- Email: [email protected]
Lyapunov analysis of collective behavior in a network of chaotic elements
17 Febxuary 1997 PHYSICS LETTERS A Physics Letters A 226 (1997) 172-178
Lya~~~Qvanalysis of collective behavior in a network of ~~a~ti~ elements Receivedt7 October1996;revised rn~use~pt received 3 December t996; accepted for publication 3 December 1996 Co~unica~
by C.R. Doering
Abstract Lyapunov analysis of collective behavior of glohaliy coupled maps is presented. There exists a phase transition on account of a competition between the local instability and the coliective motion ins~bi~ty, The rate of exponential dive~ence of the trajectories of the nonlinear ~roben~us-Penn equation is used here. PACS:05.45.+b; 05.70,Lu; 82.4O.Bj K~ords: ~lob~ly coup&d maps; Chaos; ~ro~ius-Penn
~uation; Lyapunov an~iysi~ CoIl~~ve behavior
1, Introduction Coupled chaotic maps recently attracted particular attention as an excelIent model for spatially extended dissipative systems. The number of positive Lyapunov ex~nents of such systems is often propo~on~ to the system size. This is called the extensive chaos, and statistical analysis is possible to some extent in that case, In a two-Dimensions lattice of chaotic coupled maps, an Isi~g-like phase position was ~s~overed [l-3]. O’Hem et al. investigate the relationship between the phase transition and the Lyapunov spectrum. It was found that the phase transition occurs near the minimum of the Lyapunov dimension density. We shall concentrate on globally coupled maps (GCM) below. The all-to-all nature of the coupling may be regarded as either an idealization of longrange coupling or an approximation to short-range
i E-Ml: [email protected].~oto-u.ac.jp.
coupling in high-dimensional lattices. In GCM, the concept of space loses meaning. In spite of its simplicity, GCM exhibits extremely rich behavior [ 131, The explicit form of GCM we will work with is given by x~+I(~) = (l-E)f(x,(i))+K~~(x,(~~)). i’=i
(I)
Here n repr~en~ discrete time steps, i specifies the elements (i= 1,2,3,..., N), and E gives the coupling strength. The system is thus a mean-field version of coupled map lattices (CML) [ 121. It is assumed that the individual map f(x) gives rise to chaotic dynamics. We specifically consider a tent map f(x)
= 1 - @I,
(2)
with p~~eter a satisfying 1 < a 6 2, Let Q be fixed as a = 1.99. The initial condition is such that the elements are ~nifo~ly dist~but~ over the interval
0375-9601/97/$17.~ Copyright @ 1997 Elsevier Science B.V. All rights reserved Pfrs037s-9601(96)00940-1
S. Moriia/Ph~~~~~s Letters A 226 (1997) 172-i 78
[ - I,1 1. The variable x of each map is then bounded in the same interval. The coupling is only through the mean-field h, defined by
Collective behavior of the system can be described in terms of the motion of the mean-field. In a completely synchronized state, the motion of h, is reduced to the one-dimensional map f(n). In this case, the system has only one positive Lyapunov exponent which is equal to the Lyapunov exponent log(a) of the individual map. It is clear that the completely synchronized state is stable under the condition a( 1 - E) < 1 (E > > 1 (E< Ecs= 0.4975 . . .fora= 1,99).Fora(l--E) E,,) , the elements are scattered and behave chaotically in time. In that case, the N Lyapunov exponents are all positive as we see in the next section, so that the dimension of the motion becomes N. Thus, the dimension of the motion changes abruptly at a( 1 - E) = 1, i.e. E = efS.
It was argued by Just [ 21 J that there exists a sequence of bifurcations taking place at a( 1-E) = 2’/2’. These bifurcations correspond to a qualitative change in the statistical properties in the limit of large system size. Due to the presence of the mean-field fluctuation, however, this sequence of bifurcations is truncated and a one-band state takes place. The corresponding phase diagram is shown in Fig. 17 of Ref. [ 151. In the desynchronized state, the existence of non-trivial collective motions in spite of the highdimensionality of motion was reported [ 14-243. Mechanisms have been studied with various tools such as the mean-field return map [ 1617, I9 1, correlation between the individual maps [ 181, and the FrobeniusPerron operator [ 19-241. The collective motion is expected to be reduced to a lower-dimensional motion in the thermodynamic limit N --+ cc. Such collective motion appears to be generic in coupled dynamical systems [25,26], and the significance of the Lyapunov spectra for its unde~tanding has been discussed [ 27-291. This paper presents some results of our numerical investigation of the Lyapunov analyses in relation to collective motions. The dependence of the Lyapunov exponents on the coupling parameter E and the sys-
173
-
a=l.7
-
a-l .8 a=l.S a=139
-
Fig. I. The maximumLyapunov exponents as a function of coupling strength for some values of II, N = 100. They are obtained from an average over 10’ steps after discarding IO’ steps as transient.
tern size N will be studied. In Section 2, we classify the desynchronized state into two classes according to the feature of the Lyapunov exponents. In Section 3, we present a finite-dimension approximation to the nonlinear Fro~nius-Pe~on equation, which enables a deeper understanding of the properties of the Lyapunov exponents. We compute the expansion rate of two nearby trajectories of the nonlinear FrobeniusPerron equation. Some discussions and conclusions will be given in Section 4.
