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Statistics of local expansion rates for chaotic systems
PhysicsLettersA 183 (1993) 63-68 North-Holland
PHYSICS LETTERS A
Statistics of local expansion rates for chaotic systems W. Je~.ewski Institute of MolecularPhysics,PolishAcademyof Sciences, Smoluchowskiego 17/19, PL-60-179Poznan, Poland Received 18 May 1993;revisedmanuscript received6 August 1993;acceptedfor publication 30 September1993 Communicatedby A.R. Bishop
Statistical propertiesof the fluctuationdynamicsof local expansionrates are studied for one-dimensionalsystems.The probability density for these finite-time rates is found to be, in general, asymmetric,even for values of local expansion rates close to their ensemble-averagevalue and for longtime intervals. It is shownthat the probabilitydensity takes on an exponentialform in a tail regionassociatedwith localexpansionrates less than their ensemble-averagevalue.
1. Introduction
One of the important global approaches to the study of chaotic systems is the fluctuation-spectrum theory [ 1-4 ]. This approach concerns temporal and spatial fluctuations of local (finite-time) averages lk-I
The dependence of 2q on q reflects different characteristics of the large-scale dynamics embedded in time series. As k-,ov, the time average ak approaches an ensemble average, but for finite, even very large k, it is a fluctuating variable. The probability density that O~ktakes a value a can be expressed as
Otk(j) = ~ ~ U(Xi+jk), j=O, 1, ...,
[/2--1
iffi0
pk(Ot)= l i r a - ~: J(O~--Olk(j)) .
of time series {u(x~); i=0, 1.... } generated by the chaotic dynamics x~+t =f~(x~) with a a control parameter, and xieJ, where J denotes an interval of length dependent, in general, on a (for applications of the theory to higher-dimensional systems, see ref. [ 5 ] ). The central role in the fluctuation-spectrum theory plays the characteristic function 2q defined by
J
(1.1) where q is the so-called filtering parameter and
p ( x ) = lim -1 ~ 1
~(x-x~)
is the natural invariant measure, satisfying the Frobenius-Perron equation
p ( x ) = ~ J ( x - f ~ ( y ) ) p ( y ) dy. J
Accordingly, 2¢ can be written as [ 1,3 ]
2q=q -1 lira k - l In ~ exp(qka)pk(a) dot.
(1.2) For sufficiently large k, the probability density function (PDF) has been argued to take near its peak position the Gaussian form [6], i.e.
2 ¢ = q -1 lim k -1 In f exp[qkOtk(X)]p(x) d r , k~oo
n ~ o o n jffiO
pk(Ot) "" ~
exp[- (a-2o)2k/4D] ,
(1.3)
with 2o= (tXk), where ( ) denotes the ensemble average, and D the diffusion coefficient, given for large k by D,,- ½k( (¢tk--~,0)2 > .
(1.4)
The Gaussian approximation is appropriate for small Iq I, much smaller than the convergence radius x of the cumulant expansion [2 ] of 2¢. For a given small q, the condition Iq[ ~ x is satisfied i f x is sufficiently
0375-9601/93/$ 06.00 © 1993 ElsevierSciencePublishers B.V. All rights reserved.
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large, i.e. when the amplitude of fluctuations of otk is small [ 1,2,4 ]. Consequently, the small- Iq I branch of 4q reflects the diffusive character of the dynamics of ak. The non-Gaussianity ofPk (ot), which arises in tail regions, manifests itself in the higher-order statistics when Iql >> x. This branch of 4q refers to the intermittent (laminar) structure of the dynamics [ 1,2,4]. Thus, the characteristic function is related with both the Gaussian and non-Gaussian regimes of the PDF for long-time averages, and characterizes two complementary processes i.e. diffusion and intermittency, connected respectively with these two regimes [ 1,7 ]. Here, it is shown that the PDF of local expansion rates (LERs) is, in general, asymmetric. The asymmetry of the PDF is found to be especially distinct in tail regions, but appears also in the vicinity of the position of the ensemble average of these rates. It is also shown that in a tail region associated with LERs less than their ensemble-average value, the PDF evolves from a Gaussian form to an exponential one as ot--,- oo. To investigate the shape of the PDF, the statistics of LERs is analyzed for tail regions of different scales. Contrary to the fluctuation-spectrum theory, this method does not involve the concept of the characteristic function.
2. Properties of the probability density of local expansion rates In this section general properties of the probability density for LERs, in particular the validity of the Gaussian approximation (1.3) for small deviations of LERs from their ensemble-average value, are discussed. The results obtained for various one-dimensional nonlinear maps have turned out to be qualitatively consistent. Therefore, the detailed numerical analysis presented below refers to one, representative, dynamical system, namely the quadratic map x,+ 1= 1 - axe. In order to examine the form of the PDF Pk(ot), let us introduce the asymmetry function, defined (for a given k) by
z'k(e) = n _ (k, e)/n+ (k, e ) , with 64
(2.1)
29 November 1993 ).q:~ /t
n:~ (k, e) = _+ j Pk(ot) dot,
(2.2)
where ~1>O, ot denotes a value of the LER, represented by
otk(.J) = ~: i~=ol n l f " (xi+jk) l ,
(2.3)
( f " ( X) --=dfa (X) / dx) and 4 - 4 o is the Lyapunov exponent. The function Vk(e) characterizes the deviation of the shape of pk(ot) from a symmetric form with respect to the position of 4 (note that, for any symmetric probability density with peak located at the ensemble-average value, vk(e) = 1 for all e). The asymmetry function can be determined by counting the PDF Pk(ot) over the regions ot < 4 and ot > 4 + e. Numerical results obtained for typical chaotic systems show that, for each finite k, Vk(E) < 1 when e is sufficiently small (including e = 0 ) , and Vk(e)> 1 when E is large. Moreover, it proves that vk(0)--, 1 as k--,~. However, for any constant e>0, Vk(E) increases to infinity as k grows. Consequently, the range of e, in which the deviation of vk(e) from the limit value v~ (0) = 1 is negligible, shrinks when k increases. As an illustrative example of a typical behavior of Vk(E), values of this function computed for three chosen values of the control parameter of the quadratic map are shown in tables 1 and 2 for k = 102 and k = 103, respectively. Accordingly, the PDF Pk(ot) exhibits asymmetry over the whole range of ot and for every k. However, for each sufficiently large k, there exists a range of ot, dependent on k, in which the asymmetry is insignificant. It appears that, when k grows, this range shrinks, but the height of the PDF at the ensembleaverage value 4 increases, so that most of the ensemble members belong to the range. The Gaussian region, determined by [7] lot-41 ~ O ( 2 ~ ) is, in general, wider than the region in which the asymmetry Of Pk(ot) is negligible. Nevertheless, for ot sufficiently close to the ensemble-average value 4, the Gaussian approximation still holds. Numerical results found for the characteristic function defined for LERs show that the exponent [ 1 ] a = 3 (43--42)/4(44--42) which describes the relation of the skewness with the flatness [ 8 ], takes for k--,~ the value a = ½, which one gets in the in,
Volume 183, number 1
PHYSICS LETTERS A
Table 1 Values of the function ~k(¢) with k= 102for three chaotic parameter values of the quadratic map: a= 1.5, 1.7, and 1.9. The quantities n:~( k, e), see eq. ( 2.2 ), were calculated over 10~ensemble members ak for each ¢. The following values of the Lyapunov exponent were used: 2=0.24135 (for a=l.5), 0.43472 (for a= 1.7), and 0.54931 (for a= 1.9). a
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08
1.5
1.7
1.9
0.895 0.872 0.890 1.001 1.322 2.138 4.736 12.47 63.25
0.904 0.892 0.914 1.018 1.288 1.869 3.282 6.793 21.67
0.786 0.754 0.778 0.911 1.303 2.489 6.215 25.53 147.2
Table 2 Values ofvk(e) with k= 10s for the same parameter values of the quadratic map and for the same number of ensemble members as in the case of k= 102. a
0 0.005 0.010 0.015 0.020 0.025 0.030
1.5
1.7
1.9
0.976 0.967 0.970 1.123 1.423 1.934 3.840
0.977 0.974 1.021 1.203 1.707 2.052 3.875
0.913 0.980 0.976 1.185 2.122 3.610 30.67
termittency-branch approximation [ 1 ]. Since for a symmetric approximation o f the P D F the n u m b e r o f ensemble members contained in the Gaussian region (connected with the diffusion-branch approximation of 2q) increases as k grows [ 3 ], this suggests that the asymmetry of the true P D F for LERs can also be important for calculating relatively low-order moments o f LERs. Thus, the P D F for LERs has two non-Gaussian contributions. The first one is symmetric (with respect to the position o f the ensemble-average value 2), being distinct in tail regions ofpk (at), and thereby being o f importance for calculating higher-order moments o f LERs. The second contribution has an
29 November 1993
asymmetric form. This contribution manifests itself most distinctly also in tail regions o f Pk(at ), although it is visible for values o f LERs close to their ensemble-average value and can be o f consequence for calculating even lower-order moments o f LERs. The question that arises is whether or not there are interesting quantities which visibly reflect the strong asymmetry o f the probability density for LERs in tail regions. It seems obvious that the asymmetry should be significant for calculating higher-order moments o f LERs and thereby for calculating the characteristic function ;tq for LERs as q ~ + oo. However, these moments are determined by an averaging over the whole range of the variable at, and therefore it would be difficult to extract the asymmetry from the shape o f their dependences on q. Nevertheless, the nonGaussian character o f the probability distribution tails has been shown for various physical quantities. An important example is provided by spatial velocity gradients or local velocity differences (over small scales) in turbulence [ 9-13 ]. It should be noted that, in general, non-Gaussian properties of various statistical quantities generated by chaotic flows are associated with features or probability distributions o f LERs determined by these flows [ 14 ].
3. Determination of the probability density for local expansion rates The shape o f P D F pk(a) for LERs, in particular the evolution o f this function from a Gaussian form to a non-Gaussian one as at--, :g oo, can easily be studied by calculating numerically the partial probabilities n:~ (k, e). Numerical results obtained for various nonlinear maps prove that n_ (k, e) decays for large k as n_ (k, e),,, [ c o ( A - e ) ] -k ,
(3.1)
where the decay rate c o ( 2 - e ) is independent o f k (note that co > 1 for each chaotic parameter value o f a given m a p ) . For any finite e, the decay rate appears to be a continuous function o f e. Hence, using (2.2) and (3.1), one obtains
k[co(2--e)]-(k+l) d---co(2--e),~pk(2--e) . de
(3.2)
Accordingly, for all e > 0, the P D F satisfies also a 65
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PHYSICS LETTERS A
nearly constant (independent of ot) when # = 1 (the exponential tail of P(ot)). It is also seen that the Gaussian region of P(ot) is rather narrow, and the PDF takes an exponential form even for LERs relatively close to A. Figure 1 illustrates the difference between the Gaussian and exponential tails corresponding to ot < 2, for the quadratic map with a = 1.5. It is remarkable that, for various chaotic systems, the tail associated with LERs ot << 4 is exponential (with the coefficient c depending on the kind of the map as well as on the control parameter of a given m a p ) , and not of different non-Gaussian shapes. Such a universality of forms of tails of PDFs is also observed in cases of other statistical quantities [ 9-16 ]. As concerns the tail of the P D F associated with LERs ot > 4, the results found for typical nonlinear one-dimensional maps indicate that the power-decay law for n+ (k, ~) is not satisfied, even for relatively small deviations ot-A. Moreover, it proves that the PDF decreases with ot--,~ more rapidly than any Gaussian distribution. It is to be noted that for physically relevant maps f~(xi) considered here, (d/dx;)f~(xi)
power-decay law. Obviously, it is convenient to investigate the statistics of LERs using the function P ( A - - E ) = [ p k ( A - - ~ ) ] Ilk rather than the PDF Pk(4--~). Then, for asymptotically large k, one has
P ( a ) = [ o g ( o t ) ] -~ ,
29 November 1993
(3.3)
where ot = 4 - ~ . Thus, the shape of the function P(ot) for ot < A can be studied through determining the dependence of the decay rate 09 on ot. It should be noted that the decay rate can be calculated with fairly good accuracy for rather small k. This renders the above described procedure of determining the shape of the PDF for ot<4 very useful. Numerical results show that, for ot<<4, P(ot) ~ exp ( - c Iot - 41 ~') with c a constant, and #--- 1. It proves that, as ot increases, the exponent # grows, and takes the v a l u e / t = 2 for ot very close to 4. However, the coefficient c also changes as ot increases, taking the value c = 1/4D for ot~A. Table 3 lists the results obtained in the case of the quadratic map for the exponent #, on assuming that c = 1/4D, and for the coefficient c, on taking the exponent to b e / ~ = 1. It is seen that, assuming that c = 1/4D, one obtains # = 2 (the Gaussian approximation of the function P (ot)) only for small deviations Iot - Z 1 . However, for relatively large deviations I ot - X I, the coefficient c is
Table 3 Results obtained in cases of two chaotic parameter values a = 1.5, 1.7 of the quadratic m a p for the exponent #, assuming that c = 1 / 4D, and for the coefficient c, assuming that #--- 1. Calculations were performed using eq. ( 3.3 ). The following values of the diffusion coefficient were used: D = 0 . 0 4 2 for a = 1.5 and 0 = 0 . 0 3 5 for a = 1.7. exp(ct)
a = 1.5 # (c= 1/40)
1.4 1.3 1.2 1.1 1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1
66
1.924 2.156 2.434 2.851 3.473 4.622 7.423 27.514 - 11.22 -3.806 - 1.892 - 0.886
a = 1.7 c (/.t-- 1 )
/z (c-- 1 / 4 0 )
c (,u= 1 )
0.435 0.644 0.775 0.838 0.894 0.925 0.956 0.988 0.995 1.014 1.004 1.023
2.054 2.286 2.573 2.955 3.500 4.347 5.886 9.742 37.86 - 16.10 - 5.861 -3.154 - 1.840 - 1.003
0.619 0.745 0.814 0.864 0.890 0.908 0.923 0.924 0.907 0.912 0.907 0.918 0.937 0.950
Volume 183,number 1 1.2
,
,
,
,
PHYSICS LETTERSA ,
,
,
,
,
,
.
,
,
29 November 1993 j--I
Lj, k(Xo)= I-I If Ak)'(x,)l •
i=O
(Note that Lj+sj, k(Xo)/Lj, k(Xo)< 1.) The existence of STFs is typical for chaotic trajectories. On the STFs, a given chaotic trajectory behaves like a stable one, although the infinite trajectory is unstable. It proves that, for large k, the number mn.k of STFs can be written as [ 17 ]
0.8 I
0.4
0.0 -2.0
- I .0
0.0
mn., , ran,k:- pk-I
1.0
of the Gaussian-approximation
line) of the function
P(t~-A)
and the exponential
form
(solid
tail (dashed
line) of this function for a <2 in the case of the chaotic parameter value a= 1.5 of the quadratic map. Valuesof the coefficientc of the exponentialfunction werecalculatedusing eq. (3.3). The arrow indicatesthe upper limit a ~ - A (see the text). d x ) f o ( x ) I takes at points x = x , i= 1, 2 .... , generated by a given map. Contrary to the upper limit, the lower limit for LERs does not exist, since the points of each chaotic trajectory can lie arbitrary close to the critical point (or critical points) of a given map. This is an essential reason by the PDF of LERs for the studied systems is asymmetric and decreases very fast in the tail region a >/1, as a grows. It should be noted that the asymmetry of the PDF can also be measured by the rate p describing the decay of the number of stable trajectory fractions (STFs) of chaotic trajectories as the number k of functional composition of a given map increases [ 17 ]. The STFs are defined as stable time sequences of trajectory points xj+,, xj+2 .... , xj+sj (with sj= 1, 2 .... being sequence lengths), generated b y f ~ k), i.e. x i + , = f ~ k~ (xi), i=0, 1..... w h e r e f ~ k) denotes the kfold functional composition, k = 1, 2, .... These sequences (or fractions) are determined by the conditions [ 17 ]
Lj, k ( X o ) ) L i , k ( X O ) ,
i=0, 1..... j - 1 ,
Lj.k(Xo) > Lj+~,k(Xo),
Lj, k( XO) ~'~Lj+ sj+ l , k ( X o ) where
i= 1, 2, ..., Sj , ,
(3.6)
2.0
cx-~ F i g . 1. I l l u s t r a t i o n
(3.5)
(3.4)
with p > 1 and n being the total number of chaotic trajectory points taken into account. Essentially, the method of characterization of chaotic trajectories by counting the STFs [ 17 ] involves a whole spectrum of decay rates p~, l= 1, 2, ..., associated with numbers of STFs of lengths greater than or equal to l. The analysis presented in this paper concerns only one decay rate prop1. It should be pointed out that, since the average length of STFs, i.e. the number of points from which a given time sequence consists, tends to one when k increases, one has p=to(O) .
