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Symbolic approach to intermittency
Volume 120. number
I
PHYSICS
26 January
LETTERS A
1987
SYMBOLIC APPROACH TO INTERMITTENCY J. DIAS DE DEUS and A. NORONHA
DA COSTA
CFMC, Av. Prof: Gama Pinto 2, 1699 Lisbon Codex, Portugal Received
18 August 1986; revised manuscript
received
3 November
1986; accepted
for publication
25 November
1986
We propose the use of the MSS universal sequences in numerical and experimental studies of type I intermittency. Pattern repetitions in these symbolic sequences can be easily measured and provide a reliable and unambiguous way of testing intermittency scaling laws and the predictions of one-dimensional maps. The limitations of the conventional acceptance gate method in intermittency are discussed. Universality of MSS sequences makes our approach, in contrast with the previous one, independent of particular features of the map.
However, the possibilities of the symbolic approach have not been fully exploited in neither numerical nor real experiments on intermittency. For instance, until now, laminar phases have been identified by requiring the response of the system to fall within an arbitrarily pre-established gate. In fact, we will see that this method is not fully reliable as the power in the E-“~ scaling is affected by the size of the gate. On the contrary, the use of symbolic sequences, as asserted in ref. [ 71, provides an experimentally accessible, well-defined method to select laminar phases. In the present paper we establish a symbolic characterization of laminar phases and briefly discuss the consequences of the usual gate procedure.
Intermittency is a common route to chaos in nonlinear dissipative systems [ 1,2]. It is characterized by the presence of sections of regular response (laminar phases) interrupted by turbulent burst. This behaviour has been succesfully modelled by Pomeau and Manneville [ 31 using one-dimensional maps of the interval in the neighbourhood of a tangent bifurcation, in the case of type I intermittency. This simple model accounts for the observed E-“~ scaling behaviour of the length of the laminar phase as the control parameter varies, where E measures the distance to the bifurcation point. Later, Hirsch, Huberman and Scalapino [4] have derived the scaling of the laminar phase in the presence of noise. Finally, Hirsch, Nauenberg and Scalapino [ 51 obtained the same results by renormalization group analysis, and showed that the Feigenbaum functional equation, with appropriate boundary conditions, also applies to intermittency. This provides a unified framework to discuss both period doubling and intermittency. On the other hand, from the point of view of symbolic sequences, these routes to chaos also show close similarities. In both cases, sequences of periodic orbits exist with definite symbolic patterns, and converge to a limiting regime with a well-determined rate of convergence [ 6,7]. There is now a well-developed theory establishing the rules of these symbolic MSS sequences [ 8,9] and they have been observed in many real systems [ lo]. 03759601/87/$ (North-Holland
03.50 0 Elsevier Science Publishers Physics Publishing Division)
‘O’ ; IO-
_.
IO‘
IO’
&
Fig. 1. The dots show I,,,, the largest number of consecutive blocks (RLR) in the MSS sequence, versus E in a log-log plot, for the logistic map. Straight line corresponds to the theoretical results for this map l,,x=C~-~‘~, with C=O.374 [ 71.
B.V.
19
Volume 120, number
1
PHYSICS
The universality of the MSS sequences makes the symbolic approach a more transparent way of testing the predictions of one-dimensional maps in the analysis of real systems. Taking the period 3 as an example. experiments should count blocks of the form (RLR). where R ( L) means the response is to the right (left) of the critical point. In general, an E dependent probability distribution P( I, E), where I is the number of consecutive blocks, can be constructed by iteration. On the other hand, according to the theory [ 71, for a given c, the MSS sequence contains a repetition of ,Y blocks. (RLR)-‘, corresponding to the largest, critical point. itinerary [ 91. The identification between experiment and theory then requires: IMA* =A..
(1)
where I, -zXis the largest value of 1. As I: -0 the length of laminar phases, contrary to bursts, increases and 171 &-+E / .,j
LETTERS
26 January
A
numerical h’= I,,,
test of the law = C’i: I’:
(3)
for the logistic map. Not only the power is right (0.500 * 0.001) but the coefficient C agrees with the predicted one: C= 0.373 i 0.005 (C= 0.374 in theory). In fig. 2 we present the probability distribution P( I. E) where I is the length of the observed sequence of blocks RLR and E the parameter value. It is remarkable that the distribution in the small-/ region is e independent. In contrast, in the large-l region, the distribution approximately satisfies an asymptotic & ‘I’ law: HI. E)------t P(I%l‘XX-1)
(2)
In the present approach it is then iy=iMAI and not the average of 1, (I), that has a well-defined meaning in connection with the E scaling. In fig. 1 we show a
(4)
’ ‘. Eq. (4) is tested in fig. 3. with lM4x=N-& From the plots of figs. 2 and 3, it is suggestive to write the full distribution approximately as a sum: P(/,E)=PH([)+P~(I.t.),
If2
I987
(5)
where Pa(I) is independent of & (apart from a minor normalization correction), corresponding to the bursts at small I. and Pi_. satisfying (4)) rises to a maximum at I,.,, and represents the true laminar behaviour, at large 1.
