Oscillations, Waves, and Interactions - GWDG

418 U. Parlitz
Figure 9. Correlation sum Eq. (15)
vs. radius r for the Poincaré cross
section Fig. 4 of the chaotic attractor
of Duffing’s equation computed using
100000 points. The slope of ≈ 1.34
represented by the (black) line segment
is an approximation of the correlation
dimension DC, Eq. (16).
of the correlation sum
C(r) =
2
(N)(N − 1)
ln C(r)
−5
−10
−15
−20
−15 −10 −5 0
ln r
N� �i−1
H(r − �y i − y j �) (15)
i=1 j=1
that counts the relative number of neighbouring points y i and y j closer than r (H is
the Heaviside function with H(x) = 1 for x > 0 and zero elsewhere). The correlation
dimension
ln C(r)
DC = lim
(16)
r→0 ln r
describes the scaling in the limit r → 0 and equals the Rényi dimension D2.
Since any numerical simulation or experimental measurement provides finite data
sets, only, the limit r → 0 cannot be carried out in numerical computations of DC,
but only the corresponding scaling behaviour
ln C(r) ≈ DC ln r (17)
can be exploited to estimate DC as a slope in a log-log-diagram. This approach is
illustrated in Fig. 9 showing ln C(r) vs. ln r for the Poincaré section of the chaotic
Duffing attractor in Fig. 4. The slope in a suitable intermediate range of r gives
the fractal dimension of the Poincaré section of the chaotic Duffing attractor (here
estimated as DC ≈ 1.34).
Each point in the Poincaré cross section corresponds to a one-dimensional trajectory
(segment) starting at that point. Therefore, the dimension estimate in the
Poincaré section has to be increased by one to obtain the (correlation) dimension of
the full chaotic attractor (here DC ≈ 2.34).
The definition of the correlation dimension (16) is based on a given radius r and
therefore a fixed size approach. As an alternative one may also use a fixed mass
method to estimate the dimension of the attractor as was suggested by Badii and
Politi [27,28] and Grassberger [29]. In this case the k nearest neighbours of each
reference point yn and the radius rn = r(k) of this cloud of k points are determined.
For the limit k/N → 0 one obtains, for example, an approximation of the boxcounting
dimension
DB ≈ −
log 1
N
log N
� . (18)
N
n=1
rn