Information Technology Reference
In-Depth Information
algorithm, which minimizes the average of the squared error norm between
z
i
and
A
j
y
i
with respect to all components of
A
j
as follows:
N
i
=
1
z
i
−
A
j
y
i
1
N
2
min
A
j
S
=
,
(16.16)
a
kl
∈
R
,
subject to
(16.17)
where
a
kl
denotes the
t
in the
renormalization and the approximation process do not necessarily have to be the
same, but are chosen equal for convenience.
Each invertible
n
(
k
,
l
)
component of matrix
A
j
. The evolution times
Δ
n
matrix can be split uniquely into the product of an upper
triangular matrix
R
and an orthogonal matrix
Q
, such that
×
=
=
,
A
j
E
j
Q
j
R
j
E
j
+
1
R
j
(16.18)
e
j
,...,
e
j
)
with
E
j
. The matrix
Q
j
serves as the new basis
E
j
+
1
and the loga-
rithms of the diagonal elements of
R
j
are local expanding coefficients, whose time-
averaged values are the Lyapunov exponents. Using
=(
k
j
=
0
A
(
x
j
,
Δ
t
)
E
0
=
Q
k
−
1
k
−
1
−
1
j
=
0
R
j
A
(
x
0
,
k
Δ
t
)
E
0
=
(16.19)
in Equation (16.10), we obtain
k
1
j
=
0
log
r
ii
,
−
1
λ
=
lim
k
→
∞
(16.20)
i
k
Δ
t
where
r
ii
are the diagonal elements of the matrix
R
j
.
In the numerical procedure, we let
A
j
operate on an arbitrary chosen set
e
i
}
,
and then renormalize
A
j
e
i
to have unit length. Mutual orthogonality of the basis is
maintained by using the Gram-Schmidt renormalization procedure. This is repeated
for
n
iterations where Equation (16.20) is computed each time [25].
{
16.3.2.1 Phase Space Reconstruction
One important reason for using the above approach to computing
is that the sta-
bility of the dynamical system can be determined without actually knowing and
solving the underlying differential equations explicitly. This occurs when we obtain
a chaotic time series from a dynamical system, reconstruct its strange attractor in
the corresponding phase space, and then compute the Lyapunov exponents from the
reconstructed strange attractor directly, without its explicit mathematical model.
The most important phase space reconstruction technique is the
method of de-
lays
. The basic idea is very simple. We use the time series data of a single variable
λ