+ t =.,9x 10” . Id -
x
E=16’
.
E= hc~d’
Fig. 2. Probability distribution P( I, E) of blocks of the form (RLR)! tn the MSS sequence, versus I. for three drfferent values of the parameter for the logistic map. P(0, E) is the probability of a length-three block with a pattern different from RLR. Note the independence of E rn the small-l region (bursts) and the similarity of the distribution at large 1. for the different values of the parameter (see fig. 3).
20
Volume 120. number
1
PHYSICS
9
4
I
Fig. 3. A test of scaling law (4): the probability similar when plotted as a function of I-I,,,.
LETTERS
26 January
A
;,,.e ’
6
3
distributions
a
16 1
P(I, E) at large values of I, for different
In a real experiment with a forced RCL circuit, the E independent burst distribution is easily observed, while the peak at large laminar phases tends to be washed out by noise [ 41. However, for moderately small values of E, good agreement with the E - “* law, (2),isfound[ll]. We next would like to discuss the acceptance gate method that has been used to select laminar phases [ 1,2,4]. According to this procedure, a sequence of signals beginning and ending into a pre-established gate of values is considered as a laminar phase and
1987
0
values of the parameter
E, are very
its length measured. The E scaling behaviour is then tested for the mean length (I), keeping the size of the gate constant. However, for each value of the parameter E, this mean shows a clear dependence on the width of the gate (fig. 4). As a consequence, different exponents are found for the E dependence of (1). For example, a gate of 1O-*, as used in ref. [ 41, yields an exponent - 0.565 + 0.008 while a gate of 10P3 would give - 0.823 ? 0.0 17. This clearly shows that this method is not fully reliable in the study of intermittency. The method can perhaps be improved by using a gate varying as EI’*, as discussed in ref.
1111. e = 16* 4 .
o=.o,
+
0=.003
+
.
. +
0
+++
. +
+* * +++ +++:;:+++. * . . . ..--
.
-
10
. .
We finally summarize our conclusions. A symbolic approach to intermittency is presented permitting a characterization of laminar phases in terms of the experimentally accessible MSS sequences. In contrast with the acceptance gate method that has been used so far, universality of the MSS sequences makes this approach independent of particular features of the map and therefore specially suited for the analysis of real intermittency experiments.
. . 30
References Fig. 4. Probability distributions PG( I) versus I, calculated by the acceptance gate procedure for E = lo-“ and two different value of the gate G for the logistic map. Arrows indicate in each case the mean length of laminar phases, (I) G, showing its dependence on gate width, for the same value of the parameter.
[ 1 ] Y. Pomeau, J.C. Roux, A. Rossi, S. Bachelart and C. Vidal, J. Phys. (Paris) Lett. 42 (1981) L271; P. Berg& M. Dubois, P. Manneville and Y. Pomeau, J. Phys. (Paris) Lett. 41 (1980) L341.
21
Volume
120, number
1
PHYSICS
[ 21 C. Jeffries and J. Perez, Phys. Rev. A 26 (1982) 2 I 17. [ 3) P. Manneville and Y. Pomeau. Phys. Lett. A 75 (I 979) I : Y. Pomeau and P. Manneville. Commun. Math. Phys. 74 (1980) 189. [4] J.E. Hirsch, B.A. Huberman and D.J. Scalaptno. Phys. Rex. A25 (1982) 519. [ 51 J.E. Hirsch, M. Nauenberg and D.J. Scalapino. Phys. Lett. A87 (1982) 331. [6] M. Feigenbaum, J. Stat. Phys. 19 (1978) 25; 21 (1979) 669. [ 71 J. Dias de Deus, R. Dilao and .4. Noronha da Costa. Phys. Lett. A 101 (1984) 459.
LETTERS
A
26 January
1987
[8] N. Metropolis, M.L. Stein and P.R. Stein. J. Comb. Theory A 15 (1973) 25. [ 91 J. Guckenheimer, Commun. Math. Phys. 70 C1979) 133. [ IO) J.S. Testa. J. Perez and C‘. Jeffries, Phys. Rev. Lett. 48 (1982) 714; R. Simoyi. -2. Wolf and H.L. Swtnney. Phys. Rev. Lett. 49 (1982) 245: A. Libchaber. C’. Laroche and S. Fauve. J. Phys. (Paris) Lett. 43 (1982) L2I I. [ I 1 ] A. Noronha da C‘osta. thests ( 1986): and to be publtshed.