2. Lyapunov analyses To begin with, we consider the maximum Lyapunov exponent. Fig. 1 shows the maximum Lyapunov exponent with varying coupling strength E for some fixed values of a. The flat part corresponds to complete synchronization. For each a, the maximum Lyapunov exponent exhibits a minimum at a certain E. Let h, be viewed as an external field acting on the individual maps. In this picture, only the first part of the r.h.s. of ( 1) contributes to the value of the Lyapunov exponents. All Lyapunov exponents Ai thus degenerate to a single value log[ a( 1-E) ] , which is identical with the expansion rate Atocatof the effective local dynamics. The characteristics of the collective motion are indicated by the differences Ahi = Ai- hiocairather than the Lyapunov exponents Ai themselves. In Fig. 2, the difference AAi is plotted as a function of e for some values of the system size N. It is seen
S, Morita/Physics
Letters A 226 (1997) 172-l 78
N=lO
10’
n N=lO’ + N=103 A N=104
10-l
0
.
. .1 .
“d
1O-3 10e4
N
N=IO N=lO’ N=103 N=104
fig. 3. AAl as a function of system size for some values of E and N = 1.99.
Fig, 2. AAl as a function of coupling strength for some values of N and n = 1.99. (a} Log-log plot and (b) linear-linear plot. The maximum Lyapunov exponent is obtained from an average over IO5 steps after discarding lo5 steps as transient.
that the relationship KE
(4)
holds in the region of small E. This e-dependence may naturally be expected from the fact that the offdiagonal elements of the Jacobi matrix of ( 1) are proportional to E. As E increases, however, this scaling law becomes broken. From Fig. 2b, the transition is seen to occur at about E = Et N 0.41. By comparison with Fig. I, it is seen that this transition occurs when the m~imum Lyapunov exponent takes a minimum value. Fig. 3 is reconstructed from Fig. 2, where the difference AA1 is plotted as a function of the system size for some values of 6. Two different phases are observed in Fig. 3. Namely, in the weak coupling phase (E < Et), the difference obeys AAt cc $
..___,, . 8~0.45 &=0.43 s=O.42 &=0.41 E=O.4 E=O.35 &=0.3 &=0.25 &=O.Z
.1o-2
AA1
..__,
(5)
which may be unde~tood from the fact that the offdiagonal elements of the Jacobi matrix of ( 1) are proportional to N-* I This phase roughly corresponds to turbulent or two-band states [ 19,211 in the limit of large system size. The difference between the turbulent state and the two-band state is most clearly seen from the profile of the density distribution (see Figs. 4a,4b). The density distribution p(x) will be defined in the next section. The transition between these states originates qu~itatively from the band-splitting of the local maps which occurs at a( 1 - tz) = t/z (E = 0.2893. . . for a = 1.99). In the strong coupling phase (et < E < ecs), Ah, decreases down to a certain value with the system size N, and the maximum Lyapunov exponent At approaches a value clearly different from Atocat.The ~ymptotic value of the m~imum Lyapunov exponent corresponds to the rate of exponential divergence of the evolution of the density distribution, as we shall show in the next section. The strong coupling phase roughly corresponds to what we call intermittent or one-band states in the limit of large system size. In the inte~ittent state, there are inte~ittent switchings between one-band and two-band states. In the one-band state, the elements form a broad band as in Fig. 4c and the band moves chaotically [ 191. We now consider other Lyapunov exponents Ai, 1.In each phase discussed above, all Lyapunov exponents Ai are larger than Alocal.In Fig. 5, Ahi (i = 1,2,3) are plotted against E, where N = 1000. These Lyapunov exponents do not change significantly with E. In the weak coupling phase, A& CCN-’ for any i as is seen
S. Morita/Physics Letters A 226 (I 997) 172-I 78
175
6
0
Ah,
Ah, + A.h, q
T4 z 2 0 :, -1
0
X 6
X
9-----l
N
ST4 Q
Fig. 6. AAi as a function of system size. (a) Weak coupling phase ( l = 0.35). (b) Strong coupling phase (E = 0.45).
2 0 i -1
0
1
X Fig. 4. Snapshot of the density distribution p,,(x) obtained from the approximationto the NFPE (see Section 3) after IO5steps as transient. (a) Turbulent state; a = 1.99, l = 0.25. (b) Two-band state; (I = 1.99, l = 0.35. (c) One-band state; a = 1.99, l = 0.45.
from Fig. 6a. Even in the strong coupling phase, all Lyapunov exponents except AI approach htocniwith the system size, which is seen in Fig. 6b. This decrease, however, does not obey the N-’ law but exhibits an anomalous dependence on N, i.e., A&i cc N-p with p< 1.
3. Rate of exponential divergence of NFF’E 0.8
0
0.6
+A&
A& 0 Ah,
0.4
In the thermodynamic limit N -+ cc, statistical states of an ensemble of maps governed by ( 1) are characterized by the snapshots of its density distribution P,,(X) . The time evolution of the density P,,(X) obeys
0.2
Pn+l(X) = 0.3
0.4 &
0.5
Fig. 5. A& as a function of coupling strength for a = 1.99, Iv= 1000.
P,(Y+) + Pn(Y-) a(1 -6)
’
where y+ and y_ are the two preimages of the same n, i.e., x = ( 1 - E) ( 1 - a/y&l) + eh,,. This is called the nonlinear Frobenius-Pen-on equation (NFPE) [ 201. The mean-field h, is given by
S. Morita/Physics
176
hn
=
s
f(x) p,(x) dx.
Letters A 226 (1997)
(7)
Let its initial condition be given by a uniform distribution pa(x) = 3. In considering finite-dimension approximations to the NFPE, the density distribution is expanded with a series of M -t 1 step functions as the basis functions [ 241. The density dis~ibution is thus expressed as
172-I
78
0 A (M=300) V A (M=500)
0.6 x‘-
0.4
z c
0.2
-
0.0 0.0 0.1 0.2 0.3 0.4
(8) j=O
where i?(x) is a step function 6(x) = I (x > O), =o
(x < 0).
(9)
In the above equation, b’, and c’, represent the height and the location of the jumps in pn ( x), respectively. The bigger value of the superscript j implies that the jump occurred earlier. The evolution equations for b// and c’, are given by Ref. [ 241. In the rest of this section, we investigate the rate A of exponential divergence of two nearby trajectories generated by the nonlinear Frobenius-Perron equation. This rate A describes the degree of orbital instability of the motion of the density distribution and corresponds to the maximum Lyapunov exponent of the motion of the mean-field. The trajectory is given by the time series of (2M + 2)-dimensional vectors (bft,@t ,..., bf,clf,ci ,..., c~).Weconsidertheseparation vectors (/3$ /3;, . . . , j-3,“. -y:, y:, . . . , $1 that evolve according to the equations linearized around the trajectory
j'=O
- a( 1 - E)&20(c9
- 1).
(10)
+ 4 log a(1 4)
0.5
& Fig. 7. Rate of exponential divergence of NFPE as a function of the coupling strength a for a = I.99, for the two cases, i.e. M = 300 (circle) and M = 500 (triangle). For comparison’s sake, the maximum Lyapunov exponent calculated from the original maps (I) for N = IO4 is also plotted (cross). All these results were obtained from the average over IO5 steps after discarding IO5 steps as transient. The line superposed on the data indicates the divergence rate of the local map, that is, log[ a( I - e) 1. The above set of equations can be expressed in terms of a coefficient matrix D, as
(11) The time series of the matrix D, determines A which is a quantity analogous to the usual Lyapunov exponents. In Fig. 7, A is plotted against E for the cases of M = 300 and M = 500. These two results are in good mutual agreement except for the weak coupling region where the value calculated for M = 500 is little larger than that for M = 300. When the coupling is su~ciently strong, the approximation becomes superior. In Fig. 7, we superpose the line of Atocai,i.e., log[ a( 1 -E) 1 and plot the maximum Lyapunov exponent hi estimated for N = 10000 by direct simulation. In the weak coupling phase, the maximum Lyapunov exponent A\I approaches log[a( 1 - E)] with the increase of N as mentioned in Section 2 and deviates from A even if A is positive. When A is positive, the system has a non-trivial collective motion. Consequently, this non-trivial collective motion has no effect on Lyapunov exponents of the system in the weak coupling phase. On the contrary, in the strong coupling phase, A is nearly equal to the maximum Lyapunov exponent Al. As is seen in Fig. 8, the maximum Lyapunov exponent approaches the rate of exponential divergence as
S. Morita/Physics
Letters A 226 (1997) 172-178
0 ~=a.42 0 E=O.43 + E=O.44 l E=O.45
Fig. 8. Difference between Ai and ,4 as a function of system size for some values e and n = i .99 in the strong coupling phase. hl and n are obtained from an average over 106 steps after discarding Id steps as transient.
the system size increases. At comes to obey the law ,& - A N N-p for large N. Exponent /3 is obviously smaller than 1 near the transition point. This anomalous dependence disagrees with the conjecture (/3 = 1) in the case of diffuse-type coupled maps in Ref. [30]. This disagreement suggests a singular behavior exists near the transition. It is worthwhile studying further. The transition between strong and weak coupling phases corresponds to the coupling strength at which the local expansion rates Atocatand the expansion rates A of NFPE coincide. In consequence, the m~imum Lyapunov exponent has the smallest value at the transition point.
177
deed, the difference A& is proportional to N-l. In the strong couphng phase (et < E < E&, AAt decreases to a finite value. The maximum Lyapunov exponent in the thermodynamic limit coincides with the rate of exponential divergence of NFPE. & - n decreases with the system size and behaves like A1- n 0: N-p with /3 < 1. The value of n refers to the rate of ex~ne~tial divergence of the collective motion in the the~ody~ namic limit. The Lyapunov exponent exhibits anomalous dependence on N near the transition point Et. These results imply that there is a com~tition between the local divergence rate htocat and the divergence rate A of NFPE. The asymptotic value of the Lyapunov exponent will be identical to the larger one of these. In the weak coupling phase, the Lyapunov exponents reflect the local motion rather than the collective motion. These results indicate that the usual Lyapunov analysis is not so usefui to understand the feature of the collective motion for weak coupling. The system can have a non-trivial collective motion here. Indeed A is positive in the wide parameter region. On the other hand, the Lyapunov exponents are essential in the strong coupling phase. The maximum Lyapunov exponent gives the rate of exponential divergence of the collective motion. In the limit of large system size, the weak coupling phase roughly corresponds to a turbulent or two-band state, and the strong coupling phase to an inte~ittent or one-band state. In finite systems, however, this correspondence is not so sharp because temporal switching occurs due to a finite size effect.
4. Conclusions Acknowledgement In the present paper, we have investigated the dynamics of glob~ly coupled tent maps with a Lyapunov analysis. In order to characterize the collective behavior, we introduced the rate A of exponential divergence of the nonlinear Frobenius-Perron equation. This quantity describes the degree of the instability associated with the density p,,(x). We showed that the app~nce of collective behavior has some influence on the characteristics of the Lyapunov exponents. Both the weak coupling and the strong coupling phase exist in the desynchroni~ state where the system exhibits an N-dimensional behavior. In the weak coupling phase (c; < et), the Lyapunov exponents decrease and approach &+I with the system size. In-
The author would like to acknowledge valuable advice from Y. Kur~oto and fruitful discussions with 1: Chawanya, T. Muzuguchi, K. Kaneko, and the members of the Nonlinear Dynamics group of Kyoto University. Numerical computation in this work was supported by the Yukawa Institute for Theoretical Physics.
Refemnces [ 113. Miller and D.A. Huse, Phys. Rev. E. 48 ( 1993) 2528. [ 21 H. Chat& A. Lemaitre, Ph. Marcq and P. Matmeville, Physica A 224 ( 1996) 447.
178
S. MoritdPhysics
Letters A 226 (1997) 172-l 78
[31 C.S. O’Hem, D.A. Bgolf and H.S. Greenside, Phys. Rev. B. 53 (1996) 3374. [4] K. Wiesenfeld and l? Hadley, Pbys Rev. Lett. 62 (1989) 1335. IS] K. W~e~nfeld, C. ~mciko~kj, G. James and R. Roy, Pbys. Rev. Len. 65 (1990) 1749. [6] S.H. Strogatz, CM. Marcus, R.M. Westervelt and R.B. Mirollo. Physica D 36 (1989) 23. [ 71 T. Ikegami and K. Kaneko, Chaos 2 ( 1992) 397. [S] H. Sompolinsky, D. Golomb and D. Kleinfeld, Phys. Rev. A 43 (1991) 6990. ]9] D. Golomb, D. Hansel, B.Sharaiman and H. Sompolinsky, Phys. Rev. A 45 (1992) 3516. [ IO] K. Kaneko, Physica D 77 (t994) 456. [ 11J H. Nozawa, Chaos 2 (1992) 377. [ I2] K. Kaneko, Physica D 34 ( 1989) 1. [ 131 K. Kaneko, Phys. Rev. L&t. 63 ( 1989) 219; Physica D 41 (1990) 137. [ 141 K. Kaneko, Phys. Rev. Len. 65 (1990) 1391. [IS] K. Kaneko, Physica D 55 ( 1992) 368. [ 161 G. Perez and HA. Cerdeira, Phys. Rev. A 46 ( 1992) 7492.
