Tent map

The mapping that generates the Cantor set is known as the tent map. It is defined by... 

Fig. 2.7 shows the graph of T in A. The shape of the graph explains the mapping s name. All points with x < 0 are mapped monotonically to —oo. Points with x > 1 are first mapped to 3(1 — x) < 0 and then also to —oo. Thus, none of the points outside A will ever be mapped into A. This is an important property. It implies that whenever a point of A is mapped outside A this point will never return to A. Thus, this property is called the never-come-hackpiopeTty. It facilitates appreciably the analysis of the tent map. 

Fig. 2.8. The ionization process induced by the tent map T. (a) Initial condition, (b) remaining points after the first application of T, (c) remaining points after the second application of T.

It is by now obvious how this process continues. With every application of the tent map T, the middle third of all the disconnected intervals still present will be deleted. The total length of the surviving intervals after step number n is... 

According to (2.3.2), there will be no probability left in the unit interval in the limit of n 00. Therefore, we say that a point xq in A ionizes with probabiUty 1 according to the ionization process defined by the tent map T. But is there really nothing left in A after the application of n -i- 00 mappings The successive steps in the ionization process defined by repeated application of T (see Fig. 2.8) remind us strongly of the construction scheme of Cantor s middle thirds set C (Fig. 2.2) which was introduced and briefiy discussed in Section 2.1. And indeed, there is a whole infinity of points left in A, even in the limit of an infinite number of applications of T. What kind of infinity We can easily answer this question with the tools developed in Section 2.1. Let us introduce a notation for the set of points in A that never ionize. We call this set A" ". For the tent map T we have A" " = C. 

Indeed, the logistic map for r > 4 also leads to a middle thirds elimination process, although now the middle thirds axe not exactly thirds any more. In fact, the length of the interval that leaves A after the first application of /s is 1/ 0.45. But topologically there is no difference compared with the Cantor elimination process. Therefore, in analogy to the tent map, we expect to see exponential decay of A induced by /5. This is indeed the case. An analytical approximation to the decay constant is obtained if we assume that the fraction of points that leave A in every step is the same as the firaction that leaves after the first application of fs. With this assumption we obtain Psin) = exp(—771), where 7 = ln[- /5/(- /5 — 1)]. 

In Section 2.3 we studied the tent map, a schematic model for ionization that was able to produce fractal structures as a result of ionization. An important question is therefore whether the results presented in Section 2.3 are only of academic interest, or whether fractal structures can appear as a result of ionization in physical systems. In order to answer this question we return to the microwave-driven one-dimensional hydrogen atom. As we know from the previous chapter, this model is ionizing and realistic enough to qualitatively reproduce measured ionization data. Therefore this model is expected to be a fair representative for a large class of chaotic ionization processes. 

Because it is piecewise linear, the tent map is far easier to analyze than the logistic map. 

Show that A = In r for the tent map, independent of the initial condition jCg. 

No windows forthe tent map) Prove that, in contrast to the logistic map, the tent map does not have periodic windows interspersed with chaos. 

The human A locus is located on chromosome 22. As in the case of the human Igh locus, the A genes were first tentatively mapped by studies measuring immunoglobulin gene expression in somatic cell hybrids [91] and the chromosome assignment later confirmed by direct hybridization studies [99]. 

FIGURE 15.122 Tentative map of regimes of operation of various forced convective critical heat flux mechanisms (from Hewitt and Semaria [300], with permission from Taylor Francis, Washington, DC. All rights reserved). 

Fig. 2. Illustration of the general trend of the faults. Please note that some faults are only tentatively mapped. Courtesy of the Norwegian Petroleum Directorate (NPD).

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