Birkhoff George D. - Dynamical Systems

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Dynamical Systems
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Many mathematicians will welcome the new edition of G. D. BirkhofTs book on Dynamical Systems. It represents essentially a continuation of Poincare's profound and extensive work on Celestial Mechanics. Altogether BirkhofT was strongly influenced by Poincare and devoted a major part of his mathematical work to subjects arising from Poincare's tradition. The present book contains BirkhofTs views and ideas of his earlier period of life-it appeared when BirkhofT was 43. To the modern reader the style of this book may appear less formal and rigorous than it is now звичайний. But just the informal and lively manner of writing has been inspiring to many mathematicians. The effect of this inspiration is visible in a number of later papers. For example, Morse's theory on geodesies on a closed manifold originated directly in BirkhofTs ideas in dynamical systems. The recent work by Anosov on (/-Systems answers the question of ergodicity and density of periodic solutions for a wide class of differential equations-a problem in which BirkhofTs book was studied for a single model system. These and other examples (given below) justify the hope that the reprinting of this book again will stimulate further progress. Of course, 40 years after nearly a number of statements are outdated. For this reason I have selected a list of references to pertinent literature after 1927. Naturally, the list cannot be complete further and references can be found in the books and survey articles quoted. In the Addendum section at the back of this book I added some general remarks to various chapters. More specific comments are supplied as footnotes at the end of the book. References to these footnotes are given as small numbers in the margin of the text. * T . . JURGEN MOSER.