(3.7)
The number of stable time sequences is equal, for each k, to the number of unstable time sequences. However, when k grows, the average length of STFs decreases, whereas the average length of the unstable time sequences increases [ 17 ]. Determining numerically the PDF, each of these long unstable time sequences (much longer than k) is treated as a series of unstable time sequences of the length k. These unstable time sequences are associated with ensemble members ak>/1 (eq. (2.3)). Accordingly, for sufficiently small E> 0, more ensemble members t~k belong to the interval [/1,/1+~] than to the interval [ / l - e , / l ] . This implies that, for sufficiently large e, more LERs take values in the region [ - o r , / 1 - e ] than in the region [/1+e, oo]. Since the majority of ak give values near/1, the difference in the occupation of the two tail regions becomes more visible as grows. Consequently, the asymmetry of pk(O0 is expected to be important for describing statistical properties of LERs in cases when the average length of unstable time sequences grows rapidly as k tends 67
Volume 183, number 1
PHYSICS LETTERS A
to infinity, i.e. when the decay rate o f STFs is large.
4. Conclusions T h e a s y m m e t r y o f the p r o b a b i l i t y density function for local expansion rates with respect to their ensemble average value has been s t u d i e d for nonlinear oned i m e n s i o n a l maps. It has been argued that, in contrast to the s y m m e t r i c n o n - G a u s s i a n c o n t r i b u t i o n to the p r o b a b i l i t y density for local expansion rates, which is significant only in tail regions o f the density, the a s y m m e t r i c c o n t r i b u t i o n can be i m p o r t a n t also for values o f local expansion rates close to their ensemble-average value. The p r o b a b i l i t y density o f local expansion rates has been shown to take a universal exponential f o r m in the tail region associated with negative deviations o f local expansion rates from their ensemble average.
References [ 1] H. Fujisaka, Prog. Theor. Phys. 71 (1984) 513. [2] H. Fujisaka and M. Inoue, Prog. Theor. Phys. 77 (1987) 1334;79 (1988) 758.
68
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[ 3 ] H. Fujisaka and M. Inoue, Phys. Rev. A 39 ( 1989 ) 1376. [4] H. Fujisaka and M. Inoue, Phys. Rev. A 38 (1988) 3680; S. Yanagida, H. Fujisaka and M. Inoue, Prog. Theor. Phys. 87 (1992) 1087; H. Shibata, S. Ando and H. Fujisaka, Phys. Rev. A 45 (1992) 7049. [5 ] T. Yamada and H. Fujisaka, Prog. Theor. Phys. 76 (1986) 582; H. Fujisaka, H. Ishii, M. Inoue and T. Yamada, Prog. Theor. Phys. 76 (1986) 1198; H. Fujisaka and T. Yamada, Prog. Theor. Phys. 77 ( 1987 ) 1045. [6] H. Fujisaka, Prog. Theor. Phys. 70 (1983) 1264. [7] H. Fujisaka and M. Inoue, Prog. Theor. Phys. 74 (1985) 20;78 (1987) 268. [8] U. Frisch, P.-L. Sutem and M. Nelkin, J. Fluid Mech. 87 (1978) 719. [9] R.H. Kraichnan, Phys. Rev. Lett. 65 (1990) 575. [ 10] Z.S. She, Phys. Rev. Lett. 66 ( 1991 ) 600. [ 11 ] Z.S. She and S.A. Orszag, Phys. Rev. Lett. 66 ( 1991 ) 1701. [12]R. Benzi, L. Biferale, G. Paladin, A. Vulpiani and M. Vergassola, Phys. Rev. Lett. 67 ( 1991 ) 2299. [ 13 ] P. Kailasnath, K.R. Sreenivasan and G. Stolovitzky, Phys. Rev. Lett. 68 (1992) 2766. [ 14 ] D. Beigie, A. Leonard and S. Wiggins, Phys. Rev. Lett. 70 (1993) 275. [ 15 ] Jayesh and Z. Warhaft, Phys. Rev. Lett. 67 ( 1991 ) 3503. [16]J.P. Gollub, J. Clarke, M. Gharib, B. Lane and O.N. Mesquita, Phys. Rev. Lett. 67 ( 1991 ) 3507. [ 17 ] W. Je~ewski, Phys. Lett. A 164 (1992) 274.
PHYSICS LETTERS A
Statistics of local expansion rates for chaotic systems W. Je~.ewski Institute of MolecularPhysics,PolishAcademyof Sciences, Smoluchowskiego 17/19, PL-60-179Poznan, Poland Received 18 May 1993;revisedmanuscript received6 August 1993;acceptedfor publication 30 September1993 Communicatedby A.R. Bishop
Statistical propertiesof the fluctuationdynamicsof local expansionrates are studied for one-dimensionalsystems.The probability density for these finite-time rates is found to be, in general, asymmetric,even for values of local expansion rates close to their ensemble-averagevalue and for longtime intervals. It is shownthat the probabilitydensity takes on an exponentialform in a tail regionassociatedwith localexpansionrates less than their ensemble-averagevalue.
1. Introduction
One of the important global approaches to the study of chaotic systems is the fluctuation-spectrum theory [ 1-4 ]. This approach concerns temporal and spatial fluctuations of local (finite-time) averages lk-I
The dependence of 2q on q reflects different characteristics of the large-scale dynamics embedded in time series. As k-,ov, the time average ak approaches an ensemble average, but for finite, even very large k, it is a fluctuating variable. The probability density that O~ktakes a value a can be expressed as
Otk(j) = ~ ~ U(Xi+jk), j=O, 1, ...,
[/2--1
iffi0
pk(Ot)= l i r a - ~: J(O~--Olk(j)) .
of time series {u(x~); i=0, 1.... } generated by the chaotic dynamics x~+t =f~(x~) with a a control parameter, and xieJ, where J denotes an interval of length dependent, in general, on a (for applications of the theory to higher-dimensional systems, see ref. [ 5 ] ). The central role in the fluctuation-spectrum theory plays the characteristic function 2q defined by
J
(1.1) where q is the so-called filtering parameter and
p ( x ) = lim -1 ~ 1
~(x-x~)
is the natural invariant measure, satisfying the Frobenius-Perron equation
p ( x ) = ~ J ( x - f ~ ( y ) ) p ( y ) dy. J
Accordingly, 2¢ can be written as [ 1,3 ]
2q=q -1 lira k - l In ~ exp(qka)pk(a) dot.
(1.2) For sufficiently large k, the probability density function (PDF) has been argued to take near its peak position the Gaussian form [6], i.e.
2 ¢ = q -1 lim k -1 In f exp[qkOtk(X)]p(x) d r , k~oo
n ~ o o n jffiO
pk(Ot) "" ~
exp[- (a-2o)2k/4D] ,
(1.3)
with 2o= (tXk), where ( ) denotes the ensemble average, and D the diffusion coefficient, given for large k by D,,- ½k( (¢tk--~,0)2 > .
(1.4)
The Gaussian approximation is appropriate for small Iq I, much smaller than the convergence radius x of the cumulant expansion [2 ] of 2¢. For a given small q, the condition Iq[ ~ x is satisfied i f x is sufficiently
0375-9601/93/$ 06.00 © 1993 ElsevierSciencePublishers B.V. All rights reserved.
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large, i.e. when the amplitude of fluctuations of otk is small [ 1,2,4 ]. Consequently, the small- Iq I branch of 4q reflects the diffusive character of the dynamics of ak. The non-Gaussianity ofPk (ot), which arises in tail regions, manifests itself in the higher-order statistics when Iql >> x. This branch of 4q refers to the intermittent (laminar) structure of the dynamics [ 1,2,4]. Thus, the characteristic function is related with both the Gaussian and non-Gaussian regimes of the PDF for long-time averages, and characterizes two complementary processes i.e. diffusion and intermittency, connected respectively with these two regimes [ 1,7 ]. Here, it is shown that the PDF of local expansion rates (LERs) is, in general, asymmetric. The asymmetry of the PDF is found to be especially distinct in tail regions, but appears also in the vicinity of the position of the ensemble average of these rates. It is also shown that in a tail region associated with LERs less than their ensemble-average value, the PDF evolves from a Gaussian form to an exponential one as ot--,- oo. To investigate the shape of the PDF, the statistics of LERs is analyzed for tail regions of different scales. Contrary to the fluctuation-spectrum theory, this method does not involve the concept of the characteristic function.
2. Properties of the probability density of local expansion rates In this section general properties of the probability density for LERs, in particular the validity of the Gaussian approximation (1.3) for small deviations of LERs from their ensemble-average value, are discussed. The results obtained for various one-dimensional nonlinear maps have turned out to be qualitatively consistent. Therefore, the detailed numerical analysis presented below refers to one, representative, dynamical system, namely the quadratic map x,+ 1= 1 - axe. In order to examine the form of the PDF Pk(ot), let us introduce the asymmetry function, defined (for a given k) by
z'k(e) = n _ (k, e)/n+ (k, e ) , with 64
(2.1)
29 November 1993 ).q:~ /t
n:~ (k, e) = _+ j Pk(ot) dot,
(2.2)
where ~1>O, ot denotes a value of the LER, represented by
otk(.J) = ~: i~=ol n l f " (xi+jk) l ,
(2.3)
( f " ( X) --=dfa (X) / dx) and 4 - 4 o is the Lyapunov exponent. The function Vk(e) characterizes the deviation of the shape of pk(ot) from a symmetric form with respect to the position of 4 (note that, for any symmetric probability density with peak located at the ensemble-average value, vk(e) = 1 for all e). The asymmetry function can be determined by counting the PDF Pk(ot) over the regions ot < 4 and ot > 4 + e. Numerical results obtained for typical chaotic systems show that, for each finite k, Vk(E) < 1 when e is sufficiently small (including e = 0 ) , and Vk(e)> 1 when E is large. Moreover, it proves that vk(0)--, 1 as k--,~. However, for any constant e>0, Vk(E) increases to infinity as k grows. Consequently, the range of e, in which the deviation of vk(e) from the limit value v~ (0) = 1 is negligible, shrinks when k increases. As an illustrative example of a typical behavior of Vk(E), values of this function computed for three chosen values of the control parameter of the quadratic map are shown in tables 1 and 2 for k = 102 and k = 103, respectively. Accordingly, the PDF Pk(ot) exhibits asymmetry over the whole range of ot and for every k. However, for each sufficiently large k, there exists a range of ot, dependent on k, in which the asymmetry is insignificant. It appears that, when k grows, this range shrinks, but the height of the PDF at the ensembleaverage value 4 increases, so that most of the ensemble members belong to the range. The Gaussian region, determined by [7] lot-41 ~ O ( 2 ~ ) is, in general, wider than the region in which the asymmetry Of Pk(ot) is negligible. Nevertheless, for ot sufficiently close to the ensemble-average value 4, the Gaussian approximation still holds. Numerical results found for the characteristic function defined for LERs show that the exponent [ 1 ] a = 3 (43--42)/4(44--42) which describes the relation of the skewness with the flatness [ 8 ], takes for k--,~ the value a = ½, which one gets in the in,
Volume 183, number 1
PHYSICS LETTERS A
Table 1 Values of the function ~k(¢) with k= 102for three chaotic parameter values of the quadratic map: a= 1.5, 1.7, and 1.9. The quantities n:~( k, e), see eq. ( 2.2 ), were calculated over 10~ensemble members ak for each ¢. The following values of the Lyapunov exponent were used: 2=0.24135 (for a=l.5), 0.43472 (for a= 1.7), and 0.54931 (for a= 1.9). a
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08
1.5
1.7
1.9
0.895 0.872 0.890 1.001 1.322 2.138 4.736 12.47 63.25
0.904 0.892 0.914 1.018 1.288 1.869 3.282 6.793 21.67
0.786 0.754 0.778 0.911 1.303 2.489 6.215 25.53 147.2
Table 2 Values ofvk(e) with k= 10s for the same parameter values of the quadratic map and for the same number of ensemble members as in the case of k= 102. a
0 0.005 0.010 0.015 0.020 0.025 0.030
1.5
1.7
1.9
0.976 0.967 0.970 1.123 1.423 1.934 3.840
0.977 0.974 1.021 1.203 1.707 2.052 3.875
0.913 0.980 0.976 1.185 2.122 3.610 30.67
termittency-branch approximation [ 1 ]. Since for a symmetric approximation o f the P D F the n u m b e r o f ensemble members contained in the Gaussian region (connected with the diffusion-branch approximation of 2q) increases as k grows [ 3 ], this suggests that the asymmetry of the true P D F for LERs can also be important for calculating relatively low-order moments o f LERs. Thus, the P D F for LERs has two non-Gaussian contributions. The first one is symmetric (with respect to the position o f the ensemble-average value 2), being distinct in tail regions ofpk (at), and thereby being o f importance for calculating higher-order moments o f LERs. The second contribution has an
29 November 1993
asymmetric form. This contribution manifests itself most distinctly also in tail regions o f Pk(at ), although it is visible for values o f LERs close to their ensemble-average value and can be o f consequence for calculating even lower-order moments o f LERs. The question that arises is whether or not there are interesting quantities which visibly reflect the strong asymmetry o f the probability density for LERs in tail regions. It seems obvious that the asymmetry should be significant for calculating higher-order moments o f LERs and thereby for calculating the characteristic function ;tq for LERs as q ~ + oo. However, these moments are determined by an averaging over the whole range of the variable at, and therefore it would be difficult to extract the asymmetry from the shape o f their dependences on q. Nevertheless, the nonGaussian character o f the probability distribution tails has been shown for various physical quantities. An important example is provided by spatial velocity gradients or local velocity differences (over small scales) in turbulence [ 9-13 ]. It should be noted that, in general, non-Gaussian properties of various statistical quantities generated by chaotic flows are associated with features or probability distributions o f LERs determined by these flows [ 14 ].
3. Determination of the probability density for local expansion rates The shape o f P D F pk(a) for LERs, in particular the evolution o f this function from a Gaussian form to a non-Gaussian one as at--, :g oo, can easily be studied by calculating numerically the partial probabilities n:~ (k, e). Numerical results obtained for various nonlinear maps prove that n_ (k, e) decays for large k as n_ (k, e),,, [ c o ( A - e ) ] -k ,
(3.1)
where the decay rate c o ( 2 - e ) is independent o f k (note that co > 1 for each chaotic parameter value o f a given m a p ) . For any finite e, the decay rate appears to be a continuous function o f e. Hence, using (2.2) and (3.1), one obtains
k[co(2--e)]-(k+l) d---co(2--e),~pk(2--e) . de
(3.2)
Accordingly, for all e > 0, the P D F satisfies also a 65
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nearly constant (independent of ot) when # = 1 (the exponential tail of P(ot)). It is also seen that the Gaussian region of P(ot) is rather narrow, and the PDF takes an exponential form even for LERs relatively close to A. Figure 1 illustrates the difference between the Gaussian and exponential tails corresponding to ot < 2, for the quadratic map with a = 1.5. It is remarkable that, for various chaotic systems, the tail associated with LERs ot << 4 is exponential (with the coefficient c depending on the kind of the map as well as on the control parameter of a given m a p ) , and not of different non-Gaussian shapes. Such a universality of forms of tails of PDFs is also observed in cases of other statistical quantities [ 9-16 ]. As concerns the tail of the P D F associated with LERs ot > 4, the results found for typical nonlinear one-dimensional maps indicate that the power-decay law for n+ (k, ~) is not satisfied, even for relatively small deviations ot-A. Moreover, it proves that the PDF decreases with ot--,~ more rapidly than any Gaussian distribution. It is to be noted that for physically relevant maps f~(xi) considered here, (d/dx;)f~(xi)
power-decay law. Obviously, it is convenient to investigate the statistics of LERs using the function P ( A - - E ) = [ p k ( A - - ~ ) ] Ilk rather than the PDF Pk(4--~). Then, for asymptotically large k, one has
P ( a ) = [ o g ( o t ) ] -~ ,
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(3.3)
where ot = 4 - ~ . Thus, the shape of the function P(ot) for ot < A can be studied through determining the dependence of the decay rate 09 on ot. It should be noted that the decay rate can be calculated with fairly good accuracy for rather small k. This renders the above described procedure of determining the shape of the PDF for ot<4 very useful. Numerical results show that, for ot<<4, P(ot) ~ exp ( - c Iot - 41 ~') with c a constant, and #--- 1. It proves that, as ot increases, the exponent # grows, and takes the v a l u e / t = 2 for ot very close to 4. However, the coefficient c also changes as ot increases, taking the value c = 1/4D for ot~A. Table 3 lists the results obtained in the case of the quadratic map for the exponent #, on assuming that c = 1/4D, and for the coefficient c, on taking the exponent to b e / ~ = 1. It is seen that, assuming that c = 1/4D, one obtains # = 2 (the Gaussian approximation of the function P (ot)) only for small deviations Iot - Z 1 . However, for relatively large deviations I ot - X I, the coefficient c is
Table 3 Results obtained in cases of two chaotic parameter values a = 1.5, 1.7 of the quadratic m a p for the exponent #, assuming that c = 1 / 4D, and for the coefficient c, assuming that #--- 1. Calculations were performed using eq. ( 3.3 ). The following values of the diffusion coefficient were used: D = 0 . 0 4 2 for a = 1.5 and 0 = 0 . 0 3 5 for a = 1.7. exp(ct)
a = 1.5 # (c= 1/40)
1.4 1.3 1.2 1.1 1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1
66
1.924 2.156 2.434 2.851 3.473 4.622 7.423 27.514 - 11.22 -3.806 - 1.892 - 0.886
a = 1.7 c (/.t-- 1 )
/z (c-- 1 / 4 0 )
c (,u= 1 )
0.435 0.644 0.775 0.838 0.894 0.925 0.956 0.988 0.995 1.014 1.004 1.023
2.054 2.286 2.573 2.955 3.500 4.347 5.886 9.742 37.86 - 16.10 - 5.861 -3.154 - 1.840 - 1.003
0.619 0.745 0.814 0.864 0.890 0.908 0.923 0.924 0.907 0.912 0.907 0.918 0.937 0.950
Volume 183,number 1 1.2
,
,
,
,
PHYSICS LETTERSA ,
,
,
,
,
,
.
,
,
29 November 1993 j--I
Lj, k(Xo)= I-I If Ak)'(x,)l •
i=O
(Note that Lj+sj, k(Xo)/Lj, k(Xo)< 1.) The existence of STFs is typical for chaotic trajectories. On the STFs, a given chaotic trajectory behaves like a stable one, although the infinite trajectory is unstable. It proves that, for large k, the number mn.k of STFs can be written as [ 17 ]
0.8 I
0.4
0.0 -2.0
- I .0
0.0
mn., , ran,k:- pk-I
1.0
of the Gaussian-approximation
line) of the function
P(t~-A)
and the exponential
form
(solid
tail (dashed
line) of this function for a <2 in the case of the chaotic parameter value a= 1.5 of the quadratic map. Valuesof the coefficientc of the exponentialfunction werecalculatedusing eq. (3.3). The arrow indicatesthe upper limit a ~ - A (see the text). d x ) f o ( x ) I takes at points x = x , i= 1, 2 .... , generated by a given map. Contrary to the upper limit, the lower limit for LERs does not exist, since the points of each chaotic trajectory can lie arbitrary close to the critical point (or critical points) of a given map. This is an essential reason by the PDF of LERs for the studied systems is asymmetric and decreases very fast in the tail region a >/1, as a grows. It should be noted that the asymmetry of the PDF can also be measured by the rate p describing the decay of the number of stable trajectory fractions (STFs) of chaotic trajectories as the number k of functional composition of a given map increases [ 17 ]. The STFs are defined as stable time sequences of trajectory points xj+,, xj+2 .... , xj+sj (with sj= 1, 2 .... being sequence lengths), generated b y f ~ k), i.e. x i + , = f ~ k~ (xi), i=0, 1..... w h e r e f ~ k) denotes the kfold functional composition, k = 1, 2, .... These sequences (or fractions) are determined by the conditions [ 17 ]
Lj, k ( X o ) ) L i , k ( X O ) ,
i=0, 1..... j - 1 ,
Lj.k(Xo) > Lj+~,k(Xo),
Lj, k( XO) ~'~Lj+ sj+ l , k ( X o ) where
i= 1, 2, ..., Sj , ,
(3.6)
2.0
cx-~ F i g . 1. I l l u s t r a t i o n
(3.5)
(3.4)
with p > 1 and n being the total number of chaotic trajectory points taken into account. Essentially, the method of characterization of chaotic trajectories by counting the STFs [ 17 ] involves a whole spectrum of decay rates p~, l= 1, 2, ..., associated with numbers of STFs of lengths greater than or equal to l. The analysis presented in this paper concerns only one decay rate prop1. It should be pointed out that, since the average length of STFs, i.e. the number of points from which a given time sequence consists, tends to one when k increases, one has p=to(O) .
(3.7)
The number of stable time sequences is equal, for each k, to the number of unstable time sequences. However, when k grows, the average length of STFs decreases, whereas the average length of the unstable time sequences increases [ 17 ]. Determining numerically the PDF, each of these long unstable time sequences (much longer than k) is treated as a series of unstable time sequences of the length k. These unstable time sequences are associated with ensemble members ak>/1 (eq. (2.3)). Accordingly, for sufficiently small E> 0, more ensemble members t~k belong to the interval [/1,/1+~] than to the interval [ / l - e , / l ] . This implies that, for sufficiently large e, more LERs take values in the region [ - o r , / 1 - e ] than in the region [/1+e, oo]. Since the majority of ak give values near/1, the difference in the occupation of the two tail regions becomes more visible as grows. Consequently, the asymmetry of pk(O0 is expected to be important for describing statistical properties of LERs in cases when the average length of unstable time sequences grows rapidly as k tends 67
Volume 183, number 1
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to infinity, i.e. when the decay rate o f STFs is large.
4. Conclusions T h e a s y m m e t r y o f the p r o b a b i l i t y density function for local expansion rates with respect to their ensemble average value has been s t u d i e d for nonlinear oned i m e n s i o n a l maps. It has been argued that, in contrast to the s y m m e t r i c n o n - G a u s s i a n c o n t r i b u t i o n to the p r o b a b i l i t y density for local expansion rates, which is significant only in tail regions o f the density, the a s y m m e t r i c c o n t r i b u t i o n can be i m p o r t a n t also for values o f local expansion rates close to their ensemble-average value. The p r o b a b i l i t y density o f local expansion rates has been shown to take a universal exponential f o r m in the tail region associated with negative deviations o f local expansion rates from their ensemble average.
References [ 1] H. Fujisaka, Prog. Theor. Phys. 71 (1984) 513. [2] H. Fujisaka and M. Inoue, Prog. Theor. Phys. 77 (1987) 1334;79 (1988) 758.
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[ 3 ] H. Fujisaka and M. Inoue, Phys. Rev. A 39 ( 1989 ) 1376. [4] H. Fujisaka and M. Inoue, Phys. Rev. A 38 (1988) 3680; S. Yanagida, H. Fujisaka and M. Inoue, Prog. Theor. Phys. 87 (1992) 1087; H. Shibata, S. Ando and H. Fujisaka, Phys. Rev. A 45 (1992) 7049. [5 ] T. Yamada and H. Fujisaka, Prog. Theor. Phys. 76 (1986) 582; H. Fujisaka, H. Ishii, M. Inoue and T. Yamada, Prog. Theor. Phys. 76 (1986) 1198; H. Fujisaka and T. Yamada, Prog. Theor. Phys. 77 ( 1987 ) 1045. [6] H. Fujisaka, Prog. Theor. Phys. 70 (1983) 1264. [7] H. Fujisaka and M. Inoue, Prog. Theor. Phys. 74 (1985) 20;78 (1987) 268. [8] U. Frisch, P.-L. Sutem and M. Nelkin, J. Fluid Mech. 87 (1978) 719. [9] R.H. Kraichnan, Phys. Rev. Lett. 65 (1990) 575. [ 10] Z.S. She, Phys. Rev. Lett. 66 ( 1991 ) 600. [ 11 ] Z.S. She and S.A. Orszag, Phys. Rev. Lett. 66 ( 1991 ) 1701. [12]R. Benzi, L. Biferale, G. Paladin, A. Vulpiani and M. Vergassola, Phys. Rev. Lett. 67 ( 1991 ) 2299. [ 13 ] P. Kailasnath, K.R. Sreenivasan and G. Stolovitzky, Phys. Rev. Lett. 68 (1992) 2766. [ 14 ] D. Beigie, A. Leonard and S. Wiggins, Phys. Rev. Lett. 70 (1993) 275. [ 15 ] Jayesh and Z. Warhaft, Phys. Rev. Lett. 67 ( 1991 ) 3503. [16]J.P. Gollub, J. Clarke, M. Gharib, B. Lane and O.N. Mesquita, Phys. Rev. Lett. 67 ( 1991 ) 3507. [ 17 ] W. Je~ewski, Phys. Lett. A 164 (1992) 274.