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25
3.5
3
20
2.5
15
2
10
1.5
5
λ
λ
1
0
0.5
−5
0
−10
−0.5
−15
−1
−20
−1.5
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.1
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.1
s
s
(a)
(b)
Fig. 16.6: The Lyapunov exponent values as a function of the scaling factor s for ( a )
the Lorenz attractor and ( b )theRossler attractor.
scale rather than to translation. The signals are often mixed with noise, and the sep-
aration may be very difficult if both the signal and the noise are broadband. The
problem becomes inherently difficult when the signal is chaotic because its power
spectrum is indistinguishable from a broadband noise, as in our case. Since there
is a strong relationship between these fractal signals and the wavelet transform,
the latter appears to be the natural signal processing technique, just as the Fourier
transform is natural for the linear time-invariant signals [14]. Many new filtering
techniques to handle these problems are still under development.
16.6 Summary and Conclusion
The Lyapunov exponents are conceptually the most basic indicators of determinis-
tic chaos of dynamical systems. For the analysis of such dynamics, many numer-
ical algorithms to determine the spectrum of the Lyapunov exponents have been
proposed. In this chapter, we considered an optimization technique for calculating
tangent maps with the aim of developing a robust algorithm. We have described
a method which is shown to behave well in the perturbation of certain parameter
values, but slightly sensitive in the presence of noise. This method uses the Gauss-
Newton algorithm to solve the least-squares problem that arises, which is no more
complicated to implement than the linear method. By using the new optimization
method, we could obtain good estimates of the Lyapunov spectrum from the ob-
served time series in a very systematic way.
References
1. Abarbanel, H., Brown, R., Kennel, M. Variation of Lyapunov exponents in chaotic systems:
Their importance and their evaluation using observed data. J Nonlinear Sci 2 , 343-365 (1992)
2. Chen, G., Lai, D. Making a dynamical system chaotic: Feedback control of Lyapunov expo-
nents for discrete-time dynamical systems. IEEE Trans Circuits Syst I Fundam Theory Appl
44 , 250-253 (1997)
 
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