Algebraic topology: a first course by Marvin J. Greenberg

By Marvin J. Greenberg

Great first e-book on algebraic topology. Introduces (co)homology via singular theory.

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Both types of behaviors have been observed in the experiment discussed in Chapter 1. 1 'k--kI 2, wc -1 0 20 60 40 80 100 n Fig. 0. What makes the study of the logistic map so important is not only that the organization in parameter space of these periodic and chaotic regimes can be completely understood with simple tools, but that despite of its simplicity it displays the most BIFURCATION DIAGRAMS 21 important features of low-dimensional chaotic behavior. By studying how periodic and chaotic behavior are interlaced, we will learn much about the mechanisms responsible for the appearance of chaotic behavior.

The response of many in the field of dynamical systems to the lack ofrapid understanding and developmentofthe field in low dimensions was to look for and to describe more complicated problems in higher dimensions! If we don’t have a theory in low dimensions, how are we to find one in higher dimensions? Chapter 12 addresses this problem in an indirect way. There are many similarities between the older fields of Lie group theory and singularity theory (catastrophe theory) and the newer field of dynamical systems theory.

There is a straightfonvard procedure for extracting the signature of a strange attractor from experimental data. This consists of a number of simple and easily implementable steps. The input to this analysis procedure consists of experimental time series. The output consists of a branched manifold, or more abstractly, a set of integers. The results of this analysis 14 INTRODUCTION are subject to a follow-on rejection or “confirmation”test. In Chapter 6 we present a step-by-step account of the topological analysis method.

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