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exponent values depend on
Δ
t and
τ
. The two largest Lyapunov exponents,
λ 1 and
λ 2 , seem to be rather stable in the vicinity of the chosen values
Δ
t
=
5 and
τ =
9
while
40. What
is important to be reminded of here is that the systems are extremely sensitive due
to their chaotic nature, and the observed “errors” due to perturbations of parameter
values do not have to be entirely blamed on this specific algorithm. The science of
choosing the right parameter values for these kind of problems is not general and
depends on which system you are examining.
Figure 16.5 shows how the Lyapunov exponents for the R ossler attractor depend
λ 3 is unstable for all values in the interval 1
< Δ
t
<
30, 1
< τ <
on
τ
and
Δ
t .Again,
λ 1 and
λ 2 are stable in the neighborhood of the chosen values
τ =
λ 3 is very irregular throughout the interval. This behavior
is given a deeper theoretical explanation in [28].
6 and
Δ
t
=
5, while
1
0.2
0
0.5
−0.2
0
−0.4
λ
λ
−0.6
−0.5
−0.8
−1
−1
−1.2
−1.5
−1.4
0
5
10
15
20
25
30
35
40
0
5
10
15
20
25
30
τ
Δt
(a)
(b)
Fig. 16.5: The Lyapunov exponents as a function of ( a )
τ
and ( b )
Δ
t for the Rossler
attractor.
The algorithm has also been tested for noise contaminated data. We have added
Gaussian white noise w
(
t
)
to the solutions of the Lorenz and R ossler systems ac-
cording to
x noise =
x clean +
w
(
t
) σ (
x
)
s
y noise =
y clean +
w
(
t
) σ (
y
)
s
z noise =
z clean +
w
(
t
) σ (
z
)
s
,
where s is a scaling factor for the standard deviation
.
Figure 16.6 shows the dependence between the Lyapunov exponents and the scal-
ing factor within the interval 0
σ
s
0
.
1. We see that the exponents
λ 1 and
λ 2 for
both systems are quite stable within the interval 0
<
s
<
0
.
01, i.e., they are not sen-
sitive to data contaminated with up to 0
of Gaussian white noise. Once again
we can confirm the sensitive nature of the smallest Lyapunov exponent
.
01
σ
λ 3 of the
Lorenz attractor.
Many physical signals, including the time series studied in this chapter, are fun-
damentally different from linear time-invariant signals in that they are invariant to
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