection of papers is excellent ... . it is a book for mathematics graduates specializing in non-linear dynamics. ... is a well-presented and well-deserved tribute to the life's work of Professor Yorke ... . might well be as that benchmark that mathematicians of the future will view its publication hundreds of years from now." (Dennis Morris, The Mathematical Gazette, March, 2005)

Spis treści

Contents: Preface.-Introduction.- E.N. Lorenz, Deterministic nonperiodic flow.- K. Krzyzewski and W. Szlenk, On invariant measures for expanding differentiable mappings.- A. Lasota and J.A. Yorke, On the existence of invariant measures for piecewise monotonic transformations.- R. Bowen and D. Ruelle, The ergodic theory of Axiom A flows.- T.-Y. Li and J.A. Yorke, Period three implies chaos.- R.M. May, Simple mathematical models with very complicated dynamics.- M. Henon, A two- dimensional mapping with a strange attractor.- E. Ott, Strange attractors and chaotic motions of dynamical systems.- F. Hofbauer and G. Keller, Ergodic properties of invariant measures for piecewise monotonic transformations.- D. J. Farmer, E. Ott and J.A. Yorke, The dimension of chaotic attractors .- P. Grassberger and I. Procaccia, Measuring the strangeness of strange attractors.- M. Rychlik, Invariant measures and variational principle for Lozi applications.- P. Collet and Y. Levy, Ergodic properties of the Lozi mappings .- J. Milnor, On the Concept of Attractor.-L.-S. Young, Bowen-Ruelle Measures for certain Piewise Hyperbolic Maps.-J.-P. Eckmann and D. Ruelle, Ergodic Theory of Chaos and Strange Attractors.-M.R. Rychlik, Another Proof of Jakobson Theorem and Related Results.- C. Grebogi, E. Ott, and J.A. Yorke, Unstable periodic Orbits and the Dimensions of Multifractal Chaotic Attractors.-P. Gora and A. Boyarsky, Absolutely Continuous Invariant Measures for Piecewise Expanding C Transformation in R.-M. Benedicks and L.-S. Young, Sinai-Bowen-Ruelle Measures for Certain Henon Maps.-M. Dellnitz and O. Junge, On the Approximation of Complicated Dynamical Behavior.-M. Tsujii, Absolutely Continuous Invariant Measures for Piecewise Real-Analytic Expanding Maps on the Plane.-J.F. Alves, C.Bonatti, and M. Viana, SRB Measures for Partially Hyperbolic Systems Whose Central Direction is Mostly Expanding.- B.R. Hunt, J.A. Kennedy, T.-Y. Li, and H.E. Nusse, SLYRB Measures: Natural Invariant Measures for Chaotic Systems.- Credits

Blurb

 

The development of the theory of chaotic dynamics and its subsequent wide applicability in science and technology has been an extremely important achievement of modern mathematics. This volume collects several of the most influential papers in chaos theory from the past 40 years, starting with Lorenz's seminal 1963 article and containing classic papers by Lasota and Yorke (1973), Bowen and Ruelle (1975), Li and Yorke (1975), May (1976), Henon (1976), Milnor (1985), Eckmann and Ruelle (1985), Grebogi, Ott, and Yorke (1988), Benedicks and Young (1993) and many others, with an emphasis on invariant measures for chaotic systems.

Dedicated to Professor James Yorke, a pioneer in the field and a recipient of the 2003 Japan Prize, the book includes an extensive, anecdotal introduction discussing Yorke's contributions and giving readers a general overview of the key developments of the theory from a historical perspective.