Daniel Kubat

Introduction: To the Second Edition

The subject of this book is chaos as seen through the filter of topology. The origin of this book lies in the analysis of data generated by a dynamical system operating in a chaotic regime. Throughout this book we develop topological tools for analyzing chaotic data and then show how they are applied to experimental data sets. More specifically, we describe how to extract, from chaotic data, topological signatures that determine the stretching and squeezing mechanisms that act on flows in phase space and that are responsible for generating chaotic data. In the first section of this introductory chapter we very briefly review some of the basic ideas from the field of nonlinear dynamics and chaos. This is done to make the work as self-contained as possible. More in-depth treatment of these ideas can be found in the references provided. In the second section we describe, for purposes of motivation, a laser that has been operated under conditions in which it behaved chaotically. The topological methods of analysis that we describe in this book were developed in response to the challenge of analyzing chaotic data sets generated by this laser. In the third section we list a number of questions we would like to be able to answer when analyzing a chaotic signal. None of these questions can be addressed by the older tools for analyzing chaotic data. The older methods involve estimates of the spectrum of Lyapunov exponents and estimates of the spectrum of fractal dimensions. The question that we would particularly like to be able to answer is this: How does one model the dynamics? To answer this question we must determine the stretching and squeezing mechanisms that operate together – repeatedly – to generate chaotic data. The stretching mechanism is responsible for sensitivity to initial conditions while the squeezing mechanism is responsible for recurrent nonperiodic behavior. These two mechanisms operate repeatedly to generate a strange attractor with a self-similar structure. A new analysis method, topological analysis, has been developed to respond to the fundamental question just stated [7, 8]. At the present time this method is suitable only for strange attractors that can be embedded in three-dimensional spaces. However, for such strange attractors it offers a complete and satisfying resolution to this question. The results are previewed in the fourth section of this chapter. In the final section we provide a brief overview of the organization of this book. In particular, we summarize the organization and content of the following chapters.

International Newsletter on Migration

• Election results of our balloting are noted already on the list of officers for the next intercongress period which runs from August, 1982 through the next World Congress of Sociology in 1986. There were some 100 ballots sent out; 24 were returned of which one was blank. The nominated candidates were all elected, there were a number of write-ins but not insufficient numbers to change the slate as presented by the board. The range of votes for individual candidates was a narrow one, between 18 and 22 votes. The votes were counted on July 29, two weeks past the deadline for sending in ballots. The official counter was Professor F. Fasick, at the University of Waterloo, whose interests in our activities are minimal. The new board members can be reached at the following addresses:

Book ReviewSoziologie des Fremdarbeiterproblems: Eine theoretische und empirische Analyse am Beispiel der Schweiz. By Hoffman-NowotnyHans-Joachim. Stuttgart: Ferdinand Enke, 1973. Pp. 377.Soziale Askpekte der Auslaenderbeschaeftigung. By MehrländerUrsula. Bonn-Bad Godesberg: Verlag Neue Gesellschaft, 1974. Pp. 260.

gration in West Africa, this volume incorporates some worthwhile information. It is unfortunate that the introduction is not more carefully prepared and lacks a final summary statement to tie the sixteen articles into a heuristic framework. The weakness of the introduction and the absence of a final chapter reduces the effectiveness of this volume. Indeed, the strength of the book lies in the case studies.