between a curve and an area. It has a topological dimension of non-integer value,
between one and two, that is: a fractal dimension.
There are far too few Poincaré points in Fig. 6.5(b) to reveal the finer details of
its structure. The emergence of a fractal strange attractor on a high-resolution
computer screen can be a stunning and pleasant experience, the object slowly taking
form with new points adding still more detail to the image. As a characteristic of
fractal objects, details appear at any level of magnification. Blowing up a tiny
portion of Fig. 6.5(b) you will see the sheet-like structure repeated on a smaller
scale. Subsequent magnifications can be carried on for as long as numerical resolution allows, with fine details appearing at any scale.
There are several measures quantifying the topological dimension of chaotic
attractors, e.g., the pointwise dimension, correlation dimension, information
dimension, capacity, Hausdorff dimension and Lyapunov dimension. A fine account
of these and several other measures was given by Farmer et al. (1983). In many
cases these different measures take on just about the same value.
The Lyapunov dimension d L is a useful measure based on all but the most
negative Lyapunov exponents. We adopt the following definition (Wolf et al.
1985):
d L ¼ K þ
P K
i¼1
^ k i
^ k K þ 1
;
ð6:9Þ
where the integer K is defined by the condition that
X K
i¼1
^ k i [ 0 and
X
K þ 1
i¼1
^ k i \0:
ð6:10Þ
As an example of calculating Lyapunov dimension we consider the Duffing
system (6.1), with system parameters corresponding to, respectively, stable period-2
and chaotic motion (i.e. parameters corresponding to, respectively, the (a) and the
(b) parts of Figs. 6.3, 6.4 and 6.5).
The full Lyapunov spectra were computed using the numerical algorithm
described in Wolf et al. (1985). Equation (6.9) was then employed to yield results
as shown in Table 6.1.
The attractors of the forced Duffing system reside in three-space, since there are
three autonomous equations. What we observe in Poincaré maps are
two-dimensional cross-sections of these attractors. Hence, to characterize the
Table 6.1 Lyapunov spectrum and Lyapunov dimension for Eq. (6.1) with parameters as in the
caption for Fig. 6.3
^ k 1
^ k 3
^ k 3
K
d L
Period-2 motion
0
−0.05
−0.09
1
1
Chaotic motion
0.19
0
−0.33
2
2.58
6.3 Tools for Detecting Chaotic Vibrations
339
between one and two, that is: a fractal dimension.
There are far too few Poincaré points in Fig. 6.5(b) to reveal the finer details of
its structure. The emergence of a fractal strange attractor on a high-resolution
computer screen can be a stunning and pleasant experience, the object slowly taking
form with new points adding still more detail to the image. As a characteristic of
fractal objects, details appear at any level of magnification. Blowing up a tiny
portion of Fig. 6.5(b) you will see the sheet-like structure repeated on a smaller
scale. Subsequent magnifications can be carried on for as long as numerical resolution allows, with fine details appearing at any scale.
There are several measures quantifying the topological dimension of chaotic
attractors, e.g., the pointwise dimension, correlation dimension, information
dimension, capacity, Hausdorff dimension and Lyapunov dimension. A fine account
of these and several other measures was given by Farmer et al. (1983). In many
cases these different measures take on just about the same value.
The Lyapunov dimension d L is a useful measure based on all but the most
negative Lyapunov exponents. We adopt the following definition (Wolf et al.
1985):
d L ¼ K þ
P K
i¼1
^ k i
^ k K þ 1
;
ð6:9Þ
where the integer K is defined by the condition that
X K
i¼1
^ k i [ 0 and
X
K þ 1
i¼1
^ k i \0:
ð6:10Þ
As an example of calculating Lyapunov dimension we consider the Duffing
system (6.1), with system parameters corresponding to, respectively, stable period-2
and chaotic motion (i.e. parameters corresponding to, respectively, the (a) and the
(b) parts of Figs. 6.3, 6.4 and 6.5).
The full Lyapunov spectra were computed using the numerical algorithm
described in Wolf et al. (1985). Equation (6.9) was then employed to yield results
as shown in Table 6.1.
The attractors of the forced Duffing system reside in three-space, since there are
three autonomous equations. What we observe in Poincaré maps are
two-dimensional cross-sections of these attractors. Hence, to characterize the
Table 6.1 Lyapunov spectrum and Lyapunov dimension for Eq. (6.1) with parameters as in the
caption for Fig. 6.3
^ k 1
^ k 3
^ k 3
K
d L
Period-2 motion
0
−0.05
−0.09
1
1
Chaotic motion
0.19
0
−0.33
2
2.58
6.3 Tools for Detecting Chaotic Vibrations
339