116
5 Analysis of Simple Nonlinear Dynamical Systems
Table 5.2 Spectrum of Lyapunov exponents and LLEs computed by different methods (logistic
map)
LE spectrum
Benettin method
Neural network
(LEs): 0.69315
LEs: 0.69290
Dimension Kaplan–Yorke (DKY): 1
DKY: 1
Kolmogorov–Sinai entropy (KSE): 0.69315
EKS: 0.69290
Phase volume compression (PVC): 0.69315
PVC: 0.69290
LLE
Wolf method
Rosenstein method
Kantz method
Method of
synchronization
LLE: 0.99683
LLE: 0.690553
LLE: 0.31321
LLE: 0.696
X n+1 = 1 − a X
2
n + Y n ,
Y n+1 = bX n .
(5.3)
The following parameters are fixed for numerical experiments: a = 1.4, b = 0.3.
Since the equations (5.3) do not correspond to a real object, the parameters are
replaced with fixed values. Sprott et al. [5] computed Lyapunov spectrum and KaplanYorke dimension of the map using Benettin method [6] by solving (5.3). They
obtained the following LEs: λ 1 = 0.419217, λ 2 = −1.623190 and Kaplan-Yorke
dimension: 1.258267.
Similarly to the logistic map, the power spectrum exhibits a uniform noisy shape.
However, one can distinguish a dominating frequency (ω 1 ≈ 0.45). It is also visible
on the wavelet spectrum as a region of the largest amplitudes along with the whole
signal. Plots of the change in the LLE correlate with bifurcation diagrams for the
same interval of changes in the parameters a and b. Dynamics of the LLE changes
increases with the increase in both control parameters. Starting with the graphs of
LEs for a given set of control parameters, the system mainly remains in a periodic
regime, but it exhibits chaotic dynamics for large values of the control parameters.
Beginning from the results shown in Table 5.5, the majority of the employed
computational methods yielded good results. However, the most accurate results were
obtained by the neural network method (for the whole spectrum of LEs), Rosenstein
method and the method of synchronization (in the case of LLEs). Wolf and Kantz
methods gave decreased estimated values of the LLEs.
5.5 Hyperchaotic Generalized Hénon Map [7]
To obtain the hyperchaotic Hénon map, one needs to take a point (X n , Y n , Z n ) and
map it into the following one:
5 Analysis of Simple Nonlinear Dynamical Systems
Table 5.2 Spectrum of Lyapunov exponents and LLEs computed by different methods (logistic
map)
LE spectrum
Benettin method
Neural network
(LEs): 0.69315
LEs: 0.69290
Dimension Kaplan–Yorke (DKY): 1
DKY: 1
Kolmogorov–Sinai entropy (KSE): 0.69315
EKS: 0.69290
Phase volume compression (PVC): 0.69315
PVC: 0.69290
LLE
Wolf method
Rosenstein method
Kantz method
Method of
synchronization
LLE: 0.99683
LLE: 0.690553
LLE: 0.31321
LLE: 0.696
X n+1 = 1 − a X
2
n + Y n ,
Y n+1 = bX n .
(5.3)
The following parameters are fixed for numerical experiments: a = 1.4, b = 0.3.
Since the equations (5.3) do not correspond to a real object, the parameters are
replaced with fixed values. Sprott et al. [5] computed Lyapunov spectrum and KaplanYorke dimension of the map using Benettin method [6] by solving (5.3). They
obtained the following LEs: λ 1 = 0.419217, λ 2 = −1.623190 and Kaplan-Yorke
dimension: 1.258267.
Similarly to the logistic map, the power spectrum exhibits a uniform noisy shape.
However, one can distinguish a dominating frequency (ω 1 ≈ 0.45). It is also visible
on the wavelet spectrum as a region of the largest amplitudes along with the whole
signal. Plots of the change in the LLE correlate with bifurcation diagrams for the
same interval of changes in the parameters a and b. Dynamics of the LLE changes
increases with the increase in both control parameters. Starting with the graphs of
LEs for a given set of control parameters, the system mainly remains in a periodic
regime, but it exhibits chaotic dynamics for large values of the control parameters.
Beginning from the results shown in Table 5.5, the majority of the employed
computational methods yielded good results. However, the most accurate results were
obtained by the neural network method (for the whole spectrum of LEs), Rosenstein
method and the method of synchronization (in the case of LLEs). Wolf and Kantz
methods gave decreased estimated values of the LLEs.
5.5 Hyperchaotic Generalized Hénon Map [7]
To obtain the hyperchaotic Hénon map, one needs to take a point (X n , Y n , Z n ) and
map it into the following one: