Iterated Maps and the Chaos Phenomenon…Iterated Maps and the Chaos Phenomenon The basic idea of an iterated map is to take a number between 0 and 1, and then in a sequence of steps to update this number according to a fixed rule or "map". Many of the maps I will consider can be expressed in terms of standard mathematical functions, but in general all that is needed is that the map take any possible number between 0 and 1 and yield some definite number that is also between 0 and 1. The pictures on the next two pages [ 150 , 151 ] show examples of behavior obtained with four different possible choices of maps.

Iterated Maps and the Chaos Phenomenon…And if one then uses this number as the initial condition for a shift map, the results will also be correspondingly random—just like those on the previous page . In the past this fact has sometimes been taken to indicate that the shift map somehow fundamentally produces randomness. … And indeed—as I will discuss in Chapter 7 —if one looks only at systems like the shift map then it is not clear any new randomness can ever actually be generated.

Iterated Maps and the Chaos Phenomenon…[No text on this page] The same iterated maps as on the facing page , but now started from the initial condition π/4 —a number with a seemingly random digit sequence. … Case (d) is the so-called shift map—a classic example of a system that exhibits the sensitive dependence on initial conditions often known as chaos.

Iterated Maps and the Chaos Phenomenon…And indeed, for the shift map what we have seen is that randomness will occur only when the initial conditions that are given happen to be a number whose digit sequence is random. … For if one thinks about numbers The effect of making a small change in the initial conditions for the shift map—shown as case (d) on pages 150 and 151 .

Iterated Maps and the Chaos Phenomenon…And the point is that these maps actually do intrinsically generate complexity and randomness; they do not just transcribe it when it is inserted in their initial conditions. … And the shift map shown as case (d) on the previous two pages [ 150 , 151 ] turns out to be a classic example of this.

Iterated Maps and the Chaos Phenomenon…[No text on this page] Examples of iterated maps starting from simple initial conditions.

Note (a) for Iterated Maps and the Chaos Phenomenon…Higher-dimensional generalizations [of iterated maps] One can consider so-called Anosov maps such as {x, y}  Mod[m . … Any initial condition containing only rational numbers will then yield repetitive behavior, much as in the shift map.

Note (b) for Iterated Maps and the Chaos Phenomenon…Distribution of chaotic behavior For iterated maps, unlike for discrete systems such as cellular automata, one can get continuous ranges of rules by varying parameters. With maps based on piecewise linear functions the regions of parameters in which chaotic behavior occurs typically have simple shapes; with maps based, say, on quadratic functions, however, elaborate nested shapes can occur.

Note (c) for Iterated Maps and the Chaos Phenomenon…History of iterated maps Newton's method from the late 1600s for finding roots of polynomials (already used in specific cases in antiquity) can be thought of as a smooth iterated map (see page 920 ) in which a rational function is repeatedly applied (see page 1101 ). … In the 1890s Henri Poincaré studied so-called return maps giving for example positions of objects on successive orbits. … Some detailed analytical studies of logistic maps of the form x  a x (1 - x) were done in the late 1950s and early 1960s—and in the mid-1970s iterated maps became popular, with much analysis and computer experimentation on them being done.

Note (e) for Iterated Maps and the Chaos Phenomenon…[Iterated maps from] bitwise operations Cellular automata can be thought of as analogs of iterated maps in which bitwise operations such as BitXor are used instead of ordinary arithmetic ones.