Analytical results for coupled map lattices with long-range interactions

0.14 MB·English

by  C. Anteneodo

Analytical results for coupled map lattices with long-range interactions
Celia Anteneodo1, Sandro E. de S. Pinto2, Antoˆnio M. Batista3 and Ricardo L. Viana2
1. Centro Brasileiro de Pesquisas F´ısicas, Rua Dr. Xavier Sigaud 150, 22290-180, Rio de Janeiro, RJ,
2. Departamento de F´ısica, Universidade Federal do Parana´, 81531-990, Curitiba, PR,
3. Departamento de Matem´atica e Estat´ıstica, Universidade Estadual de Ponta Grossa, Ponta Grossa, PR, Brazil.
(February 8, 2008)
We obtain exact analytical results for lattices of maps with couplings that decay with distance
asr−α. Weanalyzetheeffect ofthecouplingrangeonthesystem dynamicsthroughtheLyapunov
spectrum. For lattices whose elements are piecewise linear maps, we get an algebraic expression
for the Lyapunov spectrum. When the local dynamics is given by a nonlinear map, the Lyapunov
4 spectrum for a completely synchronized state is analytically obtained. The critical lines character-
0 izing the synchronization transition are determined from the expression for the largest transversal
0 Lyapunovexponent. Inparticular,itisshownthatinthethermodynamicallimit,suchtransitionis
2 only possible for sufficiently long-range interactions, namely, for α≤αc <d, where d is the lattice

n dimension.
a
J PACS numbers: 05.45.Ra,05.45.-a,05.45.Xt

2
2
Synchronization between coupled chaotic systems is (appearingatthelevelofthemacroscopicthermodynam-

] one of the most intriguing nonlinear phenomena [1]. It ical description as well as in the underlying microscopic
D has attracted much interest since two decades ago [2] as dynamics),whichstillrequiredeeperunderstanding[19].
C it appears in a wide range of real systems such as in ar- Simple dynamical models, such as CMLs, may add new
. raysofJosephsonjunctions[3],oscillatingchemicalreac- knowledgeonnon-equilibriumlongrangesystems. How-
n
tions[4],physiologicalprocesses[5],andhasapplications ever, there is a lack of analytical results for CMLs with
i
l as in communications [6] and control theory [7]. There arbitrary range couplings. Exact analytical results are
n
are many types of synchronized behavior [8], but we are particularly crucial because the occurrence of phenom-
[
particularly interested in the completely synchronized ena such as shadowing breakdown [20] or spurious syn-

2 states (CSSs) of coupled map lattices (CMLs), where all chronization [21] set difficulties in numerical approaches
v
maps present the same amplitude at all times. Com- due to the unavoidable finite precision of numerical sim-
4
plete synchronization is an example of non-equilibrium ulations.
1
phase transition[9], which may be relatedto actualcrit- Hereweexamineaformofcouplingwhoseintensityde-
0
8 ical phenomena like the superconducting-normal transi- cayswiththedistancerbetweensitesas1/rα,withα≥0
0 tion in Josephson junctions [10]. [22]. It has also been considered in biological networks
3 [23], in ferromagnetic spin models [24], many-particle
0 CMLs, which are dynamical systems with discrete conservative (Hamiltonian time evolution) classical sys-
n/ space and time, and a continuous state variable, have tems [25,26], large populations of limit cycle oscillators
i beeninvestigatedastheoreticalmodelsofspatiotemporal [27] and a generalization of the Kuramoto model [28],
l
n phenomena in a variety of problems in condensed mat- amongotherexamples. Explicitly,weconsiderachainof
: ter physics, neuroscience and chemical physics [11]. The N coupled one-dimensional chaotic maps x7→f(x) such
v
spatiotemporal behavior is governed by two simultane- that the coupling prescription is
i
X ousmechanisms: theintrinsicnonlineardynamicsofeach
ar map, and diffusion due to the spatial coupling between x(i) =(1−ε)f(x(i))+ ε N′ f(xn(i−r))+f(xn(i+r)),
maps; the dynamical pattern being the outcome of the n+1 n η(α) rα
competition between them. This applies, in particular, Xr=1
to the problem of synchronization of chaotic maps [12]. (1)
The effective coupling range is a crucial factor to de-
(i)
where x represents the state variable for the site i
termine whether or not chaotic maps mutually synchro- n
(i = 1,2,...,N) at time n, ε ≥ 0 and α ≥ 0 are the
nize. Nearest-neighborcouplings(shortrange)donotfa-
coupling strength and effective range, respectively, and
vorsynchronization,since the coupling effect is typically
too weak to overcome the intrinsic randomness of map η(α) = 2 N′ r−α, is the normalization factor, with
r=1
dynamics [13]. On the other hand, long-range couplings N′ = (N P− 1)/2 for odd N. In conservative systems
tend to facilitate synchronization, as exemplified by the [25,26], scaling by η plays an important role in making
limiting case of global (mean-field) coupling [14]. Lat- the systems pseudo-extensive. Here periodic boundary
tices of non-locally coupled maps appear in neural net- conditions x(i) = x(i±N) and random initial conditions
n n
workswithlocalproductionofinformation[15],modelsof are assumed. The coupling term is a weighted average
physico-chemical reactions [16], assemblies of biological of discretized spatial second derivatives, the normaliza-
cells with oscillatoryactivity [17],and diffusion coupling tionfactorsbeingthesumofthecorrespondingstatistical
in nucleation kinetics [18]. Beyond CMLs, systems with weights. It is straightforwardto prove that in the limits
many degrees of freedom with long-range couplings are α = 0 and α → ∞ Eq. (1) reduces to the global mean-
an interesting object of study because of their anomalies fieldandthelocalLaplacian-typecouplings,respectively.
1
We characterize the spatio-temporal synchronization sults Λˆ =βBˆ. So,inordertoobtainthe LS,itis enough
dynamics by means of the Lyapunov spectrum (LS) of to diagonalize B. Because of its periodicity, B can be
the lattice, that enables one to estimate, for instance, diagonalized in Fourier space [33], the eigenvalues being
theKolmogorov-SinaientropythroughthePesinformula
[29] and the Lyapunov dimension, which gives an up- N′
cos(2πkm/N)
per bound on the effective number of degrees of freedom bk =2 mα , k =1,...,N, (5)
neededtocharacterizethesystemdynamics[30]. Besides mX=1
characterizing a CSS, when it exists at all, we must in-
where we considered odd N. Finally, from Eq. (3), tak-
vestigateitsstabilitywithrespecttosmallperturbations.
If the CSS turns out to be dynamically unstable, we are ing into account the special form of Λˆ, the LS is given
faced with two possibilities: either the CSS presents the by
so called bubbling attractor, and in this case the CSS
ε
only lasts for a finite time, or the CSS loses transversal λ = lnβ+ln 1−ε+ b . (6)
stability through a blowout bifurcation [8]. k (cid:12) η(α) k(cid:12)
(cid:12) (cid:12)
In this work we will present exact analytical results (cid:12) (cid:12)
(cid:12) (cid:12)
This expressionis consistentwith previousnumericalre-
for the CML (1). We will show that for a 1D lattice
sults [34]. In the extreme cases α → ∞ and α = 0 the
of N coupled piecewise linear maps it is possible to ob-
known expressions [14,29]are recovered.
tainanexactanalyticalexpressionfortheLS,theresults
Now we will consider lattices of nonlinear maps. An
shown in [14,29] being recoveredin the limits α=0 and
importantcasethatcanbetackledeasilyistheonewhere
α→∞. Whenthemapsx7→f(x)arenonlinear,wewill
themapsareintheCSS.Asitwillbecomeclearsoon,this
show that analytical results are still possible for CSSs.