[ 171 T. Shinbmt, Phys. Rev. E 50 ( 1994) 3230. [IS] M. Griniasty and V. Hakim, Phys. Rev. E 49 ( 1994) 2661. [ 191 K. Kaneko, Physica D 86 (1995) 158. ]20] A.S. Pikovsky and J. Kurths, Physica D 76 (1994) 41 I; Phys. Rev. J&t. 72 ( 1994) 1644. 121J W. Just, J. Stat. Phys. 79 (1995) 429; Pbysica D 81 (1995) 317. f22] J. Losson, J. Milton and MC Mackey, Pbysica D 81 ( 1995) 177‘ 1231 S.V. Brshov and A.B. Potapov, Physica D 86 (1995) 523. [24] S. Morita, Phys. L.&t. A 21 I (1996) 258. [ 251 H. Chatd and P Manneville, Eumphys. L&t. 17 ( 1992) 291; Prog. Theor. Phys. 87 ( 1992) I. ]26] N. Nakagawa and K. Kuramoto, Prog. Theor. Phys. 89 (1993) 313. 1271 H. Chat& Burophys. lett. 21 (1993) 419. [28] N, Nakagawa and K. Kuramoto, Physica D 80 ( 1995) 307. 1291 R. Livi, A. Pohti, S. Ruffo and A. Valpiani, J. Stat. Phys. 46 (1987) 147. 1303 W. Yang, E.J. Ding and M. Ding, Phys. Rev. Lett. 76 ( 1996) 1808.
Lya~~~Qvanalysis of collective behavior in a network of ~~a~ti~ elements Receivedt7 October1996;revised rn~use~pt received 3 December t996; accepted for publication 3 December 1996 Co~unica~
by C.R. Doering
Abstract Lyapunov analysis of collective behavior of glohaliy coupled maps is presented. There exists a phase transition on account of a competition between the local instability and the coliective motion ins~bi~ty, The rate of exponential dive~ence of the trajectories of the nonlinear ~roben~us-Penn equation is used here. PACS:05.45.+b; 05.70,Lu; 82.4O.Bj K~ords: ~lob~ly coup&d maps; Chaos; ~ro~ius-Penn
~uation; Lyapunov an~iysi~ CoIl~~ve behavior
1, Introduction Coupled chaotic maps recently attracted particular attention as an excelIent model for spatially extended dissipative systems. The number of positive Lyapunov ex~nents of such systems is often propo~on~ to the system size. This is called the extensive chaos, and statistical analysis is possible to some extent in that case, In a two-Dimensions lattice of chaotic coupled maps, an Isi~g-like phase position was ~s~overed [l-3]. O’Hem et al. investigate the relationship between the phase transition and the Lyapunov spectrum. It was found that the phase transition occurs near the minimum of the Lyapunov dimension density. We shall concentrate on globally coupled maps (GCM) below. The all-to-all nature of the coupling may be regarded as either an idealization of longrange coupling or an approximation to short-range
i E-Ml: [email protected].~oto-u.ac.jp.
coupling in high-dimensional lattices. In GCM, the concept of space loses meaning. In spite of its simplicity, GCM exhibits extremely rich behavior [ 131, The explicit form of GCM we will work with is given by x~+I(~) = (l-E)f(x,(i))+K~~(x,(~~)). i’=i
(I)
Here n repr~en~ discrete time steps, i specifies the elements (i= 1,2,3,..., N), and E gives the coupling strength. The system is thus a mean-field version of coupled map lattices (CML) [ 121. It is assumed that the individual map f(x) gives rise to chaotic dynamics. We specifically consider a tent map f(x)
= 1 - @I,
(2)
with p~~eter a satisfying 1 < a 6 2, Let Q be fixed as a = 1.99. The initial condition is such that the elements are ~nifo~ly dist~but~ over the interval
0375-9601/97/$17.~ Copyright @ 1997 Elsevier Science B.V. All rights reserved Pfrs037s-9601(96)00940-1
S. Moriia/Ph~~~~~s Letters A 226 (1997) 172-i 78
[ - I,1 1. The variable x of each map is then bounded in the same interval. The coupling is only through the mean-field h, defined by
Collective behavior of the system can be described in terms of the motion of the mean-field. In a completely synchronized state, the motion of h, is reduced to the one-dimensional map f(n). In this case, the system has only one positive Lyapunov exponent which is equal to the Lyapunov exponent log(a) of the individual map. It is clear that the completely synchronized state is stable under the condition a( 1 - E) < 1 (E > > 1 (E< Ecs= 0.4975 . . .fora= 1,99).Fora(l--E) E,,) , the elements are scattered and behave chaotically in time. In that case, the N Lyapunov exponents are all positive as we see in the next section, so that the dimension of the motion becomes N. Thus, the dimension of the motion changes abruptly at a( 1 - E) = 1, i.e. E = efS.
It was argued by Just [ 21 J that there exists a sequence of bifurcations taking place at a( 1-E) = 2’/2’. These bifurcations correspond to a qualitative change in the statistical properties in the limit of large system size. Due to the presence of the mean-field fluctuation, however, this sequence of bifurcations is truncated and a one-band state takes place. The corresponding phase diagram is shown in Fig. 17 of Ref. [ 151. In the desynchronized state, the existence of non-trivial collective motions in spite of the highdimensionality of motion was reported [ 14-243. Mechanisms have been studied with various tools such as the mean-field return map [ 1617, I9 1, correlation between the individual maps [ 181, and the FrobeniusPerron operator [ 19-241. The collective motion is expected to be reduced to a lower-dimensional motion in the thermodynamic limit N --+ cc. Such collective motion appears to be generic in coupled dynamical systems [25,26], and the significance of the Lyapunov spectra for its unde~tanding has been discussed [ 27-291. This paper presents some results of our numerical investigation of the Lyapunov analyses in relation to collective motions. The dependence of the Lyapunov exponents on the coupling parameter E and the sys-
173
-
a=l.7
-
a-l .8 a=l.S a=139
-
Fig. I. The maximumLyapunov exponents as a function of coupling strength for some values of II, N = 100. They are obtained from an average over 10’ steps after discarding IO’ steps as transient.
tern size N will be studied. In Section 2, we classify the desynchronized state into two classes according to the feature of the Lyapunov exponents. In Section 3, we present a finite-dimension approximation to the nonlinear Fro~nius-Pe~on equation, which enables a deeper understanding of the properties of the Lyapunov exponents. We compute the expansion rate of two nearby trajectories of the nonlinear FrobeniusPerron equation. Some discussions and conclusions will be given in Section 4.
2. Lyapunov analyses To begin with, we consider the maximum Lyapunov exponent. Fig. 1 shows the maximum Lyapunov exponent with varying coupling strength E for some fixed values of a. The flat part corresponds to complete synchronization. For each a, the maximum Lyapunov exponent exhibits a minimum at a certain E. Let h, be viewed as an external field acting on the individual maps. In this picture, only the first part of the r.h.s. of ( 1) contributes to the value of the Lyapunov exponents. All Lyapunov exponents Ai thus degenerate to a single value log[ a( 1-E) ] , which is identical with the expansion rate Atocatof the effective local dynamics. The characteristics of the collective motion are indicated by the differences Ahi = Ai- hiocairather than the Lyapunov exponents Ai themselves. In Fig. 2, the difference AAi is plotted as a function of e for some values of the system size N. It is seen
S, Morita/Physics
Letters A 226 (1997) 172-l 78
N=lO
10’
n N=lO’ + N=103 A N=104
10-l
0
.
. .1 .
“d
1O-3 10e4
N
N=IO N=lO’ N=103 N=104
fig. 3. AAl as a function of system size for some values of E and N = 1.99.
Fig, 2. AAl as a function of coupling strength for some values of N and n = 1.99. (a} Log-log plot and (b) linear-linear plot. The maximum Lyapunov exponent is obtained from an average over IO5 steps after discarding lo5 steps as transient.
