Characterization of a system described by Kuramoto–Sivashinsky equation with Lyapunov exponent

Physica A 269 (1999) 314–321

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Characterization of a system described by Kuramoto–Sivashinsky equation with Lyapunov exponent a Department b Faculty

Hiroshi Shibataa; ∗ , Ryuji Ishizakib of General Education, Kumamoto Institute of Technology, Kumamoto 860-0082, Japan of Integrated Humane Studies and Social Science, Fukuoka Prefectural University, Tagawa 825-8585, Japan Received 22 February 1999

Abstract The characteristics of a system described by Kuramoto–Sivashinsky equation are obtained through the statistics of a mean Lyapunov exponent. This mean Lyapunov exponent takes large values and uctuates large when the system is disordered temporally and spatially. This behavior of the spatially extended system is captured clearly by the probability distribution function for c 1999 Elsevier Science B.V. All the time averaged one of the mean Lyapunov exponent. rights reserved. PACS: 05.45.Ac; 05.70.Ln; 47.54.+r Keywords: Nonlinear partial dierential equation; Kuramoto–Sivashinsky equation; Spatiotemporal pattern; Mean Lyapunov exponent; Local Lyapunov exponent

1. Introduction Spatiotemporal chaos and turbulence have attracted great interests recently. One of this reason is that the approach of the linear stability to chaos succeeded [1–5] and the tendency to apply this approach to spatially extended systems has been growing [6]. The theory of the linear stability was applied to coupled map lattices and the eciency of it was con rmed [7–9]. The quantitative characterization of the discrete systems was done by considering the Lyapunov’s matrix. But the Lyapunov’s matrix ∗

Corresponding author. Fax: +81-96-326-3000. E-mail addresses: [email protected] (H. Shibata), [email protected] (R. Ishizaki)

c 1999 Elsevier Science B.V. All rights reserved. 0378-4371/99/$ - see front matter PII: S 0 3 7 8 - 4 3 7 1 ( 9 9 ) 0 0 0 9 9 - 0

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cannot be de ned for the continuous systems. However, this problem proved to be avoided when we treat it numerically [10,11]. We describe it rst. Let us consider the partial dierential equation written by @ u(r; t) = f(u) ; @t

(1)

where u(r; t) is a variable de ned at the location r and the time t. One of the methods to solve Eq. (1) numerically is the Crank–Nicolson [12]. We rewrite Eq. (1) to the nite-dierence equation as follows: ujn+1 − ujn = 12 fjn+1 (u) + 12 fjn (u) :
t

(2)

j runs from 1 to N .
t is the time mesh width we take and ujn is the variable de ned at discrete time step n and discrete space site j. Then we consider the virtual deviation. Putting ujn + ujn into ujn in Eq. (2) we get the equation (Bij )n+1; N un+1 = (Aij )n; N un :

(3)

(Bij )n+1; N and (Aij )n; N are N × N matrices and un is written down as un = Col:(u1n ; u2n ; : : : ; uNn ) :

(4)

Last, we get the mean Lyapunov exponent (MLE) n =

1 {ln | (Aij )n; N | − ln | (Bij )n+1; N |} ;
t × N

(5)

where |(Cij )| means the determinant of matrix (Cij ). We observed that the time series of the MLE re ects the characteristics of the spatiotemporal pattern for discrete systems [7–9]. The time series of the MLE takes large and random values when the spatiotemporal pattern is disordered. On the other hand, the time series of the MLE takes small and periodic values when the spatiotemporal pattern is ordered. We can expect that the same characteristics hold for continuous systems. One of the methods which capture these characteristics is introducing the probability distribution function of the local Lyapunov exponent. The local Lyapunov exponent (LLE) is de ned by n−1

n ≡

1X j ; n

(6)

j=0

where n is appropriately long time step number and j is the MLE at jth step. This paper is organized as follows. In Section 2 the MLE for a system described by KS equation is given. In Section 3 the characteristics of the time series of the MLE for the system are stated. These characteristics are expressed by the probability distribution function of the LLE. Conclusions are given in Section 4.

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2. Mean Lyapunov exponent for Kuramoto–Sivashinsky equation In this section we write down the elements of matrices (Aij ) and (Bij ) in Eq. (3). Kuramoto–Sivashinsky (KS) equation is a well-known model [13–15] that describes not only spatiotemporal chaos but also turbulence [6]. Intensive and extensive studies were done and this equation has attracted many investigators [16 –21]. We consider here the one-dimensional version of KS equation written down as @t + Á + @xx + @xxxx + 2 @x = 0 ;

(7)

where Á is the damping constant and there is an unstable mode for Á ¡ 14 . The control parameters of this system are the system size length L and the damping constant Á. We set the rigid boundary conditions @x (0) = @x (L) = 0 :

(0) = (L) = 0;

(8)

Let us derive the MLE on the basis of the Crank–Nicolson method for this KS equation. The KS Equation (1) is rewritten to nite-dierence equation ! k+1 k+1 k+1 k+1 k k k + j−1 − jk Á j+1 + j + j−1 j+1 + j j + +
t 2 3 3 ! k+1 k+1 k+1 k k k + j−1 1 j+1 − 2 j + j−1 j+1 − 2 j + + 2 (
x)2 (
x)2 +

k+1 j+2

1 2

+

k j+2

−4

−4

k+1 j+1

k j+1

k+1 j (
x)4

+6

+6

k j

−4

−4 k j−1

k+1 j−1

+

k+1 j−2

+

k j−2

!

(
x)4 k+1 k+1 ( j+1 j

1 + 2

−


x

k+1 j−1 )

+

k k j ( j+1

−

k j−1 )

! =0;


x

(9)

where j runs from 1 to N − 1. We substitute +  into in Eq. (9) and linearize it with respect to  . Then we obtain the elements of matrices (Aij )n; N −1 and (Bij )n+1; N −1 as follows:
t
t
t
t k ( k − i−1 −3 − ); (i = 1; : : : ; N − 1) ; Á+ Ai; i = 1 − 2 4 6 (
x) (
x) 2(
x) i+1 A1;2 = −


t
t 3
t
t Á− + − 2 4 6 2(
x) 2 (
x) 2(
x)

k 1

;

Ai; i+1 = −


t
t
t
t +2 − Á− 6 2(
x)2 (
x)4 2(
x)

i

Ai; i−1 = −


t
t
t
t +2 + Á− 6 2(
x)2 (
x)4 2(
x)

i

k

;

(i = 2; : : : ; N − 2) ;

k

;

(i = 2; : : : ; N − 2) ;

H. Shibata, R. Ishizaki / Physica A 269 (1999) 314–321

AN −1; N −2 = −


t
t 3
t
t Á− + + 6 2(
x)2 2 (
x)4 2(
x)

Ai; i+2 = −


t ; 2(
x)4

(i = 1; : : : ; N − 3) ;

Ai+2; i = −


t ; 2(
x)4

(i = 1; : : : ; N − 3) ;

Ai; j = 0;

(the other elements) ;

B1; 2 =

k+1 i+1

−


t
t 3
t
t Á+ − + 6 2(
x)2 2 (
x)4 2(
x)

k+1 1

;

Bi; i+1 =


t
t
t
t Á+ −2 + 6 2(
x)2 (
x)4 2(
x)

i

Bi; i−1 =


t
t
t
t Á+ −2 − 2 4 6 2(
x) (
x) 2(
x)

i

BN −1; N −2 =


t ; 2(
x)4

(i = 1; : : : ; N − 3) ;

Bi+2; i =


t ; 2(
x)4

(i = 1; : : : ; N − 3) ;

k+1 i−1 );

(i = 1; : : : ; N − 1) ;

k+1

;

(i = 2; : : : ; N − 2) ;

k+1

;

(i = 2; : : : ; N − 2) ;


t
t 3
t
t Á+ − − 6 2(
x)2 2 (
x)4 2(
x)

Bi; i+2 =

Bi; j = 0

;

(10)


t
t
t
t +3 + Á− ( 6 (
x)2 (
x)4 2(
x)

Bi; i = 1+

k N −1

317

k+1 N −1

(the other elements) :

;

(11)

The rigid boundary conditions (8) are added in the form k 0

=

k N

= 0;

k −1

=

k 1 ;

k N +1

=

k N −1 :

(12)

Then we obtain the MLE n =

1 {ln |(Aij )n; N −1 | − ln |(Bij )n+1; N −1 |}:
t × (N − 1)

(13)

Notice that we excluded the N th point because the xed boundary conditions are imposed. We need to know the solution of Eq. (9) in order to calculate the elements

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of matrices (Aij )n; N −1 and (Bij )n+1; N −1 . So we solve Eq. (9) and calculate Eq. (13) simultaneously. 3. Characteristics of a system described by KS equation We show the spatial patterns and the values of the MLE for a system described by KS equation. Then we introduce the local Lyapunov exponent (LLE) (6) in order to capture the characteristics of the system [22,23]. The system size length L and the mesh size space
x are set at 16 and 16=320, respectively. The time mesh size
t is set at 0.00001. The initial condition is set as   0 i : (14) i = 1:0 sin N We used the second-order Adams–Bashforth in stead of Eq. (9) in order to get the solution only for the sake of convenience. The spatial patterns and the values of the MLE for Á = 0:0; 0:1; and 0.13 are shown in Fig. 1 at the rst 5 × 107 th time step. It is clear that the spatial pattern is more disordered and the value of MLE is larger when the value of Á is smaller.

Fig. 1. Spatial patterns and MLEs for KS equation. Figures are for the rst 5 × 107 th step. The values of Á for (a), (b), and (c) are 0.0, 0.1, and 0.13, respectively. L,
x, and
t are 16, 16=320, and 0.00001, respectively. The initial condition is given by Eq. (14).

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Fig. 2. The PDFs of the LLE for KS equation for Á = 0:0, 0.1, and 0.13. The rst 5 × 107 steps are cut and 1:5 × 107 ensmbles are used. The average number of time steps n for n is 1 × 105 . L,
x, and
t are same as in Fig. 1.

We study the uctuation of the LLE through the probability distribution function (PDF) de ned by P(; T ) ≡ h( − n )i ; T = n ×
T;

(15)

where (·) is the Dirac’s distribution and h· · ·i means the long time average. The PDFs of the LLE are shown in Fig. 2 for Á = 0:0, 0.1, and 0.13. The average number of time steps n for n is 1 × 105 . The rst 5 × 107 steps are cut and 1:5 × 107 ensembles are used. Fig. 2 shows the characteristics of the system described by KS equation clearly. When the damping constant Á is small, the LLE takes large values and its uctuation is large. As Á increases, the LLE takes smaller values and its uctuation is smaller. It is considered that the most important characteristics of spatiotemporal chaos are that the LLE for the observable takes large values and the uctuation of it is large when the spatiotemporal chaos develops. At present we do not know precisely what kind of shape these PDFs are. The shape of these PDFs for the LLE seems like ames. So we study the contribution of each succesive 1 × 106 steps after the rst 5 × 107 steps to this Fig. 2(a). This is shown in Fig. 3. We understand that the PDF of the LLE is constructed by many PDFs in the shape of ‘U’.

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Fig. 3. The PDFs of the LLE for KS equation for Á = 0:0. (a) – (f ) are obtained from successive 106 steps after the rst 5 × 107 steps. L,
x, and
t are same as in Fig. 1.

4. Conclusions We studied a system described by KS equation through statistics for a MLE. We introduced the LLE for the MLE by averaging the MLE over appropriately long time steps. The PDF of the LLE clearly shows that the LLE takes large values and its

uctuation is large when the spatiotemporal chaos develops. On the contrary, the LLE takes small values and its uctuation is small when the spatiotemporal chaos is weak. We have to study many systems described by various nonlinear dierential equations extensively and compare them. After that, we will know the common characteristics among various nonlinear dierential equations and the dierent aspects between them.

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Note added in proof The mean Lyapunov exponent is equivalent to entropy production rate, i.e.
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