instance provides relevant information on the synchro-
By means of the algebraic formulas for the LS, one can
nization transition. In the CSS, the dynamical variables
find the synchronizationregionsin theε×αspace, since
the second largest Lyapunov exponent (belonging to the of all maps coincide, i.e, x(n1) = x(n2) = ...,xn(N) ≡ xn(∗),
directiontransversaltotheSM)equaltozeroindicatesa ateachtimestepn. TheLSfortheCCSwhenα=0has
transition to the synchronized state [16,31]. Finally, the already been found by Kaneko [14]. Now, for arbitrary
results obtained for a chain of maps will be extended to α, we have Dn = f′(x(n∗))11N, thus, Tn = f′(xn(∗))Bˆ and
d-dimensional hypercubic lattices. TTT = ( n−1[f′(x(∗))]2)Bˆ2n. Therefore, for the CSS,
n n j=0 j
In order to calculate the LS one has to consider the
following EQq. (3), one arrives at
tangent dynamics. By differentiating the equations of
the original maps (1), one obtains the evolution equa- n
1 ε
tions for tangent vectors ξ = (δx(1),δx(2),...,δx(N))T, λ∗ = lim ln|f′(x(∗))|+ln|1−ε+ b |, (7)
that in matrix form read ξn+1 = Tnξn, with the Jaco- k n→∞nXi=1 i η(α) k
bian matrix T given by
n whereb arethe eigenvaluesofB defined in(5). Assum-
k
T = (1−ε)+ ε B D , (2) ingergodicity,thetime-averagein(7)canbesubstituted
n n
(cid:20) η(α) (cid:21) by an averageoverthe single-mapattractor. In this way
where the matrices D and B are defined, respectively, one gets
n
by Dnjk = f′(xn(j))δjk and Bjk = 1/rjαk(1−δjk) , being λ∗ = λ +ln 1−ε+ ε b , (8)
rjk = minl∈Z|j − k + lN|. Notice that the particular k U (cid:12) η(α) k(cid:12)
choice of the interaction law is embodied in the matrix (cid:12) (cid:12)
(cid:12) (cid:12)
B which is time independent. where λ = hln|f′(x(∗))|(cid:12)i is the Lyapun(cid:12)ov exponent of
U
Once specified the initial conditions, the LS is ex- an uncoupled map. This expression is general: it ap-
tracted from the evolution of the initial tangent vector
plies to any lattice of nonlinear 1D maps coupled with
ξ : ξ = T ξ , where T ≡ T ...T T is product
0 n n 0 n n−1 1 0 the scheme here considered, the parameters that define
of n Jacobian matrices calculated at successive points of
the particular uncoupled map affecting only λ . For in-
U
the discrete trajectory. If Λ ,...,Λ are the eigenvalues
1 N stance, for the logistic map x 7→ f(x) = ax(1−x), with
of Λˆ = nl→im∞(TnTTn)21n (that are real and positive), the a = 4 and x ∈ [0,1], λU = hln|4(1−2x(∗))|i = ln2 [35]
Lyapunov exponents are obtained as [32] andthe contributionof the power-lawcoupling is always
ln|1−ε+ ε b |. Notice that the LS in CSSs has the
λ =lnΛ , k =1,...,N. (3) η(α) k
k k same structure as the LS obtained for piecewise linear
We start by applying the expressions above to the maps [Eqs. (5)-(6)].
piecewise linear maps x 7→ f(x) = βx (mod 1), with The synchronization transition can be characterized
β ≥1. In this case we have f′(x)=β =constant, there- by a complex order parameter [4] defined, for time n,
fore Dn =β11N, and Tn becomes as Rn = |N1 Nj=1e2πixn(j)|. A time-averaged amplitude
ε R¯ is computePd over an interval large enough to warrant
Tn =β (1−ε)11N + B ≡βBˆ , (4) thatthelatticehasattainedtheasymptoticstate. Inthe
(cid:20) η(α) (cid:21)
CSS, one has R¯ =1. On the opposite case of completely
the rightmost identity defining the matrix Bˆ. Since the non-synchronized maps, the site state variables x(j) are
n
symmetric tangent map does not depend on time, it re- so uncorrelated that R¯ ≈0.
