114
5 Analysis of Simple Nonlinear Dynamical Systems
one to obtain information about higher order features hidden in the investigated
signal.
The eighth-order Gauss wavelets are defined in the following form:
g 8 (x) = −(105 − 420x
2
+ 210x
4
− 28x
6
+ x
8
)exp
−x 2
2 .
(5.1)
5.3 Logistic Map [2]
In this section, we study simple classical systems (Tables 5.1, 5.4, 5.7, 5.10, 5.13) with
emphasis put on a comparison of the LEs (Tables 5.2, 5.5, 5.8, 5.11, 5.14) obtained
using Wolf, Rosenstein, Kantz and neural network methods. The convergence of the
mentioned methods, depending on the number of iteration steps, is illustrated and
discussed (Tables 5.3, 5.6, 5.9, 5.12, 5.15).
A logistic map describes how the population changes with respect to time:
X n+1 = R X n (1 − X n ).
(5.2)
Here, X n takes the values from 0 to 1 and presents the population in the nth year, whereas X 0 denotes the initial population (in the year 0); R is a positive
parameter characterizing an increase in the population (computations were carried
out for R = 4).
The first Lyapunov exponent and Kaplan-Yorke dimension were estimated by
Sprott [3]. He obtained: λ 1 = 0.693147181, and Kaplan-Yorke dimension: 1.0.
Tables 5.1, 5.4, 5.7, 5.10, 5.13 report the following results: (a) signal; (b) signal
window; (c) Poincaré pseudo-map; (d) Fourier power spectrum; (e) Gauss 8 wavelet;
(f) bifurcation diagram with LLE; (g) graphs of LEs on the control parameters plane.
The power spectrum is noisy and it is not possible to distinguish the dominating
frequency. A similar situation is exhibited by Gauss wavelet, where a large set of
frequencies is visible. They are varied with respect to power, the whole interval of
the signal changes and the estimated LLEs correlate with the bifurcation diagram for
the same interval of the control parameter r .
As can be seen in Table 5.2, all computational methods were compared with
Benettin’s original results. A good coincidence was exhibited by the neural network
method, Rosenstein method and the method of synchronization. Kantz/Wolf method
gave decreased/increased value of LLE in comparison to the original value.
5.4 Hénon Map [4]
Hénon map takes a point (X n , Y n ) and maps it into another point by the following
formulas:
5 Analysis of Simple Nonlinear Dynamical Systems
one to obtain information about higher order features hidden in the investigated
signal.
The eighth-order Gauss wavelets are defined in the following form:
g 8 (x) = −(105 − 420x
2
+ 210x
4
− 28x
6
+ x
8
)exp
−x 2
2 .
(5.1)
5.3 Logistic Map [2]
In this section, we study simple classical systems (Tables 5.1, 5.4, 5.7, 5.10, 5.13) with
emphasis put on a comparison of the LEs (Tables 5.2, 5.5, 5.8, 5.11, 5.14) obtained
using Wolf, Rosenstein, Kantz and neural network methods. The convergence of the
mentioned methods, depending on the number of iteration steps, is illustrated and
discussed (Tables 5.3, 5.6, 5.9, 5.12, 5.15).
A logistic map describes how the population changes with respect to time:
X n+1 = R X n (1 − X n ).
(5.2)
Here, X n takes the values from 0 to 1 and presents the population in the nth year, whereas X 0 denotes the initial population (in the year 0); R is a positive
parameter characterizing an increase in the population (computations were carried
out for R = 4).
The first Lyapunov exponent and Kaplan-Yorke dimension were estimated by
Sprott [3]. He obtained: λ 1 = 0.693147181, and Kaplan-Yorke dimension: 1.0.
Tables 5.1, 5.4, 5.7, 5.10, 5.13 report the following results: (a) signal; (b) signal
window; (c) Poincaré pseudo-map; (d) Fourier power spectrum; (e) Gauss 8 wavelet;
(f) bifurcation diagram with LLE; (g) graphs of LEs on the control parameters plane.
The power spectrum is noisy and it is not possible to distinguish the dominating
frequency. A similar situation is exhibited by Gauss wavelet, where a large set of
frequencies is visible. They are varied with respect to power, the whole interval of
the signal changes and the estimated LLEs correlate with the bifurcation diagram for
the same interval of the control parameter r .
As can be seen in Table 5.2, all computational methods were compared with
Benettin’s original results. A good coincidence was exhibited by the neural network
method, Rosenstein method and the method of synchronization. Kantz/Wolf method
gave decreased/increased value of LLE in comparison to the original value.
5.4 Hénon Map [4]
Hénon map takes a point (X n , Y n ) and maps it into another point by the following
formulas: