Quantitative characterization of spatiotemporal patterns II

Physica A 260 (1998) 374–380

Quantitative characterization of spatiotemporal patterns II Hiroshi Shibata ∗ Department of General Education, Kumamoto Institute of Technology, Kumamoto 860, Japan Received 27 April 1998

Abstract Disorderness of spatiotemporal patterns which are obtained by nonlinear partial dierential equations is characterized quantitatively. The mean Lyapunov exponent for a nonlinear partial dierential equation is given. The local Lyapunov exponent which is a nite time average of the mean Lyapunov exponent is shown to have close relation to the spatiotemporal patterns. It is suggested that the systems which are described by nonlinear partial dierential equations are characterized statistically through the probability distribution function of the local Lyapunov c 1998 Elsevier Science B.V. All rights reserved. exponent. Keywords: Spatiotemporal chaos; Spatiotemporal pattern; Mean Lyapunov exponent; Local Lyapunov exponent; Nonlinear partial dierential equation

1. Introduction The Lyapunov exponent has attracted considerable attention for many years. The Lyapunov exponent has been calculated for many systems which show the temporal chaos [1 –4]. Especially the local Lyapunov exponent which is a nite time average of the local expansion rate was used to characterize the temporal chaos statistically [5 – 7]. But the spatiotemporal chaos had not been characterized along this approach. Recently it was shown that the local Lyapunov exponent which is a kind of extension to the spatiotemporal chaos expresses the disorderness of the observable [8]. This means that the spatiotemporal chaos is characterized by the local Lyapunov exponent. This was con rmed mainly for the coupled map lattices [8,9]. But it has been a puzzle that we obtain the Lyapunov exponent for the systems which are described by the nonlinear partial dierential equations. We give an answer to it and characterize quantitatively ∗

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c 1998 Elsevier Science B.V. All rights reserved. 0378-4371/98/$ – see front matter PII: S 0 3 7 8 - 4 3 7 1 ( 9 8 ) 0 0 3 1 1 - 2

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the systems described by the nonlinear partial dierential equations. In this short paper the author gives a method to obtain the local Lyapunov exponent and characterizes a concrete system quantitatively. In Section 2 the method to obtain the mean Lyapunov exponent for the nonlinear partial dierential equations is given. In Section 3 a concrete system is taken up and the mean Lyapunov exponent and the local Lyapunov exponent are calculated numerically. Then the statistics of the local Lyapunov exponent is stated. Conclusions are given in Section 4.

2. Mean Lyapunov exponent and local Lyapunov exponent First we describe the mean Lyapunov exponent and the local Lyapunov exponent for discrete systems brie y. Then the counterparts for continuum systems which are described by the nonlinear partial dierential equations are stated. Let us take up the system Xn (N ) composed of N components at time step n. For example, Xn (N ) can be written down as 

xn (1)



   xn (2)     x (3)    n   Xn (N ) =  ·  :    ·     ·    xn (N )

(1)

Xn+1 (N ) is assumed to be determined by the discrete map as Xn+1 (N ) = g(Xn (N ); Xn−1 (N ); : : : ; Xn−k (N )) :

(2)

g(An ; An−1 ; : : : ; An−k ) is a function which is dierentiable at least once with respect to An . Then the Jacobi matrix 

Bn; N

@xn+1 (1)  @xn (1)   @x (2)  n+1   @xn (1)   · =   ·   ·    @xn+1 (N ) @xn (1)

@xn+1 (1) @xn (2)

@xn+1 (1) @xn (3)

@xn+1 (2) @xn (2) ·

@xn+1 (2) @xn (3) ·

·

·

· · ·

·

·

· · ·

@xn+1 (N ) @xn (2)

@xn+1 (N ) @xn (3)

· · ·

· · · · · · · · ·

 @xn+1 (1) @xn (N )   @xn+1 (2)    @xn (N )    ·    ·   ·   @xn+1 (N )  @xn (N )

(3)

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exists. As stated in the previous paper [9], the mean Lyapunov exponent n ≡

1 ln|Bn; N | N

(4)

expresses the spatial instability over the whole system at time step n. |Bn; N | is the value of the determinant of the Jacobi matrix Bn; N . If we quantify the disorderness of the spatiotemporal patterns, we need to introduce the local Lyapunov exponent n−1

n ≡

1X j ; n

(5)

j=0

where n is a properly large number. When we identify that the system is disordered spatially and temporally, the mean Lyapunov exponent n takes large values successively, i.e., the local Lyapunov exponent n is large. This was con rmed before [8,9]. The most important thing is that this formalism can be extended to the partial dierential equations directly. When we solve the partial dierential equations numerically, we reduce the degrees of freedom for the system from in nite to nite. For example, if we use the dierence method to solve them numerically, we make the degrees of freedom for the system decrease from in nite to nite [10,11]. Let us take up the system described by @ v(r; t) = f(r; v(r; t); t) ; @t

(6)

where v(r; t) is the eld observable at locus r and time t. When we solve it by the use of dierence method numerically, we rewrite the dierential equation (6) into the dierence equation vjk+1 − vjk = h(
x; {vjk }) ;
t

(7)

where
t is the time step width and k is the discrete time.
x is the cutting size of space and j is the discrete locus. {vjk } is a set of eld observable. In this case we solve the dierence equation instead of the dierential equation. The key point is that the dierence equation can be treated just as the discrete map.

3. Lyapunov exponent of nonlinear partial dierential equation In this section we take up a concrete system and calculate the mean Lyapunov exponent and the local Lyapunov exponent. Let us take up the nonlinear partial dierential equation @2 @ u(x; t) = d 2 u + a(u − u2 ) ; @t @x

(8)

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that has the xed boundary condition u(0; t) = u(1; t) = 0 :

(9)

We use the dierence method in order to solve Eqs. (8) and (9) by rewriting to k k vj+1 − 2vjk + vj−1 vjk+1 − vjk =d + a(1 − vjk )vjk ;
t (
x)2

(10)

where k = 0; 1; : : : and j = 1; 2; : : : ; J . We set at a = 4:0, J = 1=
x = 33, and
t = 0:55. The initial condition  j−1 J +1   ; ; j = 2; : : : ; J 2 0 (11) vj =   1 − j − 1 ; j = J + 1 + 1; : : : ; J J 2 is set. Fixed boundary condition is written as v1k = vJk +1 = 0;

k = 0; 1; 2; : : : :

(12)

Eq. (10) is transformed to vjk+1 = vjk +


t · d k k (v − 2vjk + vj−1 ) +
t · a(1 − vjk )vjk : (
x)2 j+1

Then the Jacobi matrix in this  k+1 @v2 @v2k+1  @v k @v3k  2   @v k+1 @v k+1  3 3  k  @v2k @v 3   · Bk; J =  ·   · ·    · ·   @v k+1 @v k+1  J J @v2k @v3k

case @v2k+1 @v4k

· · ·

@v3k+1 @v4k

· · ·

·

· · ·

·

· · ·

·

· · ·

@vJk+1 @v4k

· · ·

 @v2k+1 @vJk    k+1  @v3   @vJk   ·  ;  ·    ·   @vJk+1   @vJk

(13)

(14)

where @vjk+1 @vjk @vjk+1 k @vj−1

@vjk+1 k @vj+1

=1 −

2
t · d +
t · a(1 − 2vjk ); (
x)2

=


t · d ; (
x)2

( j = 3; : : : ; J );

=


t · d ; (
x)2

( j = 2; : : : ; J − 1);

the other elements = 0:

(j = 2; : : : ; J );

(15)

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H. Shibata / Physica A 260 (1998) 374–380

Fig. 1. Space-amplitude plots for Eq. (8). Figures are obtained by successive ve time step spatial patterns. a is set at 4.0. In (a) – (d) the values of d are set at 0:000113, 0:000110, 0:000060, and 0:000010, respectively.

We can calculate the mean Lyapunov exponent k =

1 ln |Bk; J | J −1

(16)

and the local Lyapunov exponent n−1

T =

1X j ; n

(17)

j=0

where T = n ×
t. d is set at various values to get various spatiotemporal patterns and the rst 107 iterates of Eq. (13) are cut. Fig. 1a–d show the spatiotemporal patterns when we set d at 0.000113, 0.000110, 0.000060, and 0.000010, respectively. These gures are obtained by drawing the successive ve time step spatial patterns. Their values of the local Lyapunov exponent are −0:21414, −0:39333, −0:74241, and −0:93777, respectively. n is set at 16. The time series of the mean Lyapunov exponent in Fig. 1 are shown in Fig. 2. Fig. 2 shows that the time series of the mean Lyapunov exponent is random and takes large values when the spatiotemporal patterns are disordered. On the other hand, the time series of the mean Lyapunov exponent takes regular and small values when the spatiotemporal patterns are ordered. Then we introduce the probability distribution function of the local Lyapunov exponent in order to characterize quantitatively the spatiotemporal patterns. The probability

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Fig. 2. The time series of the mean Lyapunov exponent for Eq. (8). a is set at 4.0. In (a) – (d) the values of d are set at 0:000113, 0:000110, 0:000060, and 0:000010, respectively.

Fig. 3. Probability distribution function P(; T ) for Eq. (10). a and d are set at 4.0 and 0.000113, respectively. The ensemble number is 107 and the rst 107 iterates are cut.

distribution function can be written formally as P(; T ) = h( − T )i ;

(18)

where (·) is the Dirac’s distribution and h· · ·i is a long time average. The probability distribution function of the local Lyapunov exponent at d = 0:000113 is shown in Fig. 3. The shape of it is strange. One of its reasons is that n is not so large. But this gure shows how the local Lyapunov exponent is uctuating. This function captures the

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characteristics of the spatiotemporal patterns quantitatively. In this way, the probability distribution function of the local Lyapunov exponent characterizes quantitatively the systems described by the nonlinear partial dierential equations. 4. Conclusions The method to calculate the local Lyapunov exponent for nonlinear partial dierential equations was given. It was shown that the spatiotemporal chaos which are described by the nonlinear partial dierential equations have the same characteristics as in the case of the coupled map lattices. The characteristics are stated that the time series of the mean Lyapunov exponent takes large and random values when the spatiotemporal patterns are disordered. On the other hand, the time series of the mean Lyapunov exponent takes small and regular values when the spatiotemporal patterns are ordered. Although only one concrete example was taken up, the author believes that almost all systems have the same characteristics. The spatiotemporal patterns are described by many dierential equations. So we have to study and characterize many spatiotemporal patterns quantitatively. We will probably nd new laws underlying the nonlinear and nonequilibrium systems. Note added in proof In this paper, a method to calculate the local Lyapunov exponent is given. However, the model used in this paper, Eq. (8), is not a good example. After all, the spatiotemporal chaos obtained by the dierence equation (10) cannot be obtained by the original equation (10). The author con rmed that the Kuramoto–Sivashinsky equation can be studied in the same way extended in the present paper and the mean Lyapunov exponent decreases as the contribution of the damping term increases. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11]

P. Berge, Y. Pomeau, C. Vidal, Order within Chaos, Wiley, New York, 1984. H.G. Schuster, Deterministic Chaos, VCH, Weinheim, 1988. H.-O. Peitgen, H. Jurgens, D. Saupe, Chaos and Fractals, Springer, New York, 1992. I. Shimada, T. Nagashima, Prog. Theor. Phys. 61 (1979) 1605. H. Hata, T. Horita, H. Mori, T. Morita, K. Tomita, Prog. Theor. Phys. 80 (1988) 809. T. Horita, H. Hata, H. Mori, T. Morita, K. Tomita, Prog. Theor. Phys. 80 (1988) 923. H. Mori, H. Hata, T. Horita, T. Kobayashi, Prog. Theor. Phys. (Suppl. 99) (1989) 1 and references cited therein. H. Shibata, Physica A 252 (1998) 428. H. Shibata, Physica A, 253 (1998) 118. G.D. Smith, Numerical Solution of Partial Dierential Equations, 3rd ed., Oxford University Press, New York, 1985. R.D. Richtmyer, K.W. Morton, Dierence Methods for Initial-Value Problems, 2nd ed., Krieger, Malabar, 1994.