TABLE OF CONTENTS As noted in Sec. 1.1, the logistic map (1.4) is the basic model for discussing the period-doubling transition to chaos. We shall not consider this map in detail here, but refer the reader to standard textbooks in nonlinear dynamics [6, 7, 8, 9, 16]. Let us recall, however, that the metric and topological properties of the logistic map have been studied in significant detail over the last two or three decades [68, 69, 70]. It is known, for instance, that f ? (x) for ? ?[1, 4] has a unique attracting state that can be periodic (a cycle of points) or chaotic (a cycle of chaotic intervals). The set of parameters ? for which f ? (x) has an attracting cycle is open and everywhere dense, and the set of parameters that produce a chaotic solution has a Cantor-like structure with a measure >0. In the considered parameter range, the basin of attraction is the unit interval x ?[0, 1].
Feigenbaum [32, 33] applied ideas (renormalization group theory) from statistical mechanics to prove the universal properties of unimodal, one-dimensional maps with quadratic maxima. This shed new light on the significance of the scaling factors ? F and ? F and initiated a broad interest in the study of nonlinear dynamic phenomena. (A map is unimodal when it has one extremum, and the extremum is quadratic if the second derivative of the map is finite). Other important contributions were...