2
A diagnostic of synchronization can also be extracted As can be observed in Fig. 1, the critical frontier de-
from the LS. It can be easily verified that, for arbitrary pends on the system size N. In the limit N → ∞ we
α, the CSS lies along the direction of the eigenvectoras- obtain
sociated to the largest exponent. This was previously
1−e−λU
observed by Kaneko for the particular case α = 0 [14]. ε = , (10)
c,∞
1−C(α)
Therefore, the CSS will be transversally stable if the
(N −1) remaining exponents are non positive, that is, where C(α) corresponds to
λ∗ < 0 (where the tilde stands for ordered exponents).
2 N′ N′ −1
Thesecondlargestexponent,λ∗,correspondsinEq. (8), cos(2πm/N) 1
e 2 C(α) = lim .
to k = 1 (or k = N −1, dueeto degeneracy) if the ar- N→∞(cid:18)(cid:20)mX=1 mα (cid:21)(cid:20)mX=1mα(cid:21) (cid:19)
gument of the modulus in Eq. (8) is positive and to
k = (N −1)/2 (or k = (N +1)/2) otherwise. One ob- (11)
tains
This limit is equal to unity for α > 1, so that Eq. (10)
N′ −1 furnishesa divergentresult. Forαoutside the domainof
2 cos(2πm/N)
ε =(1−e−λU) 1− , (9) convergence of the series, one gets
c (cid:18) η(α) mα (cid:19)
mX=1 1−α π cos(x)
2 N′ cos(πm(N −1)/N) −1 C(α) = π1−α Z xα dx. (12)
ε′ =(1+e−λU) 1− , 0
c (cid:18) η(α) mα (cid:19)
mX=1 Moreover, one has ε′c,∞ = 1+e−λU, if α < 1. Then,
in the limit N →∞, synchronizationis only possible for
where ε (ε′ ) are the coupling strengths below (above)
c c sufficiently long-range interactions (see Fig. 2), namely,
which the SM ceases to be transverselystable, such that
for α ≤ α < 1. The critical value here obtained for
synchronizedstatesarenottypicallyobserved. Itisnote- c
1D CMLs is different from the one reported for other
worthy that Eq. (9) is quite general for the coupling
1D systems with similar power-law interactions, such as
scheme here considered: the parameters that define the
ferromagneticspinmodels [24]or many-particleclassical
particular uncoupled nonlinear map (embedded in λ )
U Hamiltoniansystems[25];insuchcases,thecriticalvalue
just participate through the first factor.
fortheexistenceofanorder/disordertransitionisα =2.
c
In a generalization of the Kuramoto model, the value of
α is controversial: While α =2 was first reported [28],
1.0 c c
recentanalyticalconsiderationspoint to α =1 [36]. Al-
8 c
501
51 though the generalized Kuramoto model is a continuous
0.9 SYNCHRONIZED 21 time dynamical system, our analytical result for the up-
per bound of the critical value suggests that the latter
0.8 value is the correct one.
e
The expressions we have derived for the LS of chains
of maps can be straightforwardly generalized for hyper-
0.7
cubic lattices of arbitrary dimension d. In fact, for the
general d-dimensional case, it is straightforwardto show
0.6 that the eigenvalues of the matrix B become
0.5 b = 2d cos(2πr¯k.m¯/Nd1), k =1,...,N, (13)
k mα
mX¯6=0
0.4
0.0 0.2 0.4 0.6 0.8 1.0 1.2 where r¯k is the position vector of site k, m¯ =
a (m1,...,md), with 0≤mi ≤N′, and N′ =(Nd1 −1)/2.
The normalization factor reads
FIG.1. Synchronizationinparameterplaneε×αfora1D
1 Nα/d−1
lattice of N coupled logistic maps x7→f(x)=4x(1−x) and η(α,d) = 2d ∝ . (14)
ε ≤ 1. The critical line εc(α) was determined analytically mX¯6=0mα 1−α/d
from Eq. (9) for different values of N (full lines) and numer-
ically from the conditions R¯ 6=1 (open symbols) and λ˜∗ =0 Then, following the same lines as before but now for
2
(full symbols) for N =21 and 51. the sake of generalizing Eq. (10), it is easy to see that
ε will diverge if α/d>1, which leads to α <d.
c,∞ c
Fig. 1 presents the lower critical line in the parameter Additionally, our results for the LS could be extended
space ε × α for N coupled logistic maps x 7→ f(x) = to the more general class of coupling schemes where the
4x(1 − x). The critical line was obtained analytically dependenceofthecouplingstrengthontheinter-mapdis-
from Eq. (9) and numerically by means of two proce- tance is not necessarily of the power-law type. In these
dures: either when R¯ 6= 1 is numerically detected or by cases, one should feed Eqs. (6) and (8) with the eigen-
the condition of nullity of the second largest Lyapunov values of the appropriate matrix B, that contains the
exponent in the CSS. The numerical results are in good particulardependence of the interactionstrengthon dis-
agreement with the analytical prediction. tance.
3
2 [4] Y. Kuramoto, Chemical Oscillations, waves and turbu-
8 lence (Springer-Verlag, Berlin, 1984).
501 [5] L. Glass, Nature(London) 403, 849 (2000).
21 [6] K.M.CuomoandA.V.Oppenheim,Phys.Rev.Lett.71,
9
e 'c 5 [7] K65.(P1y9r9a3g)a.s, Phys. Lett. A 181, 203 (1993).
3
[8] S.Boccaletti,J.Kurths,G.Osipov,D.L.Valladaresand,
1 C.S. Zhou,Phys. Rep.366, 1 (2002).
[9] V. Ahlers and A. Pikovsky,Phys. Rev. Lett. 88, 254101
(2002).
[10] K. Park and M.Y. Choi, Phys. Rev. B 56, 387 (1997);
e
H.R. Shea and M. Tinkham, Phys. Rev. Lett. 79, 2324
c
(1997).
l U = ln 2 [11] K. Kaneko, in Theory and Applications of Coupled Map
0 Lattices, editedbyK.Kaneko(Wiley,Chichester, 1993).
0 1 2 3 4 5 a 6 [12] R. Brown and N.Rulkov,Chaos 7, 395 (1997).
[13] K. Kaneko, Physica D 34, 1 (1989); Physica D 23, 436
(1986).
FIG. 2. Critical coupling strengths for synchronization vs [14] K. Kaneko, Physica D 41, 137 (1990).
range parameter, for different values of N indicated in the [15] H.Nozawa,Chaos2,377(1992);S.Ishii,M.Sato,Phys-
figure. The critical lines were analytically obtained from Eq. ica D 121, 344 (1998).
(9) for a 1D lattice with power-law couplings and λU =ln2. [16] P.M.GadeandC.-K.Hu,Phys.Rev.E60,4966(1999).
Synchronization occurs for εc <ε<ε′c. [17] Y. Kuramoto, H. Nakao, Physica D 103, 294 (1997).
[18] Y. Kuramoto, D. Battogtokh and H. Nakao, Phys. Rev.
Inconclusion,wehavepresentedanalyticalexpressions Lett. 81, 3543 (1998).