that the relationship KE
(4)
holds in the region of small E. This e-dependence may naturally be expected from the fact that the offdiagonal elements of the Jacobi matrix of ( 1) are proportional to E. As E increases, however, this scaling law becomes broken. From Fig. 2b, the transition is seen to occur at about E = Et N 0.41. By comparison with Fig. I, it is seen that this transition occurs when the m~imum Lyapunov exponent takes a minimum value. Fig. 3 is reconstructed from Fig. 2, where the difference AA1 is plotted as a function of the system size for some values of 6. Two different phases are observed in Fig. 3. Namely, in the weak coupling phase (E < Et), the difference obeys AAt cc $
..___,, . 8~0.45 &=0.43 s=O.42 &=0.41 E=O.4 E=O.35 &=0.3 &=0.25 &=O.Z
.1o-2
AA1
..__,
(5)
which may be unde~tood from the fact that the offdiagonal elements of the Jacobi matrix of ( 1) are proportional to N-* I This phase roughly corresponds to turbulent or two-band states [ 19,211 in the limit of large system size. The difference between the turbulent state and the two-band state is most clearly seen from the profile of the density distribution (see Figs. 4a,4b). The density distribution p(x) will be defined in the next section. The transition between these states originates qu~itatively from the band-splitting of the local maps which occurs at a( 1 - tz) = t/z (E = 0.2893. . . for a = 1.99). In the strong coupling phase (et < E < ecs), Ah, decreases down to a certain value with the system size N, and the maximum Lyapunov exponent At approaches a value clearly different from Atocat.The ~ymptotic value of the m~imum Lyapunov exponent corresponds to the rate of exponential divergence of the evolution of the density distribution, as we shall show in the next section. The strong coupling phase roughly corresponds to what we call intermittent or one-band states in the limit of large system size. In the inte~ittent state, there are inte~ittent switchings between one-band and two-band states. In the one-band state, the elements form a broad band as in Fig. 4c and the band moves chaotically [ 191. We now consider other Lyapunov exponents Ai, 1.In each phase discussed above, all Lyapunov exponents Ai are larger than Alocal.In Fig. 5, Ahi (i = 1,2,3) are plotted against E, where N = 1000. These Lyapunov exponents do not change significantly with E. In the weak coupling phase, A& CCN-’ for any i as is seen
S. Morita/Physics Letters A 226 (I 997) 172-I 78
175
6
0
Ah,
Ah, + A.h, q
T4 z 2 0 :, -1
0
X 6
X
9-----l
N
ST4 Q
Fig. 6. AAi as a function of system size. (a) Weak coupling phase ( l = 0.35). (b) Strong coupling phase (E = 0.45).
2 0 i -1
0
1
X Fig. 4. Snapshot of the density distribution p,,(x) obtained from the approximationto the NFPE (see Section 3) after IO5steps as transient. (a) Turbulent state; a = 1.99, l = 0.25. (b) Two-band state; (I = 1.99, l = 0.35. (c) One-band state; a = 1.99, l = 0.45.
from Fig. 6a. Even in the strong coupling phase, all Lyapunov exponents except AI approach htocniwith the system size, which is seen in Fig. 6b. This decrease, however, does not obey the N-’ law but exhibits an anomalous dependence on N, i.e., A&i cc N-p with p< 1.
3. Rate of exponential divergence of NFF’E 0.8
0
0.6
+A&
A& 0 Ah,
0.4
In the thermodynamic limit N -+ cc, statistical states of an ensemble of maps governed by ( 1) are characterized by the snapshots of its density distribution P,,(X) . The time evolution of the density P,,(X) obeys
0.2
Pn+l(X) = 0.3
0.4 &
0.5
Fig. 5. A& as a function of coupling strength for a = 1.99, Iv= 1000.
P,(Y+) + Pn(Y-) a(1 -6)
’
where y+ and y_ are the two preimages of the same n, i.e., x = ( 1 - E) ( 1 - a/y&l) + eh,,. This is called the nonlinear Frobenius-Pen-on equation (NFPE) [ 201. The mean-field h, is given by
S. Morita/Physics
176
hn
=
s
f(x) p,(x) dx.
Letters A 226 (1997)
(7)
Let its initial condition be given by a uniform distribution pa(x) = 3. In considering finite-dimension approximations to the NFPE, the density distribution is expanded with a series of M -t 1 step functions as the basis functions [ 241. The density dis~ibution is thus expressed as
172-I
78
0 A (M=300) V A (M=500)
0.6 x‘-
0.4
z c
0.2
-
0.0 0.0 0.1 0.2 0.3 0.4
(8) j=O
where i?(x) is a step function 6(x) = I (x > O), =o
(x < 0).
(9)
In the above equation, b’, and c’, represent the height and the location of the jumps in pn ( x), respectively. The bigger value of the superscript j implies that the jump occurred earlier. The evolution equations for b// and c’, are given by Ref. [ 241. In the rest of this section, we investigate the rate A of exponential divergence of two nearby trajectories generated by the nonlinear Frobenius-Perron equation. This rate A describes the degree of orbital instability of the motion of the density distribution and corresponds to the maximum Lyapunov exponent of the motion of the mean-field. The trajectory is given by the time series of (2M + 2)-dimensional vectors (bft,@t ,..., bf,clf,ci ,..., c~).Weconsidertheseparation vectors (/3$ /3;, . . . , j-3,“. -y:, y:, . . . , $1 that evolve according to the equations linearized around the trajectory
j'=O
- a( 1 - E)&20(c9
- 1).
(10)
+ 4 log a(1 4)
0.5
& Fig. 7. Rate of exponential divergence of NFPE as a function of the coupling strength a for a = I.99, for the two cases, i.e. M = 300 (circle) and M = 500 (triangle). For comparison’s sake, the maximum Lyapunov exponent calculated from the original maps (I) for N = IO4 is also plotted (cross). All these results were obtained from the average over IO5 steps after discarding IO5 steps as transient. The line superposed on the data indicates the divergence rate of the local map, that is, log[ a( I - e) 1. The above set of equations can be expressed in terms of a coefficient matrix D, as
(11) The time series of the matrix D, determines A which is a quantity analogous to the usual Lyapunov exponents. In Fig. 7, A is plotted against E for the cases of M = 300 and M = 500. These two results are in good mutual agreement except for the weak coupling region where the value calculated for M = 500 is little larger than that for M = 300. When the coupling is su~ciently strong, the approximation becomes superior. In Fig. 7, we superpose the line of Atocai,i.e., log[ a( 1 -E) 1 and plot the maximum Lyapunov exponent hi estimated for N = 10000 by direct simulation. In the weak coupling phase, the maximum Lyapunov exponent A\I approaches log[a( 1 - E)] with the increase of N as mentioned in Section 2 and deviates from A even if A is positive. When A is positive, the system has a non-trivial collective motion. Consequently, this non-trivial collective motion has no effect on Lyapunov exponents of the system in the weak coupling phase. On the contrary, in the strong coupling phase, A is nearly equal to the maximum Lyapunov exponent Al. As is seen in Fig. 8, the maximum Lyapunov exponent approaches the rate of exponential divergence as
S. Morita/Physics
Letters A 226 (1997) 172-178
0 ~=a.42 0 E=O.43 + E=O.44 l E=O.45
Fig. 8. Difference between Ai and ,4 as a function of system size for some values e and n = i .99 in the strong coupling phase. hl and n are obtained from an average over 106 steps after discarding Id steps as transient.
the system size increases. At comes to obey the law ,& - A N N-p for large N. Exponent /3 is obviously smaller than 1 near the transition point. This anomalous dependence disagrees with the conjecture (/3 = 1) in the case of diffuse-type coupled maps in Ref. [30]. This disagreement suggests a singular behavior exists near the transition. It is worthwhile studying further. The transition between strong and weak coupling phases corresponds to the coupling strength at which the local expansion rates Atocatand the expansion rates A of NFPE coincide. In consequence, the m~imum Lyapunov exponent has the smallest value at the transition point.
177
deed, the difference A& is proportional to N-l. In the strong couphng phase (et < E < E&, AAt decreases to a finite value. The maximum Lyapunov exponent in the thermodynamic limit coincides with the rate of exponential divergence of NFPE. & - n decreases with the system size and behaves like A1- n 0: N-p with /3 < 1. The value of n refers to the rate of ex~ne~tial divergence of the collective motion in the the~ody~ namic limit. The Lyapunov exponent exhibits anomalous dependence on N near the transition point Et. These results imply that there is a com~tition between the local divergence rate htocat and the divergence rate A of NFPE. The asymptotic value of the Lyapunov exponent will be identical to the larger one of these. In the weak coupling phase, the Lyapunov exponents reflect the local motion rather than the collective motion. These results indicate that the usual Lyapunov analysis is not so usefui to understand the feature of the collective motion for weak coupling. The system can have a non-trivial collective motion here. Indeed A is positive in the wide parameter region. On the other hand, the Lyapunov exponents are essential in the strong coupling phase. The maximum Lyapunov exponent gives the rate of exponential divergence of the collective motion. In the limit of large system size, the weak coupling phase roughly corresponds to a turbulent or two-band state, and the strong coupling phase to an inte~ittent or one-band state. In finite systems, however, this correspondence is not so sharp because temporal switching occurs due to a finite size effect.
4. Conclusions Acknowledgement In the present paper, we have investigated the dynamics of glob~ly coupled tent maps with a Lyapunov analysis. In order to characterize the collective behavior, we introduced the rate A of exponential divergence of the nonlinear Frobenius-Perron equation. This quantity describes the degree of the instability associated with the density p,,(x). We showed that the app~nce of collective behavior has some influence on the characteristics of the Lyapunov exponents. Both the weak coupling and the strong coupling phase exist in the desynchroni~ state where the system exhibits an N-dimensional behavior. In the weak coupling phase (c; < et), the Lyapunov exponents decrease and approach &+I with the system size. In-
The author would like to acknowledge valuable advice from Y. Kur~oto and fruitful discussions with 1: Chawanya, T. Muzuguchi, K. Kaneko, and the members of the Nonlinear Dynamics group of Kyoto University. Numerical computation in this work was supported by the Yukawa Institute for Theoretical Physics.
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Letters A 226 (1997) 172-l 78
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