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It can be shown that the potential supports bounded motion for the particle for
H
<
1
/
6. Thus, to fulfill this condition, the initial values are chosen as x 0 =
0, y 0 =
0
.
15, p x , 0 =
0
.
50, and p y , 0 =
0.
16.5 Computational Experiments
This section presents the main results of the implemented algorithm and compares
these with two other algorithms described in [28, 34]. Some enhancements are made
to the algorithm, which is finally tested for robustness against change in parameter
values and noise.
16.5.1 Numerical Computations
The differential equations are integrated with the Runge-Kutta (RK4) method using
afixedstep
t . This method is reasonably simple and robust, even without the adap-
tive step-size routine. The RK4 method is a fourth-order method, meaning that the
error per step is
δ
5
4
O (( δ
)
)
O (( δ
)
)
.
Table 16.2 summarizes the values for the computed Lyapunov exponents, which
has been estimated using three different implemented algorithms described in
t
, while the total accumulated error has order
t
Table 16.2: Results of the preliminary computational experiments for n = 2 , 000. For the Lorenz
attractor, the parameter values have been chosen as:
τ =
9,
Δ
t
=
5,
δ
t
=
0
.
01,
ε =
1
.
20. The
parameter values for the R ossler attractor are:
τ =
6,
Δ
t
=
5,
δ
t
=
0
.
12,
ε =
0
.
28
System with
n = 2 , 000
n = 4 , 000
initial condition
Pardalos
Sano
Wolf
Pardalos
Sano
Wolf
Lorenz
x 0
=
0
λ
1
.
08547
1
.
08546
0
.
86241
1.15313
1.15312
0.92140
1
y 0
=
1
.
0
λ
0
.
30421
0
.
30421
0
.
31226
0
.
31232
2
z 0
=
0
λ
10
.
61753
10
.
61754
10
.
57328
10
.
57319
3
R ossler
x 0 = 0 . 1
λ 1
0 . 06928
0 . 06924
0 . 06642
0.06891
0.06881
0.06593
y 0 = 0 . 1
λ 2
0 . 00132
0 . 00156
0.00122
0.00131
z 0 = 0 . 1
λ 3
1 . 27202
1 . 27183
1 . 28908
1 . 28849
Henon
x 0 = 0 . 1
λ 1
0 . 41672
0 . 41672
0 . 40445
0.41669
0.41672
0.41547
y 0 =
.
.
.
.
.
0
1
λ 2
1
57647
1
57647
1
57765
1
57773
Henon-Heiles
x 0 =
0
λ 1
0
.
15207
0
.
15209
0
.
16012
0.14875
0.14874
0.15493
y 0 =
0
.
15
λ 2
0
.
01950
0
.
01949
0.01382
0.01385
p x , 0 =
0
.
50
λ 3
0
.
04242
0
.
04245
0
.
04874
0
.
04871
p y , 0 =
0
λ 4
0
.
23309
0
.
23395
0
.
22142
0
.
22141
 
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