[19] For a recent review, see T. Dauxois, S. Ruffo, E. Ari-
for the LS of CMLs with an interaction which decays
mondo and M. Wilkens (eds.) Dynamics and thermody-
with the lattice distance as a power law, for two cases:
namicsofsystemswithlong-rangeinteractions(Springer,
(i) piecewise linear coupled maps; and (ii) the CSS of
Berlin, 2002).
lattices of one-dimensional maps. Our results enable us
[20] C. Grebogi, S.M. Hammel, J.A. Yorke,and T. Sauer,
to predict the critical values for synchronization in the
Phys. Rev.Lett. 65, 1527 (1990).
coupling parameter plane (strength versus range), and
[21] C. Zhou, Y.-C. Lai, Physica D 135, 1 (2000); C. Zhou,
also may be used to obtain related quantities of inter- J. Kurths,Phys. Rev.Lett. 88, 230602 (2002).
est, like KS-entropies and Lyapunov dimensions. Such [22] S.E. de S. Pinto, R.L. Viana, Phys. Rev. E 61, 5154
exact analytical results are crucial in order to avoid dif-
(2000).
ficulties present in numerical approaches, such as shad- [23] S. Raghavachari and J. A. Glazier, Phys. Rev. Lett. 74,
owing breakdown, due to unavoidable finite precision of 3297 (1995).
numerical simulations. In addition, we have shown that, [24] S.A. Cannas and F.A. Tamarit, Phys. Rev. B 54,
in the thermodynamical limit, the critical range for syn- R12661(1996).
chronization is equal to the lattice dimension. Many of [25] F. Tamarit and C. Anteneodo, Phys. Rev. Lett. 84, 208
our results could be extended to lattices of continuous- (2000).
time oscillators, and hence have an even wider range of [26] C. Anteneodo and C. Tsallis, Phys. Rev. Lett. 80, 5313
applicability. (1998); C. Anteneodo and R.O. Vallejos, Phys. Rev. E
65, 016210 (2001).
[27] M.S.O.MassunagaandM.Bahiana,PhysicaD168,136
ACKNOWLEDGMENTS: (2002).
[28] J.L. Rogers and L.T. Wille, Phys. Rev. E 54, R2193
(1996).
We are grateful to S.R. Lopes, C. Tsallis and R.O.
[29] S. Isola, A. Politi, S. Ruffo, and A. Torcini, Phys. Lett.
Vallejos for useful comments, and to S. Ruffo for old A 143 365 (1990).
fruitful remarks. This work was partially supported by [30] D. Ruelle, Chaotic Evolution and Strange Attractors
brazilian agencies CAPES, CNPq, FAPERJ, Fundac¸a˜o (Cambridge University Press, Cambridge, 1989).
Arauc´aria e PRONEX. [31] P.M. Gade, H.A. Cerdeira and R. Ramaswamy, Phys.
Rev. E52, 2478 (1995).
[32] J.-P Eckmann and D. Ruelle, Rev. Mod. Phys. 57, 617
(1985).
[33] A. Campa, A. Giansanti and D. Moroni, Phys. Rev. E
[1] L.M. Pecora, T.L. Carroll, G.A. Johnson, D.J. Mar and 62, 303 (2000); M.C. Firpoand S.Ruffo,J.Phys.A34,
J.F. Heagy,Chaos 7, 520 (1997) and references therein. L511 (2001).
[2] H. Fujisaka and T. Yamada, Prog. Theor. Phys. 69, 32 [34] A.M. Batista, S.E. de S. Pinto, R.L. Viana, and S.R.
(1983); L.M. Pecora and T.L. Carroll, Phys. Rev. Lett. Lopes, Phys. Rev.E 65, 056209 (2002).
64, 821 (1990). [35] E. Ott, Chaos in Dynamical Systems (Cambridge Uni-
[3] P. Hadley, M.R. Beasley and K. Wiesenfeld, Phys. Rev. versity Press, 1993).
B 38, 8712 (1988). [36] M.Marodi,F.d’Ovidio,andT.Vicsek,Phys,Rev.E66,
011109 (2002).
4