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0 9 6 0 - 0 7 7 9 ( 9 4 ) 0 0 2 5 3 - 3

Chaos, Solitons & Fraetals Vol. 6, pp. 131-135, 1995 Copyright © 1995 Elsevier Science Ltd

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T i m e and Frequency Study of Intermit tency

A. FIGLIOLA and A. S C H U S C H N Y

1)el)artamento de Ffsica, Facultad de Ciencia.s Exactas y Naturales, (UBA) Pal). 1 Ciudad Universitaria.. (1428) Buenos Aires. Argentina.

e-maih fig@d fuba.edu.ar

A b s t r a c t - Intermittency in the logistic map is studied by using Discrete Wavelet and Gabor Transforln . The evolution of the periodic zones in the appearance of bursts is ob- served as fun(:tion of the i)aranleter. The total energy of the system is found concentrated at high frequency levels, wherl more bursts are present. It is shown that the results ob- tailled with time-frequency analysis are in agreement with the prediction of the analytical model.

T I M E - F R E Q U E N C Y R E P R E S E N T A T I O N

Given a signal f(t), it is interesting to observe its frequency content locally in time, in many applications. Tile representation of a signal by means the Fourier Transform is not natural in spite of being the most commonly used. For example, music or a speech signal has a spectrum that evolves considerably in time. Fourier Transform cannot reflect the time-evolution of the frequencies.

Time localization can be achieved by using the Gabor Transform, that cuts off well-localized slices of f(t). Then, in each segment, the Fourier Transform is taken. The Gabor Transform at t ime b and frequency w of a signal f ( t ) E L2(N) is defined by [1,2]:

+co

Gg, f(b,w) = / f(t)go(t - b)e-i~tdt (1) --CO30

where g,(t) can be considered as a time-frequencial window and is known as Gabor function. Usually, Gaussian functions are used to ob.tain the best sinmltaneous localization in t ime and frequency domains.

The Wavelet Transform provides a similar time-frequency description, with a few but important differences. The Wavelet Transform of a signal f( t) is:

+co

T:,~'(I) = ao 1/2 -- J .f(t)~( t - b ) d t = < f , ~b~,b > (2) a

- - 0 0

with 1/2 ~ - - b ~o,b = I.I- ~ ( - - ) (3)

a

The difference lies in the shapes of the analyzing functions g~ and ~b~,b. All fnnctions g, have the same envelope translated to the proper time location. Meanwhile, the ~b,,b have temporal widths

131

132 A. FIGLIOLA and A. SCHUSCHNY

adapted to their fi'equency: they widths are very llarrow for high frequencies while for low frequencies they are broader. Gabor analysis is badly adapted to signals which incorporate different scales and has poor resolution for short time phenomena. [1]

For a discrete set of parameters, aj = 2 - j , bj,k = 2Jk (4)

the Wavelet Transform is evaluted just by computing the values:

and

c(j, t,) = < f , Cj,k > (5)

~/ , j , k ( t ) = 2 J / ~ , / , ( 2 J t - k) (6)

with j,k eZ. For an appropiate selection of the function ~b, the set of Wavelets ~j,k is an orthonormal busis of the Hilbert space, and it is easy to prove that:

+ ~

=, / If(t)12dt ~ ~ ~ Ic(j, k)l 2 (7) II/(t)l[ 2 - o o j k

A N A L Y S I S O F T H E T Y P E - I I N T E R M I T T E N C Y I N T H E L O G I S T I C M A P

As is well know, intermittency , is the alternation of long regular or laminar phases with short irregular or turbulent bursts. It has also been observed that the occurrence frequency of chaotic bursts increases with the control parameter of the system. So, intermittency is a continuous route from regular to chaotic motion [3]. In this work we present time-frequency patterns of intermittency obtained by using Gabor and Wavelet Transform applied to time sequences of the Logistic Map.

X k + 1 -~ Axa(1 - xa)

Also, we can verify the analytical results presented by Hirsch et al [4].