I
PHYSICS
26 January
LETTERS A
1987
SYMBOLIC APPROACH TO INTERMITTENCY J. DIAS DE DEUS and A. NORONHA
DA COSTA
CFMC, Av. Prof: Gama Pinto 2, 1699 Lisbon Codex, Portugal Received
18 August 1986; revised manuscript
received
3 November
1986; accepted
for publication
25 November
1986
We propose the use of the MSS universal sequences in numerical and experimental studies of type I intermittency. Pattern repetitions in these symbolic sequences can be easily measured and provide a reliable and unambiguous way of testing intermittency scaling laws and the predictions of one-dimensional maps. The limitations of the conventional acceptance gate method in intermittency are discussed. Universality of MSS sequences makes our approach, in contrast with the previous one, independent of particular features of the map.
However, the possibilities of the symbolic approach have not been fully exploited in neither numerical nor real experiments on intermittency. For instance, until now, laminar phases have been identified by requiring the response of the system to fall within an arbitrarily pre-established gate. In fact, we will see that this method is not fully reliable as the power in the E-“~ scaling is affected by the size of the gate. On the contrary, the use of symbolic sequences, as asserted in ref. [ 71, provides an experimentally accessible, well-defined method to select laminar phases. In the present paper we establish a symbolic characterization of laminar phases and briefly discuss the consequences of the usual gate procedure.
Intermittency is a common route to chaos in nonlinear dissipative systems [ 1,2]. It is characterized by the presence of sections of regular response (laminar phases) interrupted by turbulent burst. This behaviour has been succesfully modelled by Pomeau and Manneville [ 31 using one-dimensional maps of the interval in the neighbourhood of a tangent bifurcation, in the case of type I intermittency. This simple model accounts for the observed E-“~ scaling behaviour of the length of the laminar phase as the control parameter varies, where E measures the distance to the bifurcation point. Later, Hirsch, Huberman and Scalapino [4] have derived the scaling of the laminar phase in the presence of noise. Finally, Hirsch, Nauenberg and Scalapino [ 51 obtained the same results by renormalization group analysis, and showed that the Feigenbaum functional equation, with appropriate boundary conditions, also applies to intermittency. This provides a unified framework to discuss both period doubling and intermittency. On the other hand, from the point of view of symbolic sequences, these routes to chaos also show close similarities. In both cases, sequences of periodic orbits exist with definite symbolic patterns, and converge to a limiting regime with a well-determined rate of convergence [ 6,7]. There is now a well-developed theory establishing the rules of these symbolic MSS sequences [ 8,9] and they have been observed in many real systems [ lo]. 03759601/87/$ (North-Holland
03.50 0 Elsevier Science Publishers Physics Publishing Division)
‘O’ ; IO-
_.
IO‘
IO’
&
Fig. 1. The dots show I,,,, the largest number of consecutive blocks (RLR) in the MSS sequence, versus E in a log-log plot, for the logistic map. Straight line corresponds to the theoretical results for this map l,,x=C~-~‘~, with C=O.374 [ 71.
B.V.
19
Volume 120, number
1
PHYSICS
The universality of the MSS sequences makes the symbolic approach a more transparent way of testing the predictions of one-dimensional maps in the analysis of real systems. Taking the period 3 as an example. experiments should count blocks of the form (RLR). where R ( L) means the response is to the right (left) of the critical point. In general, an E dependent probability distribution P( I, E), where I is the number of consecutive blocks, can be constructed by iteration. On the other hand, according to the theory [ 71, for a given c, the MSS sequence contains a repetition of ,Y blocks. (RLR)-‘, corresponding to the largest, critical point. itinerary [ 91. The identification between experiment and theory then requires: IMA* =A..
(1)
where I, -zXis the largest value of 1. As I: -0 the length of laminar phases, contrary to bursts, increases and 171 &-+E / .,j
LETTERS
26 January
A
numerical h’= I,,,
test of the law = C’i: I’:
(3)
for the logistic map. Not only the power is right (0.500 * 0.001) but the coefficient C agrees with the predicted one: C= 0.373 i 0.005 (C= 0.374 in theory). In fig. 2 we present the probability distribution P( I. E) where I is the length of the observed sequence of blocks RLR and E the parameter value. It is remarkable that the distribution in the small-/ region is e independent. In contrast, in the large-l region, the distribution approximately satisfies an asymptotic & ‘I’ law: HI. E)------t P(I%l‘XX-1)
(2)
In the present approach it is then iy=iMAI and not the average of 1, (I), that has a well-defined meaning in connection with the E scaling. In fig. 1 we show a
(4)
’ ‘. Eq. (4) is tested in fig. 3. with lM4x=N-& From the plots of figs. 2 and 3, it is suggestive to write the full distribution approximately as a sum: P(/,E)=PH([)+P~(I.t.),
If2
I987
(5)
where Pa(I) is independent of & (apart from a minor normalization correction), corresponding to the bursts at small I. and Pi_. satisfying (4)) rises to a maximum at I,.,, and represents the true laminar behaviour, at large 1.
