Oscillations, Waves, and Interactions - GWDG

3 Characterising complex dynamics
Complex dynamics of nonlinear systems 417
The most important quantities for characterising chaotic dynamics are attractor dimensions
and Lyapunov exponents that we shall briefly introduce in the following.
3.1 Fractal dimensions
The (fractal) dimension of an attractor quantifies its complexity and gives a lower
bound for the number of equations or variables needed for modelling the underlying
dynamical process.
The simplest concept of a fractal dimension is the box-counting dimension (or
capacity dimension). There, the point set (attractor) to be characterised is covered
with N d-dimensional hypercubes of size ε. The smaller ε is, the more cubes are
necessary and the scaling
N(ε) ∝
� �DB 1
ε
of the number of cubes N(ε) with the size ε provides, in the limit of infinitesimally
boxes, the box-counting dimension
(10)
ln N(ε)
DB = lim . (11)
ε→0 ln(1/ε)
With the box-counting dimension a hypercube is counted if it already contains just
a single point of the set to be described. In general, however, also the local density
of points is of interest, given by the probability pi to find a point in cube number i.
This probability can be estimated by the relative number of points falling in box i
and depends on the size ε. For a general point set (attractor) covered with N(ε)
d-dimensional hypercubes of size ε we obtain in this way the Rényi information of
order q
Iq = 1
1 − q ln
N(ε) �
i=1
p q
i , (12)
which is used to define the generalised (Rényi) dimension of order q
Dq = lim
ε→0
Iq
. (13)
ln(1/ε)
Note that q can be any real number, i. e., Eq. (13) describes an infinite family of
dimensions. For q = 0 the Rényi information I0 equals ln N(ε) and Eq. (13) coincides
with the definition of the box-counting dimension, D0 = DB.
From the infinite family of (generalised) dimensions Dq the correlation dimension
DC introduced by Grassberger and Procaccia [26] is often used for analysing strange
(chaotic) attractors. The correlation dimension is given by the scaling
C(r) ∝ r DC (14)