A Generic Chaotic System — The Bernoulli Shift Map

Despite bearing no direct relation to any physical dynamical system, the onedimensional discrete-time piecewise linear Bernoulli Shift map nonetheless displays many of the key mechanisms leading to deterministic chaos. The map is defined by (see figure 4.2)  

In turns out that the most convenient representation for the initial point, Xq, is as a binary decimal. That is, we write 

This binary decimal representation of xq should make obvious the reason why this map is named the Bernoulli shift. If xq 1/2, then a = 0 if xq 1/2 then ai = 1. Thus 

In other words, a single application of the map / to the point Xq discards the first digit and shifts to the left all of the remaining digits in the binary decimal expansion of Xq. In this way, the iterate is given by Xn = an+iCtn+2  

Tt is not difficult to show that such binary expansions - in fact expansions to an arbitrary base 6 1 - are complete in the unit interval see I. Niven, Irrational Numbers , The Cams Mathematical Monographs 11 (1956). 

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