Clas Blomberg - Physics of life-Elsevier Science (2007)

294 Part VII. Non-linearity
Figure 28.1 The figure shows the successive transformations with n 2, b 2.
Here, m is an integer between 1 and n. n shall be an integer, and b shall be larger than n, but
otherwise, we need no condition on b. With this transformation, the original square is divided
into n slices of width 1/b. At the next step, this will be divided into n 2 slices and so on
(Figure 28.1).
This transformation which can be repeated an arbitrary number of times, and then the
coordinates of the original square are very mixed. If b 2, they all the time fill the entire
square, but the coordinates are changed in a manner that much relates to the entropy discussions.
A sequence is ergodic in the sense that with the exception of certain values that
give rise to periodic variations (as the certain values of the logistic equation), all parts of the
square will be equally covered by the elements of the sequence. When b is larger than n, the
sequence will cover thinner and thinner sheets in the square, definitely of lower order than
the square although covering an infinite length. We get what is called a fractal structure.
Thus,
(i) The limit set after an infinite number of iterations is a fractal structure.
(ii) Any initial accuracy is lost in the x-direction, similarly to the example of multiplication
by two. This means that limiting structure is apprehended as a chaotic attractor, a kind
of archetype of a strange attractor.
We can get characteristic numbers for this limit set. A main fractal dimension, usually
referred to as Hausdorff dimension is calculated in the following way. (There are other ways
to define dimensions and we will meet another possibility later). At the N th iteration, one considers
squares of side 1/b N that covers the resulting slices. Each slice is covered by b N squares,
and as there are n N slices (n for each iteration), there are totally (bn) N squares that cover the
entire structure at that stage. The fractal (Hausdorff) dimension is defined as the limit of the
quotient of the logarithm of the number of covering units and the length size of the unit:
ln( n
b)
ln( n)
D 1
ln( b)
ln( b)
(28.4)
The one corresponds to the length in the x-direction, which certainly has dimension 1. The
last term is the dimension in the y-direction. Obviously, if n b, the transformation at each