+ t =.,9x 10” . Id -
x
E=16’
.
E= hc~d’
Fig. 2. Probability distribution P( I, E) of blocks of the form (RLR)! tn the MSS sequence, versus I. for three drfferent values of the parameter for the logistic map. P(0, E) is the probability of a length-three block with a pattern different from RLR. Note the independence of E rn the small-l region (bursts) and the similarity of the distribution at large 1. for the different values of the parameter (see fig. 3).
20
Volume 120. number
1
PHYSICS
9
4
I
Fig. 3. A test of scaling law (4): the probability similar when plotted as a function of I-I,,,.
LETTERS
26 January
A
;,,.e ’
6
3
distributions
a
16 1
P(I, E) at large values of I, for different
In a real experiment with a forced RCL circuit, the E independent burst distribution is easily observed, while the peak at large laminar phases tends to be washed out by noise [ 41. However, for moderately small values of E, good agreement with the E - “* law, (2),isfound[ll]. We next would like to discuss the acceptance gate method that has been used to select laminar phases [ 1,2,4]. According to this procedure, a sequence of signals beginning and ending into a pre-established gate of values is considered as a laminar phase and
1987
0
values of the parameter
E, are very
its length measured. The E scaling behaviour is then tested for the mean length (I), keeping the size of the gate constant. However, for each value of the parameter E, this mean shows a clear dependence on the width of the gate (fig. 4). As a consequence, different exponents are found for the E dependence of (1). For example, a gate of 1O-*, as used in ref. [ 41, yields an exponent - 0.565 + 0.008 while a gate of 10P3 would give - 0.823 ? 0.0 17. This clearly shows that this method is not fully reliable in the study of intermittency. The method can perhaps be improved by using a gate varying as EI’*, as discussed in ref.
1111. e = 16* 4 .
o=.o,
+
0=.003
+
.
. +
0
+++
. +
+* * +++ +++:;:+++. * . . . ..--
.
-
10
. .
We finally summarize our conclusions. A symbolic approach to intermittency is presented permitting a characterization of laminar phases in terms of the experimentally accessible MSS sequences. In contrast with the acceptance gate method that has been used so far, universality of the MSS sequences makes this approach independent of particular features of the map and therefore specially suited for the analysis of real intermittency experiments.
. . 30
References Fig. 4. Probability distributions PG( I) versus I, calculated by the acceptance gate procedure for E = lo-“ and two different value of the gate G for the logistic map. Arrows indicate in each case the mean length of laminar phases, (I) G, showing its dependence on gate width, for the same value of the parameter.
[ 1 ] Y. Pomeau, J.C. Roux, A. Rossi, S. Bachelart and C. Vidal, J. Phys. (Paris) Lett. 42 (1981) L271; P. Berg& M. Dubois, P. Manneville and Y. Pomeau, J. Phys. (Paris) Lett. 41 (1980) L341.
21
Volume
120, number
1
PHYSICS
[ 21 C. Jeffries and J. Perez, Phys. Rev. A 26 (1982) 2 I 17. [ 3) P. Manneville and Y. Pomeau. Phys. Lett. A 75 (I 979) I : Y. Pomeau and P. Manneville. Commun. Math. Phys. 74 (1980) 189. [4] J.E. Hirsch, B.A. Huberman and D.J. Scalaptno. Phys. Rex. A25 (1982) 519. [ 51 J.E. Hirsch, M. Nauenberg and D.J. Scalapino. Phys. Lett. A87 (1982) 331. [6] M. Feigenbaum, J. Stat. Phys. 19 (1978) 25; 21 (1979) 669. [ 71 J. Dias de Deus, R. Dilao and .4. Noronha da Costa. Phys. Lett. A 101 (1984) 459.
LETTERS
A
26 January
1987
[8] N. Metropolis, M.L. Stein and P.R. Stein. J. Comb. Theory A 15 (1973) 25. [ 91 J. Guckenheimer, Commun. Math. Phys. 70 C1979) 133. [ IO) J.S. Testa. J. Perez and C‘. Jeffries, Phys. Rev. Lett. 48 (1982) 714; R. Simoyi. -2. Wolf and H.L. Swtnney. Phys. Rev. Lett. 49 (1982) 245: A. Libchaber. C’. Laroche and S. Fauve. J. Phys. (Paris) Lett. 43 (1982) L2I I. [ I 1 ] A. Noronha da C‘osta. thests ( 1986): and to be publtshed.