A piecewise-linear [1,2,3] one-dimensional map relates the output real variable y to the input real variable x in the ranges (0,1), (0,1) by connected straight line segments in a rectangular Cartesian graph of y against x. The example taken as the starting point for this investigation is shown in Figure 1.
Figure 1: A piecewise-linear trapped map
In a region where the slope
of the straight line section has magnitude
greater than unity (ie more than +1 or less than -1)
any error in x results in a greater error in y.
A piecewise-linear map which has slope everywhere
magnitude greater than unity, and is also multiple
valued in the sense that there are regions where
given a value of y, there corresponds more than one value of
x, gives rise to unending chaos when iterated.
To iterate a map, one transfers the output to the input,
calculates a new output, and so-on. This is particularly
easy to do electronically; and the piecewise-linear
characteristic can be synthesized from a collection
of linear operational amplifiers. A circuit diagram
and a functional block diagram will be shown at the
meeting. There are a number of subtle implementation
details, including the avoidance of trapping regions
at the turning points of the transfer characteristic,
which will be communicated.
The multiple valued property is loosely referred to as folding, and the slope magnitude greater than unity is referred to as stretching. There is a set of measure zero of starting values of x which result in closed trajectories which are periodic. In such a system the starting value repeats exactly after a finite number of iterations only when the starting values lie within this very restricted set [4] Roughly speaking, the chance of entering a periodic trajectory is zero; or at least vanishingly small. Any added noise, however small, will destroy the periodicity. Such an iterating system we term a chaotic map into which we introduce traps.