- Email: [email protected]
Quantitative characterization of spatiotemporal patterns
Physica A 253 (1998) 134 –142
Quantitative characterization of spatiotemporal patterns Hiroshi Shibata ∗ Department of General Education, Kumamoto Institute of Technology, Kumamoto 860, Japan Received 10 September 1997; revised 19 November 1997
Abstract Disorderness of spatiotemporal patterns which are obtained by discrete maps and partial di erential equations is characterized quantitatively. This characterization is very generic and can be used for almost all spatiotemporal patterns. As examples, the coupled map lattices that consist c 1998 of the Henon map and one of the nonlinear partial di erential equations are studied. Elsevier Science B.V. All rights reserved
1. Introduction For many years, people have watched and been interested in spatiotemporal patterns. The ow of a river, re set to a pile of fallen leaves, smoke rising up from a chimney, etc. We have been watching many shapes which change themselves as time goes by. People have drawn pictures that capture the shapes of things every moment. Scientists often describe the sights by the use of di erential equations and so on. The disorderness of the patterns obtained by discrete maps or partial di erential equations has been described with Hausdor dimension, capacity, etc. [1,2]. On the other hand, Lyapunov exponent is used for the temporal instability of the observables [ 3 –11]. In recent years the mean Lyapunov exponent was introduced to describe the spatiotemporal instability [12]. The mean Lyapunov exponent has been used only for relatively simple models. But a kind of its generalization makes it possible to quantify the disorderness of spatiotemporal patterns written by various types of time evolutional functions. We have been looking forward to the quantity that describes the disorderness for almost all kinds of phenomena. In this paper the author proposes the local Lyapunov exponent for the description of disorderness which exists in nature. In Section 2 the quantity that characterizes the ∗
Tel.: 011-81-96-326-3111; fax: 011-81-96-326-3000; e-mail: [email protected].
c 1998 Elsevier Science B.V. All rights reserved 0378-4371/98/$19.00 Copyright PII S 0 3 7 8 - 4 3 7 1 ( 9 7 ) 0 0 6 7 3 - 0
H. Shibata / Physica A 253 (1998) 134 –142
135
spatiotemporal patterns is de ned. In Section 3 this quantity for concrete systems is given. 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. Then the counterparts for continuum systems are described. Let us consider the following system; Xn+1 (N ) = g(Xn (N ); Xn−1 (N ); : : : ; Xn−k (N ))
(n ¿ k) :
(1)
Xn (N ) is the observable at time step n composed of N components. For example, Xn (N ) is can be written down as x (1) n xn (2) xn (3) (2) Xn (N ) = · : · · xn (N )
g(An ; An−1 ; : : : ; An−k ) is a function which is di erentiable at least once with respect to An . After all, we assume that the Jacobi matrix
Bn; N
@xn+1 (1) @xn (1) @xn+1 (2) @xn (1) = · · · @xn+1 (N ) @xn (1)
@xn+1 (1) @xn (2) @xn+1 (2) @xn (2) · · · @xn+1 (N ) @xn (2)
@xn+1 (1) @xn (3) @xn+1 (2) @xn (3) · · · @xn+1 (N ) @xn (3)
· · · · · · · · · · · · · · · · · ·
@xn+1 (1) @xn (N ) @xn+1 (2) @xn (N ) · · · @xn+1 (N ) @xn (N )
(3)
exists. In this case, the time evolution is given by the function g(An ; An−1 ; : : : ; An−k ) and the spatial pattern at time step n is expressed by Xn (N ). The mean Lyapunov exponent is given by n ≡
1 ln |Bn; N | ; N
(4)
where |Bn; N | is the value of the determinant of Jacobi matrix Bn; N . The mean Lyapunov exponent expresses the spatiotemporal instabilities at one moment. But, in practice, we can realize whether the spatiotemporal patterns are disordered or not when we see a series of pictures. So we need to introduce the local Lyapunov exponent [13,14]
136
H. Shibata / Physica A 253 (1998) 134 –142
de ned by n−1
n ≡
1X j ; n
(5)
j=0
where n is a large number. The local Lyapunov exponent expresses the disorderness of spatiotemporal patterns. The spatiotemporal patterns are ordered when the local Lyapunov exponent is small, and disordered when the local Lyapunov exponent is large. Then we consider the following system, @ C(r; t) = f(r; C(r; t); t) ; @t
(6)
where C(r; t) is the eld observable at locus r and time t and @=@t is the partial di erentiation with respect to t. In this case, we assume that the derivative (@=@C)f(r; C; t) exists. Then the continuum counterpart of j is Z @C(r; t + dt) 1 dr ln (dt) ≡ |V | @C(r; t) V
=
1 |V |
Z V
@ dr ln I + f(r; C; t) dt ; @C
(7)
where V is the space and |V | is its volume, I is the identity matrix and | · | means the determinant. We also need to introduce the local Lyapunov exponent as in the case of discrete systems because of the same reason. So we introduce the local Lyapunov exponent de ned by T ≡
1 T
tZ 0 +T
(dt) ;
(8)
t0
where T is a properly long time span. For three-dimensional space and one-dimensional time, 1 T = T where
tZ 0 +T
t0
u C= v w
1 dt |V |
Z dr V
@ @ @ fx + fy + fz ; @u @v @w
(9)
H. Shibata / Physica A 253 (1998) 134 –142
and
137
fx (r; C; t) f(r; C; t) = fy (r; C; t) : fz (r; C; t)
3. Spatiotemporal patterns and local Lyapunov exponent In this section we consider good examples for the quantitative characterization of spatiotemporal patterns for the systems described by discrete maps and partial di erential equations. First, we take up coupled map lattices, where each lattice is connected with nearest lattices via di usion. One of these models is written down as xn+1 (i) = (1 − )g(xn (i); xn−1 (i)) + (g(xn (i + 1); xn−1 (i + 1)) + g(xn (i − 1); xn−1 (i − 1))) : 2
(10)
We consider two models g(x; y) = 1 − ax2 + by
(11)
g(x; y) = 1 − ax2 + by2 :
(12)
and
We set a = 2:0, b = 0:3, = 0:06, n = 32, and N = 32 in both cases, and also set periodic boundary conditions. On the other hand, probability distribution function P() is de ned as P() = h( − n )i ;
(13)
where (·) is the Dirac’s distribution and h· · ·i means the long time average. We can know the range where n exists if we see this probability distribution function P(). Of course, we have to calculate P() beforehand. Fig. 1(a) is the probability distribution function for model (11) and Fig. 1b shows the order-q moment [15] Mq (n; N ) ≡ hexp(qnNn )i
(14)
for model (11). This quantity is important in the sense that we can know what kind of time series or n are emphasized for various values of q. We can pick up the spatiotemporal patterns whose local Lyapunov exponent takes the values in the range shown in the probability distribution function P(). Two space–amplitude plots are shown in Fig. 2a and 2b. Fig. 2a is drawn for n = 0:1234 and Fig. 2b is for n = 0:2673. The value of local Lyapunov exponent expresses how the spatiotemporal patterns are disordered. Fig. 3 is the probability distribution function P() for model (12). This gure shows how the local Lyapunov exponent n spreads for model (12). Fig. 4a is drawn
138
H. Shibata / Physica A 253 (1998) 134 –142
Fig. 1. Probability distribution function P() and order-q moment Mq (n; N ) for model (11) are shown in (a) and (b), respectively. In both gures the initial 2 × 107 iterates are cut.
Fig. 2. Space–amplitude plots of model (11) are shown in (a) and (b) for n = 0:1234 and n = 0:2673, respectively. The successive ve spatial patterns are drawn for the space–amplitude plots.
Fig. 3. Probability distribution function P() for model (12) is shown. The initial 2 × 107 iterates are cut.
H. Shibata / Physica A 253 (1998) 134 –142
139
Fig. 4. Space–amplitude plots of model (12) are shown in (a) and (b) for n = 0:2022 and n = 0:3378, respectively.
for n = 0:2022 and Fig. 4b is for n = 0:3378. These two gures show how the local Lyapunov exponent indicates the disorderness of spatiotemporal patterns. Second we take up the nonlinear partial di erential equation described by @2 @ u(x; t) = d 2 u + a(u − u2 ) ; @t @x
(15)
which has the xed boundary condition u(0; t) = u(1; t) = 0 :
(16)
The local Lyapunov exponent in this case is 1 T = T
tZ 0 +T
Z1 dt
t0
dx 0
@ @2 a(1 − 2u) + d u @u @x2
:
(17)
We use di erence method to solve Eqs. (15) and (16) by rewriting as k k vj+1 − 2vjk + vj−1 vjk+1 − vjk =d + a(1 − vjk )vjk ; t (x)2
(18)
where k = 0; 1; : : : and j = 1; 2; : : : ; J . We set a = 4:0, J = 1=x = 33, and t = 0:55. The initial condition J +1 j−1 ; j = 2; : : : ; ; J 2 (19) vj0 = 1 − j − 1 ; j = J + 1 + 1; : : : ; J J 2 is set. The xed boundary condition is written as v1k = vJk +1 = 0;
k = 0; 1; 2; : : : :
(20)
140
H. Shibata / Physica A 253 (1998) 134 –142
Fig. 5. Space–amplitude plots for Eq. (15). The gures are obtained by successive ve time step spatial patterns. a is set at 4.0. The values of d are set at 0:000166; 0:000130; 0:000120; 0:000115; 0:000110, and 0:000010 for (a), (b), (c), (d), (e), and (f), respectively.
d is set at various values to get various spatiotemporal patterns and t0 is set at 107 × t. It is possible to get various spatiotemporal patterns as shown in Fig. 5. The gures are obtained to draw successive ve time step spatial patterns. d = 0:000166, 0.000130, 0.000120, 0.000115, 0.000110, and 0.000010 are used for Fig. 5a, b, c, d, e, and f, respectively. T is set at 32 × t. T ’s are calculated as T = − 1:513; −1:565; −1:616; −1:656; −1:758, and −1:801 for Fig. 5a, b, c, d, e, and f, respectively without considering the second term of the right hand side of Eq. (17). We can easily see that the spatiotemporal patterns are more disordered as T increases.
H. Shibata / Physica A 253 (1998) 134 –142
141
4. Conclusions We saw that the local Lyapunov exponent expresses the disorderness of the spatiotemporal patterns very well. It was shown that the local Lyapunov exponent takes small values when the spatiotemporal patterns are regular. On the other hand, the local Lyapunov exponent takes large values when the spatiotemporal patterns are disordered. Two ideas are very important there. One is that the mean Lyapunov exponent shows the disorderness at one time step. Another is that the coarse grain, i.e., the local Lyapunov exponent picks out various time series of the mean Lyapunov exponent. The local Lyapunov exponent can be de ned well both for continuum systems and for discrete systems. This quantity expresses the disorderness for spatiotemporal patterns obtained by various types of time evolutional functions. The author expects that a new perspective will be extended in the eld of science. Acknowledgements The author would like to thank Professor Atsuyama for stimulating and useful discussions. Note added in proof Eq. (7) in this paper is not correct accurately. Eq. (7) expresses the diagonal part of Jacobi matrix Bn; N . So Eqs. (9) and (17) are incorrect results from it. Eq. (7) has to be changed to Z Z C(r; t + dt) 1 ; dr In dr1 (dt) ≡ |V | C(r1 ; t) V
V
where C(r; t) means a virtual variation of the eld observable. Fortunately the correct local Lyapunov exponent (8) can be calculated numerically. The correct result will be published elsewhere. References [1] B.B. Mandelbrot, The Fractal Geometry of Nature, Freeman, San Francisco, 1982. [2] T.C. Halsey, M.H. Jensen, L.P. Kadano , I. Procaccia, B.I. Shraiman, Phys. Rev. A 33 (1986) 1141. [3] J. Guckenheimer, P. Holmes, Nonlinear Oscillators, Dynamical Systems, and Bifurcations of Vector Fields, Springer, New York, 1983. [4] P. Berge, Y. Pomeau, C. Vidal, Order within Chaos, Wiley, New York, 1984. [5] P. Cvitanovic (Ed.), Universality in Chaos, Adam Hilger, Bristol, 1984. [6] H.G. Schuster, Deterministic Chaos, VCH, Weinheim, 1988. [7] E.A. Jackson, Perspectives of Nonlinear Dynamics, Cambridge University Press, New York, 1991. [8] H.O. Peitgen, H. Jurgens, D. Saupe, Chaos and Fractals, Springer, New York, 1992.
142 [9] [10] [11] [12]
H. Shibata / Physica A 253 (1998) 134 –142
A.J. Lichtenberg, M.A. Lieberman, Regular and Stochastic Motion, Springer, New York, 1982. R.S. Mackay, J.D. Meiss, Hamilton Dynamical Systems, Adam Higler, Bristol, 1987. M.C. Gutzwiller, Chaos in Classical and Quantum Systems, Springer, Verlag, New York, 1990. K. Kaneko, in: K. Kaneko (Ed.), Theory and Applications of Coupled Map Lattices, Wiley, Chichester, 1993, p. 1. [13] H. Fujisaka, Prog. Theor. Phys. 70 (1983) 1264. [14] H. Fujisaka, Prog. Theor. Phys. 71 (1984) 513. [15] H. Shibata, Physica A 252 (1998) 428.
Quantitative characterization of spatiotemporal patterns Hiroshi Shibata ∗ Department of General Education, Kumamoto Institute of Technology, Kumamoto 860, Japan Received 10 September 1997; revised 19 November 1997
Abstract Disorderness of spatiotemporal patterns which are obtained by discrete maps and partial di erential equations is characterized quantitatively. This characterization is very generic and can be used for almost all spatiotemporal patterns. As examples, the coupled map lattices that consist c 1998 of the Henon map and one of the nonlinear partial di erential equations are studied. Elsevier Science B.V. All rights reserved
1. Introduction For many years, people have watched and been interested in spatiotemporal patterns. The ow of a river, re set to a pile of fallen leaves, smoke rising up from a chimney, etc. We have been watching many shapes which change themselves as time goes by. People have drawn pictures that capture the shapes of things every moment. Scientists often describe the sights by the use of di erential equations and so on. The disorderness of the patterns obtained by discrete maps or partial di erential equations has been described with Hausdor dimension, capacity, etc. [1,2]. On the other hand, Lyapunov exponent is used for the temporal instability of the observables [ 3 –11]. In recent years the mean Lyapunov exponent was introduced to describe the spatiotemporal instability [12]. The mean Lyapunov exponent has been used only for relatively simple models. But a kind of its generalization makes it possible to quantify the disorderness of spatiotemporal patterns written by various types of time evolutional functions. We have been looking forward to the quantity that describes the disorderness for almost all kinds of phenomena. In this paper the author proposes the local Lyapunov exponent for the description of disorderness which exists in nature. In Section 2 the quantity that characterizes the ∗
Tel.: 011-81-96-326-3111; fax: 011-81-96-326-3000; e-mail: [email protected].
c 1998 Elsevier Science B.V. All rights reserved 0378-4371/98/$19.00 Copyright PII S 0 3 7 8 - 4 3 7 1 ( 9 7 ) 0 0 6 7 3 - 0
H. Shibata / Physica A 253 (1998) 134 –142
135
spatiotemporal patterns is de ned. In Section 3 this quantity for concrete systems is given. 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. Then the counterparts for continuum systems are described. Let us consider the following system; Xn+1 (N ) = g(Xn (N ); Xn−1 (N ); : : : ; Xn−k (N ))
(n ¿ k) :
(1)
Xn (N ) is the observable at time step n composed of N components. For example, Xn (N ) is can be written down as x (1) n xn (2) xn (3) (2) Xn (N ) = · : · · xn (N )
g(An ; An−1 ; : : : ; An−k ) is a function which is di erentiable at least once with respect to An . After all, we assume that the Jacobi matrix
Bn; N
@xn+1 (1) @xn (1) @xn+1 (2) @xn (1) = · · · @xn+1 (N ) @xn (1)
@xn+1 (1) @xn (2) @xn+1 (2) @xn (2) · · · @xn+1 (N ) @xn (2)
@xn+1 (1) @xn (3) @xn+1 (2) @xn (3) · · · @xn+1 (N ) @xn (3)
· · · · · · · · · · · · · · · · · ·
@xn+1 (1) @xn (N ) @xn+1 (2) @xn (N ) · · · @xn+1 (N ) @xn (N )
(3)
exists. In this case, the time evolution is given by the function g(An ; An−1 ; : : : ; An−k ) and the spatial pattern at time step n is expressed by Xn (N ). The mean Lyapunov exponent is given by n ≡
1 ln |Bn; N | ; N
(4)
where |Bn; N | is the value of the determinant of Jacobi matrix Bn; N . The mean Lyapunov exponent expresses the spatiotemporal instabilities at one moment. But, in practice, we can realize whether the spatiotemporal patterns are disordered or not when we see a series of pictures. So we need to introduce the local Lyapunov exponent [13,14]
136
H. Shibata / Physica A 253 (1998) 134 –142
de ned by n−1
n ≡
1X j ; n
(5)
j=0
where n is a large number. The local Lyapunov exponent expresses the disorderness of spatiotemporal patterns. The spatiotemporal patterns are ordered when the local Lyapunov exponent is small, and disordered when the local Lyapunov exponent is large. Then we consider the following system, @ C(r; t) = f(r; C(r; t); t) ; @t
(6)
where C(r; t) is the eld observable at locus r and time t and @=@t is the partial di erentiation with respect to t. In this case, we assume that the derivative (@=@C)f(r; C; t) exists. Then the continuum counterpart of j is Z @C(r; t + dt) 1 dr ln (dt) ≡ |V | @C(r; t) V
=
1 |V |
Z V
@ dr ln I + f(r; C; t) dt ; @C
(7)
where V is the space and |V | is its volume, I is the identity matrix and | · | means the determinant. We also need to introduce the local Lyapunov exponent as in the case of discrete systems because of the same reason. So we introduce the local Lyapunov exponent de ned by T ≡
1 T
tZ 0 +T
(dt) ;
(8)
t0
where T is a properly long time span. For three-dimensional space and one-dimensional time, 1 T = T where
tZ 0 +T
t0
u C= v w
1 dt |V |
Z dr V
@ @ @ fx + fy + fz ; @u @v @w
(9)
H. Shibata / Physica A 253 (1998) 134 –142
and
137
fx (r; C; t) f(r; C; t) = fy (r; C; t) : fz (r; C; t)
3. Spatiotemporal patterns and local Lyapunov exponent In this section we consider good examples for the quantitative characterization of spatiotemporal patterns for the systems described by discrete maps and partial di erential equations. First, we take up coupled map lattices, where each lattice is connected with nearest lattices via di usion. One of these models is written down as xn+1 (i) = (1 − )g(xn (i); xn−1 (i)) + (g(xn (i + 1); xn−1 (i + 1)) + g(xn (i − 1); xn−1 (i − 1))) : 2
(10)
We consider two models g(x; y) = 1 − ax2 + by
(11)
g(x; y) = 1 − ax2 + by2 :
(12)
and
We set a = 2:0, b = 0:3, = 0:06, n = 32, and N = 32 in both cases, and also set periodic boundary conditions. On the other hand, probability distribution function P() is de ned as P() = h( − n )i ;
(13)
where (·) is the Dirac’s distribution and h· · ·i means the long time average. We can know the range where n exists if we see this probability distribution function P(). Of course, we have to calculate P() beforehand. Fig. 1(a) is the probability distribution function for model (11) and Fig. 1b shows the order-q moment [15] Mq (n; N ) ≡ hexp(qnNn )i
(14)
for model (11). This quantity is important in the sense that we can know what kind of time series or n are emphasized for various values of q. We can pick up the spatiotemporal patterns whose local Lyapunov exponent takes the values in the range shown in the probability distribution function P(). Two space–amplitude plots are shown in Fig. 2a and 2b. Fig. 2a is drawn for n = 0:1234 and Fig. 2b is for n = 0:2673. The value of local Lyapunov exponent expresses how the spatiotemporal patterns are disordered. Fig. 3 is the probability distribution function P() for model (12). This gure shows how the local Lyapunov exponent n spreads for model (12). Fig. 4a is drawn
138
H. Shibata / Physica A 253 (1998) 134 –142
Fig. 1. Probability distribution function P() and order-q moment Mq (n; N ) for model (11) are shown in (a) and (b), respectively. In both gures the initial 2 × 107 iterates are cut.
Fig. 2. Space–amplitude plots of model (11) are shown in (a) and (b) for n = 0:1234 and n = 0:2673, respectively. The successive ve spatial patterns are drawn for the space–amplitude plots.
Fig. 3. Probability distribution function P() for model (12) is shown. The initial 2 × 107 iterates are cut.
H. Shibata / Physica A 253 (1998) 134 –142
139
Fig. 4. Space–amplitude plots of model (12) are shown in (a) and (b) for n = 0:2022 and n = 0:3378, respectively.
for n = 0:2022 and Fig. 4b is for n = 0:3378. These two gures show how the local Lyapunov exponent indicates the disorderness of spatiotemporal patterns. Second we take up the nonlinear partial di erential equation described by @2 @ u(x; t) = d 2 u + a(u − u2 ) ; @t @x
(15)
which has the xed boundary condition u(0; t) = u(1; t) = 0 :
(16)
The local Lyapunov exponent in this case is 1 T = T
tZ 0 +T
Z1 dt
t0
dx 0
@ @2 a(1 − 2u) + d u @u @x2
:
(17)
We use di erence method to solve Eqs. (15) and (16) by rewriting as k k vj+1 − 2vjk + vj−1 vjk+1 − vjk =d + a(1 − vjk )vjk ; t (x)2
(18)
where k = 0; 1; : : : and j = 1; 2; : : : ; J . We set a = 4:0, J = 1=x = 33, and t = 0:55. The initial condition J +1 j−1 ; j = 2; : : : ; ; J 2 (19) vj0 = 1 − j − 1 ; j = J + 1 + 1; : : : ; J J 2 is set. The xed boundary condition is written as v1k = vJk +1 = 0;
k = 0; 1; 2; : : : :
(20)
140
H. Shibata / Physica A 253 (1998) 134 –142
Fig. 5. Space–amplitude plots for Eq. (15). The gures are obtained by successive ve time step spatial patterns. a is set at 4.0. The values of d are set at 0:000166; 0:000130; 0:000120; 0:000115; 0:000110, and 0:000010 for (a), (b), (c), (d), (e), and (f), respectively.
d is set at various values to get various spatiotemporal patterns and t0 is set at 107 × t. It is possible to get various spatiotemporal patterns as shown in Fig. 5. The gures are obtained to draw successive ve time step spatial patterns. d = 0:000166, 0.000130, 0.000120, 0.000115, 0.000110, and 0.000010 are used for Fig. 5a, b, c, d, e, and f, respectively. T is set at 32 × t. T ’s are calculated as T = − 1:513; −1:565; −1:616; −1:656; −1:758, and −1:801 for Fig. 5a, b, c, d, e, and f, respectively without considering the second term of the right hand side of Eq. (17). We can easily see that the spatiotemporal patterns are more disordered as T increases.
H. Shibata / Physica A 253 (1998) 134 –142
141
4. Conclusions We saw that the local Lyapunov exponent expresses the disorderness of the spatiotemporal patterns very well. It was shown that the local Lyapunov exponent takes small values when the spatiotemporal patterns are regular. On the other hand, the local Lyapunov exponent takes large values when the spatiotemporal patterns are disordered. Two ideas are very important there. One is that the mean Lyapunov exponent shows the disorderness at one time step. Another is that the coarse grain, i.e., the local Lyapunov exponent picks out various time series of the mean Lyapunov exponent. The local Lyapunov exponent can be de ned well both for continuum systems and for discrete systems. This quantity expresses the disorderness for spatiotemporal patterns obtained by various types of time evolutional functions. The author expects that a new perspective will be extended in the eld of science. Acknowledgements The author would like to thank Professor Atsuyama for stimulating and useful discussions. Note added in proof Eq. (7) in this paper is not correct accurately. Eq. (7) expresses the diagonal part of Jacobi matrix Bn; N . So Eqs. (9) and (17) are incorrect results from it. Eq. (7) has to be changed to Z Z C(r; t + dt) 1 ; dr In dr1 (dt) ≡ |V | C(r1 ; t) V
V
where C(r; t) means a virtual variation of the eld observable. Fortunately the correct local Lyapunov exponent (8) can be calculated numerically. The correct result will be published elsewhere. References [1] B.B. Mandelbrot, The Fractal Geometry of Nature, Freeman, San Francisco, 1982. [2] T.C. Halsey, M.H. Jensen, L.P. Kadano , I. Procaccia, B.I. Shraiman, Phys. Rev. A 33 (1986) 1141. [3] J. Guckenheimer, P. Holmes, Nonlinear Oscillators, Dynamical Systems, and Bifurcations of Vector Fields, Springer, New York, 1983. [4] P. Berge, Y. Pomeau, C. Vidal, Order within Chaos, Wiley, New York, 1984. [5] P. Cvitanovic (Ed.), Universality in Chaos, Adam Hilger, Bristol, 1984. [6] H.G. Schuster, Deterministic Chaos, VCH, Weinheim, 1988. [7] E.A. Jackson, Perspectives of Nonlinear Dynamics, Cambridge University Press, New York, 1991. [8] H.O. Peitgen, H. Jurgens, D. Saupe, Chaos and Fractals, Springer, New York, 1992.
142 [9] [10] [11] [12]
H. Shibata / Physica A 253 (1998) 134 –142
A.J. Lichtenberg, M.A. Lieberman, Regular and Stochastic Motion, Springer, New York, 1982. R.S. Mackay, J.D. Meiss, Hamilton Dynamical Systems, Adam Higler, Bristol, 1987. M.C. Gutzwiller, Chaos in Classical and Quantum Systems, Springer, Verlag, New York, 1990. K. Kaneko, in: K. Kaneko (Ed.), Theory and Applications of Coupled Map Lattices, Wiley, Chichester, 1993, p. 1. [13] H. Fujisaka, Prog. Theor. Phys. 70 (1983) 1264. [14] H. Fujisaka, Prog. Theor. Phys. 71 (1984) 513. [15] H. Shibata, Physica A 252 (1998) 428.