By Symposium in Pure Mathematics Stanford University 1976, Visit Amazon's R. James Milgram Page, search results, Learn about Author Central, R. James Milgram, , American Mathematical Society

ISBN-10: 082181432X

ISBN-13: 9780821814321

ISBN-10: 0821814737

ISBN-13: 9780821814734

Comprises sections on Algebraic $K$- and $L$-theory, surgical procedure and its purposes, staff activities

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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. It is a hope that simply asking the same questions of the newer field that were asked of the older fields will at least provide a structure in which advances can more rapidly be made. The entire chapter is devoted to comparing similar questions in these three fields, and suggestinghow some answers might hasten the development of a theory for dynamical systems which is as mature as Lie group theory and singularity theory are today.

This is a one-to-one transformation between I and itself, FULLY DEVELOPED CHAOS IN THE LOGISTIC MAP 33 which is a diffeomorphism everywhere except at the endpoints x = f 2 , where the ) not differentiable. 25) a piecewise linear map known as the tent map. 5 1 -. 5 2 xn Fig. 25). 9 shows that the graph of the tent map is extremely similar to that of the logistic map (Fig. 1). In both cases, the interval I is decomposed into two subintervals: I = I0 U 11,such that each restriction fk : I k + f ( I k ) o f f is a homeomorphism,with f ( I 0 ) = f(I1) = I .

A fascinating property of the period-doubling cascade is that we do not need to analyze directly the orbit of period 2w to determine very accurately am. Indeed, it can be remarked that the orbit of period 2O0 is formally its own period-doubled orbit. This indicates some kind of scale invariance. Accordingly, it was recognized by Feigenbaum that the transition to chaos in the period-doubling cascade can be analyzed by means of renormalizationgroup techniques [ 15,161. In this section we have analyzed how the periodic solutions of the logistic map are created.

### Algebraic and geometric topology by Symposium in Pure Mathematics Stanford University 1976, Visit Amazon's R. James Milgram Page, search results, Learn about Author Central, R. James Milgram, , American Mathematical Society

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