Biomedical Engineering Reference
In-Depth Information
FIGURE 2.22:
Phase portraits of human ECG in three-dimensional space. A two-
dimensional projection is displayed for two values of the delay τ: (a) 12 ms and (b)
1200 ms. (c) represents the phase portrait constructed from ECG of simultaneously
recorded signals from three ECG leads. From [Babloyantz and Destexhe, 1988].
system is predictable before chaotic behavior sets in. The
Lyapunov exponent
or
Lya-
punov characteristic exponent
of a dynamical system is a quantity that characterizes
the rate of separation of infinitesimally close trajectories. Quantitatively, the separa-
tion of two trajectories in phase space with initial distance Δ
Z
0
can be characterized
by the formula:
e
λ
t
|
Δ
Z
(
t
)
|≈
|
Z
0
|
(2.129)
where λ is the Lyapunov exponent. Positive Lyapunov exponent means that the tra-
jectories are diverging which is usually taken as an indication that the system is
chaotic. The number of Lyapunov exponents is equal to the number of dimensions
of the phase space.
2.5.2 Correlation dimension
The concept of generalized dimension (special cases of which are: correlation di-
mension and Hausdorff dimension) was derived from the notion that geometrical ob-
jects have certain dimensions, e.g., a point has a dimension 0, a line—1, a surface—2;
in case of chaotic trajectories dimension is not an integer.
The measure called
correlation dimension
was introduced by [Grassberger and
Procaccia, 1983]. It involves definition of the correlation sum
C
for a collection
of points
x
n
in some vector space to be the fraction of all possible pairs of points
which are closer than a given distance ε in a particular norm. The basic formula for
(
ε
)
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