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The Topology of Chaos: Alice in Stretch and Squeezeland (German Edition)

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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.
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