122
5 Analysis of Simple Nonlinear Dynamical Systems
Table 5.8 Lyapunov exponents spectrum and LLEs computed by different methods (generalized
Hénon map)
Spectrum of LEs
Benettin method
Neural network
LEs: 0.27628 0.25770–4.04053
LEs: 0.29251 0.27104–4.04583
DKY: 2.13215
DKY: 2.13929
EKS: 0.53397
EKS: 0.56355
PVC: –3.50656
PVC: –3.48227
LLEs
Wolf method
Rosenstein method
Kantz method
Method of
synchronization
LLE: 0.45214
LLE: 0.27930
LLE: 0.26601
LLE: 0.27250
5.6 Rössler Attractor [8]
The following Rössler system of ODEs is investigated as follows:
⎧
⎪ ⎨
⎪ ⎩
˙
x = −y − z,
˙
y = x + ay,
˙
z = b + z(x − c),
(5.5)
and the computations are carried out for the following fixed parameters a = b = 0.2
and c = 5.7.
The original study yielded Lyapunov spectrum: 0.0714, 0, –5:3943 and KaplanYorke dimension equals to 2.0132.
The power spectrum contains the fundamental frequency ω 1 , which is accompanied by damped bursts (frequencies ω 2 − ω 10 ). In the whole time interval, Gauss
wavelet exhibits the brightest region of the fundamental frequency with darker peaks
going to zero. Thus, the picture is analogous to the power spectrum. Contrarily to
the studied maps, the bifurcation diagrams have a more complex structure. However, there is still correlation with the changes in LLEs for the corresponding control
parameters. The parameter b has the smallest influence on the change in LLE. Graphs
of LLEs also exhibit a more complex structure. Borders of different vibration kinds
have complex forms, which illustrates the increase in the system complexity. Aside
from the chaos and hyper-chaos zones, there are drops of hyper hyper-chaos (three
positive LEs).
As far as Table 5.11 is considered, the best results were yielded by Benettin and
Rosenstein methods. The method of neural networks gave very good results in the
case of estimates of two first LEs but underestimated the third exponent. Wolf method
yielded smaller value of the first exponent compared to the reference data. The most
underestimated results were given by Kantz method.
5 Analysis of Simple Nonlinear Dynamical Systems
Table 5.8 Lyapunov exponents spectrum and LLEs computed by different methods (generalized
Hénon map)
Spectrum of LEs
Benettin method
Neural network
LEs: 0.27628 0.25770–4.04053
LEs: 0.29251 0.27104–4.04583
DKY: 2.13215
DKY: 2.13929
EKS: 0.53397
EKS: 0.56355
PVC: –3.50656
PVC: –3.48227
LLEs
Wolf method
Rosenstein method
Kantz method
Method of
synchronization
LLE: 0.45214
LLE: 0.27930
LLE: 0.26601
LLE: 0.27250
5.6 Rössler Attractor [8]
The following Rössler system of ODEs is investigated as follows:
⎧
⎪ ⎨
⎪ ⎩
˙
x = −y − z,
˙
y = x + ay,
˙
z = b + z(x − c),
(5.5)
and the computations are carried out for the following fixed parameters a = b = 0.2
and c = 5.7.
The original study yielded Lyapunov spectrum: 0.0714, 0, –5:3943 and KaplanYorke dimension equals to 2.0132.
The power spectrum contains the fundamental frequency ω 1 , which is accompanied by damped bursts (frequencies ω 2 − ω 10 ). In the whole time interval, Gauss
wavelet exhibits the brightest region of the fundamental frequency with darker peaks
going to zero. Thus, the picture is analogous to the power spectrum. Contrarily to
the studied maps, the bifurcation diagrams have a more complex structure. However, there is still correlation with the changes in LLEs for the corresponding control
parameters. The parameter b has the smallest influence on the change in LLE. Graphs
of LLEs also exhibit a more complex structure. Borders of different vibration kinds
have complex forms, which illustrates the increase in the system complexity. Aside
from the chaos and hyper-chaos zones, there are drops of hyper hyper-chaos (three
positive LEs).
As far as Table 5.11 is considered, the best results were yielded by Benettin and
Rosenstein methods. The method of neural networks gave very good results in the
case of estimates of two first LEs but underestimated the third exponent. Wolf method
yielded smaller value of the first exponent compared to the reference data. The most
underestimated results were given by Kantz method.