(8)

Figure 1: Five first level frequencies of Wavelet Coefficients for the series of the Logistic Map with A = A ~ - 10 -6

Time and frequency study of intermittency 133

In the region where the control i)arameter A < A,, = 1 + v~, intermit tency takes place. The laminar region is characterized by a well-defined spectral line, characteristic of regions of period three, and all wavelet, coefficients c(j, k) are very smooth. When A decreases, the coefficients of the Wavelet t rausform show changes among the smooth zones and the turbulent bursts, and they are very sensitive for" detecting the starting point of the lmrts. Figure 1 shows a diagram of tlte tive first levels of the c(j, k) coefficients.

r If we contpute the power P(j ) = ~ ~ Ic(j, k)[ 2, we can observe that the high frequency level is k

rising continllosly while the burts I)ecome m o r e [ ' r e q u e t l t . This is in agreeem('.nt with the theory of intermittency.

Table 1 shows the increment of tale energy for the first level of frequencies ( j = l ) for different values of A = A,o - 6 as a filnction of the parameter 6.

Table 1. l)ercentage of the total 1 ) for the highest level of tire frequencies

6 P( , i= l ) 10 -7 74.2 % 10 -6 77.1% 10 -s 80.2 %

When an intermit tent burst appears, the Gabor representation has a broad band spectrum with low values of l)ower. (See figure 2). For each given time, tlte sum of tile power spect rum values has been calculated for all fi'equencies. When the system is in the laminar region this sum is bigger than in the case of an intermit tent burst, due to tile significant power of the spectral line. The average length of both the laminar and the bursts regions have been calculated and it has been verified that the intermit tent bursts have a constant average length. On the other hand, in laminar regions the length is a decreasing function of the control parameter. Figure 3 shows the curve log < 1 > versus log(At - 5), where < I > is the average temporal length of the laminar regions.

o.°:t t I /10 t i ltl t o.1~- ~ ~ II ~ ~1/~ [~ i~tl~ d

0.05 ~

0 500 1000 1500 2000 2500 3000 3500 4000

Time

Figure 2: Time-frequency Gabor representation for the series of the Logistic Map with A = A~ - 10 -6

A

V

0 J

5

2 , i

1 - 1 0 - 9 - 8 - 7 - 6 - 5 - 4

Log x~-6)

134 A. FIGLIOLA and A. SCHUSCHNY

Figure 3: Logar i thm of the average laminar length versus log(A) = log(Ac - 6), and 10 -7 < < 10 -5 Dots are obtained from the time-frequency Gabor representation and the

solid line from the analytical model.

Each point of the curve was obtained from more than 800.000 data points of the logistic map (6). We fitted the average laminar lengths with the model expression of the tog < l > for a one-

dimensional map [4] as:

then,

< I > . . . . (9) 2 ~ . g

log(a) log(At - (5) log < l > = log(~) 2 2 (10)

In order to obtain the value a as in ref [4], the third iterate f ( f ( f ( x ) ) ) of the logistic map is fitted by a quadrat ic curve, with e = A~ - ~i and ~ = 10 -s. In this case a = 2.31 + 0.001. On the other hand, by fitting the Jog-log plot built fi'om the Gabor t ime-frequency representation, we obtained the va.lue of a in agreeemeut with (7) and (8), and a =- 2.3220 ~ 0.0001 whith a correlation coeflicient of p = 0.999.

C O N C L U S I O N S

Time-frequency representations are useful tools to analize nonlinear dynamical systems, especially for non s ta t ionary signals. There are remarkable advantages with theses techniques, since much information can be represented through graphic patterns that can be easily and quickly interpretated. The results agree with the analytical treatment. This shows the usefulness of the Wavelet and Gabor Transforms, even when analytical models are not avaible, as in many natural times series.

Acknowledgments: We thank E. Serrano for comments and J. Sheer for the redaction of this ~;ticle. This work was supported by PID 00571/88 C, ONICET Argentina

R E F E R E N C E S

1. I. Daubechies Ten Lectures on Wavelets, , SIAM, (1992).

Time and frequency study of intermittency 135

2. N. Delprat, B. Escudi~, P. Guillemain Asymptotic Wavelet and Gabor Analysis: Extraction of Instantaneous Frequencies, IEEE Trans.on Information Theory, Vol. 38, No. 2, (March 1992).

3. Y. Pomeau & P. Manneville, Intermittent Transition to Turbulence in Dissipative Dynamical Systents Commun. Math. Phys., 74, pp. 189 - 197, (1980).

4. J.E. Hirsch, B.A. Huberman, D.J. Scalapino, Theory of Intermittency Physical Review A, Vol. 25, No. 1, (January 1982).