Information Technology Reference
In-Depth Information
1
t
λ
i
=
lim
t
log
A
(
x
0
,
t
)
e
i
,
(16.11)
→
∞
then
λ
i
,
i
=
1
,...,
n
, are the Lyapunov exponents. They are ordered by their mag-
nitudes
λ
1
≥
λ
2
≥···≥
λ
n
, and if they are independent of
x
0
, the system is called
ergodic
.
Therefore, one can write
A
(
x
0
,
t
)
as the product of
n
×
n
matrices
A
(
x
j
,
Δ
t
)
, where
each one maps
x
j
=
φ
(
x
0
,
j
Δ
t
)
to
x
j
+
1
:
k
1
j
=
0
A
(
x
j
,
Δ
t
)
,
−
A
(
x
0
,
k
Δ
t
)=
(16.12)
with
k
Δ
t
=
t
.
16.3.2 Implementation Details
We often have no knowledge of the nonlinear equations of the system which pro-
duce the observed time series. But there is a possibility of estimating the linearized
flow map
A
Δ
t
=
D
φ
(
x
j
,
Δ
t
)
from a single trajectory by using the recurrent structure
of strange attractors. Let
, denote a time series of some physical
quantity measured at the discrete time interval
{
x
j
}
,
j
=
1
,
2
,...
Δ
t
, i.e.,
x
j
=
x
(
t
0
+(
j
−
1
)
Δ
t
)
. Con-
sider a small ball of radius
centered at the orbital point
x
j
, and find a set of
N
difference vectors included in this ball, i.e.,
ε
N
2
{
y
i
}
=
{
x
j
−
x
i
|
x
j
−
x
i
≤
ε
},
i
=
1
,
2
,...,
,
(16.13)
···
where
y
i
is the displacement vector between
x
j
and
x
i
. Here,
denotes a usual
w
1
+
w
2
+
...
+
w
n
)
1
/
2
Euclidean norm defined as follows:
w
=(
for some vec-
tor
w
=(
w
1
,
w
2
,...,
w
n
)
. After the evolution of a time interval
k
Δ
t
,
y
i
=
x
j
−
x
i
is
mapped to the set
{
}
=
{
x
j
+
k
−
x
i
+
k
},
=
,
,...,
.
z
i
i
1
2
N
(16.14)
If the radius
ε
and the evolution time
Δ
t
are small enough for the displacement
vectors
{
y
i
}
and
{
z
i
}
to be regarded as a good approximation of tangent vectors in
the tangent space
TM
, the evolution of
y
i
to
z
i
can be represented by some matrix
A
j
as
z
i
=
A
j
y
i
.
(16.15)
The matrix
A
j
should be a good approximation of the matrix of linearized flow in
Equation (16.9). A plausible procedure for optimal estimation is the least-square
2
In the implementation, among the
N
displacement vectors found inside the sphere of radius
ε
,
only five to seven vectors with the smallest norm are chosen.
N
is often chosen as
d
E
≤
N
≤
20 [28]
and is kept at a low value to optimize the efficiency of the algorithm.
