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

Chapter 28. Deterministic chaos 299
book, as in most experimental situations, one considers dissipative processes, where energy
is not conserved but spread out to undefined, low-level degrees of freedom (see e.g. Jackson,
1990). A dissipative system shall be contracting, which means that a region of initial points
in the variable space shall always be shrinking with time, and eventually be confined to an
attracting set with lower dimensions than the entire space.
When studying several trajectories starting close to a given one at t t 0 , one finds that there
are varying tendencies along different directions. Eq. (28.8) provides a dominating behaviour,
but there may be a weaker stretching along other directions, and there must be contractions
along further directions. In fact, a dissipative process means that there shall be an overall
contraction. As described for the baker’s transformation, there are further Lyapunov exponents,
in principle given as eigenvalues of a linearised equation around the chaotic trajectory.
The number of exponents is the same as the dimension of the variable space, i.e. equal to the
number of degrees of freedom. These shall be both positive (representing stretching) and
negative (representing contraction). Their total sum shall be negative for dissipative systems,
zero for conservative ones. (For these questions, see e.g. Chapter 7 in Jackson, 1990).
The combination of stretching and contracting in different directions leads to the typical
fractal character of a chaotic attractor as was seen for the baker’s transformation. The relation
(28.5) is suggested to be a general relation, proposed by Kaplan and Yorke (1979)
between the fractal dimension and Lyapunov exponents (see also the discussion in Jackson
(1990, Chapter 7) or (Farmer et al. (1983)). Let the number of variables (the dimension) be,
m, and that the exponents are ordered so that l 1 l 2 l 3 … l m . Further let k be the
largest integer so that k i1l i 0. k is the number of dimensions for which there is not a
contraction. Note that l k may well be negative. (For a conservative system, k is equal to m,
the total dimension, while k m for a dissipative system.). Then the Kaplan–Yorke expression
for the fractal dimension is:
∑
i
D KY k l
1
l
k
k
1
(28.9)
The motivation of this is given by consideration of mappings with well-defined stretching
and contracting. It will be shown by an explicit model in the next section. It is found to
agree with the fractal dimensions for some standard chaotic models (we say more about
this later) (see Jackson, 1990 and further references therein).
The Lyapunov exponents as well as fractal dimensions should be regarded as global values
for the entire attracting set. It is possible to make definitions that are valid locally and
vary in space. Actual characteristic values should then be regarded as some averages over such
values. Large local variations lead to difficulties of calculating proper values. Most investigations
on chaos concern what could be called “normal chaos” or “low-dimensional chaos”,
which is described by a relatively small number of variables (not more than 3–5), with just
one positive Lyapunov exponent and then a fractal character. Of course, one cannot in general
expect that chaos encountered in a realistic situation would be of a simple kind.
Another fruitful way of characterising chaos is over sequences of periodic trajectories
that become unstable. A well-known early characterisation of the onset of chaos has been