Chaos synchronization in long-range coupled map lattices

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by  C. Anteneodo

Chaos synchronization in long-range coupled map lattices
C. Anteneodo1, A.M. Batista2 and R.L. Viana3
1. Centro Brasileiro de Pesquisas F´ısicas,
Rua Dr. Xavier Sigaud 150, 22290-180, Rio de Janeiro, RJ, Brazil;
2. Departamento de Matem´atica e Estat´ıstica, Universidade Estadual de Ponta Grossa,
84030-900, Ponta Grossa, PR, Brazil;
3. Departamento de F´ısica, Universidade Federal do Parana´,
81531-990, Curitiba, PR, Brazil.
We investigate thesynchronization phenomenon in coupled chaotic map lattices where the cou-
plings decay with distance following a power-law. Depending on the lattice size, the coupling
4 strength and the range of the interactions, complete chaos synchronization may be attained. The
0 synchronizationdomaininparameterspacecanbeanalyticallydelimitedbymeansofthecondition
0 of negativity of the largest transversal Lyapunov exponent. Here we analyze in detail the role of
2 all the system parameters in the ability of the lattice to achieve complete synchronization, testing

n analytical results with the outcomes of numerical experiments.
a
J PACS numbers: 05.45.Ra,05.45.-a,05.45.Xt

2
2
Coupledmaplattices(CMLs),dynamicalsystemswith roleofallthe systemparametersinthe abilityofthe lat-

] discrete space and time, are being intensively investi- tice to synchronize. Analytical results will be compared
D
gatednowadaysas models ofspatiotemporalphenomena with the outputs of numerical simulations performed for
C occurringinawidevarietyofsystemsofphysical,biolog- diverse chaotic maps.
. ical and technical interest [1]. In this letter we will deal Complete synchronization can be characterized by a
n
withthephenomenonofsynchronizationand,inparticu- complex order parameter defined, for time n, as R =
i n
nl lar, amongstthe various kinds of synchronizedbehavior, |N1 Nj=1e2πixn(j)| [10]. Typically a time-averaged ampli-
[ with the complete synchronization (CS) [2] occurring in tudPe R¯ is computed over a time interval long enough to
CMLs with regular long-range interactions. Most work
allow the lattice attain the asymptotic state. In the CS
1
v done so far on synchronization in CMLs has focused on state, one has R¯ =1 within a small allowed deviation.
two extreme coupling types: local (nearest-neighbor) [3]
4 Anotherdiagnosticofcompletesynchronizationcanbe
and global (”mean field”) ones [4]. However, non-local
3 obtained from the Lyapunov spectrum (LS) of the CS
0 couplings are relevant to a variety of situations ranging states. If the chaotic maps are completely synchronized,
1 fromneuralnetworks[5]tophysico-chemicalreactionsys-
themaximalLyapunovexponent,inthedirectionparallel
0 tems [6].
tothesynchronizationmanifold(SM),isstrictlypositive.
4 WeconsiderchainsofN coupledone-dimensional(1D)
ThenegativityofthesecondlargestLyapunovexponent,
0
chaotic maps x7→f(x) whose evolution is given by [7]
/ whichbelongs to the directiontransversalto the SM, in-
n
dicatesthestabilityofthesynchronizedstateundersmall
nli x(i) =(1−ε)f(x(i))+ ε N′ f(xn(i−r))+f(xn(i+r)), (1) transversaldisplacements [11].
n+1 n η rα In our case the lattice dynamics given by Eq. (1) can
v: Xr=1 be writtenasx(i) = F f(x(j)), whereF isa matrix
i n+1 j ij n
X where x(i) represents the state variable for the site i of the form P
n
r (i=1,2,...,N)attime n, ε≥0 is the coupling strength, ε
a α ≥ 0 controls the effective range of the interactions F= (1−ε)11+ B , (2)
and η = 2 N′ r−α is a normalization factor, with (cid:20) η (cid:21)
N′ = (N −1P)/r2=1for odd N. The main interest in this with 11 the N × N identity matrix and B defined by
couplingschemeresidesinthefactthatitallowstoinves- Bjk =(1−δjk)/rjαk, being rjk =minl∈Z|j−k+lN|.
tigate the role of the range of the interactions, scanning TheLyapunovspectrumisobtainedfromthedynamics
from the local (α→∞) to the global (α=0) cases [8]. of tangent vectors ξ, which in turn is obtained by differ-
CS takes place when the dynamical variables that de- entiation of the original evolution equations. In matrix
fine the state of each map adopt the same value for all form the tangent dynamics reads ξn = Tnξ0, where Tn
the coupled maps at all times n, i.e, x(1) = x(2) = ... = is product of n Jacobian matrices calculated at succes-
n n
x(N) ≡ x(∗). Depending on the lattice size and on the sivepointsofagiventrajectory. IfΛ(1),...,Λ(N) arethe
rannge of tnhe interactions, there may exist an interval of eigenvalues of Λˆ = nl→im∞(TnTTn)21n, the Lyapunov expo-
values of the coupling strength ε, for which such state is nents are obtained as λ(k) = lnΛ(k), for k = 1,...,N
spontaneouslyattained,aswehaveanalyticallyshownin [12]. For the CS states, one arrives at the following ex-
previouswork[9]. Itisourpurposeheretoscrutinizethe pression for the Lyapunov spectrum [9]
1
b(k)
λ¯(k) = λ + ln 1−ε+ε , (3) 2
U
(cid:12)(cid:12) η (cid:12)(cid:12) e (a) l = ln2 8
(cid:12) (cid:12) U 501
where λ > 0 is the Lyapu(cid:12)nov exponen(cid:12)t of the uncou-
U 21
pled chaotic map, and b(k) are the eigenvalues of B that 9
can be obtained by Fourier diagonalization and, for odd 5
N, read 3
N′ 1
cos(2πkm/N)
b(k) =2 , 1≤k ≤N. (4)
mα
mX=1
For even N, summations run up to N′ = N/2 and half
of the N′th term has to be substracted. The maximal
eigenvalue is b(N) =η and the minimal one is b(N′). Ex-
cept for the cases k = N′, with even N, and k = N, 0
the remaining eigenvaluesare two-folddegenerate,being 0 1 2 3 4 5 a 6
b(k) =b(N−k).
In the calculation of Lyapunov exponents λ(k), notice
thatthe parametersthatdefinethe particularuncoupled
2
map affect only λU, while the second term in Eq. (3) is (b) l = ln1.3 8
determined by the particular cyclic dependence on dis- e U 501
tance inthe regularcouplingscheme (apowerlawinour 21
case). It can be easily verified that, for arbitrary α, the 9
CS state lies along the direction of the eigenvector asso- 5
ciated to the largest exponent λ¯(N). Therefore, the CS 3
statewillbetransversallystableifthe(N−1)remaining
1
exponents are negative, that is |1−ε+εb(k)/η|<e−λU,
∀k 6= N. This is equivalent to requiring that the sec-
ondlargest(orlargesttransversal)asymptoticexponent,
denoted by λ¯⊥, be negative. This exponent is obtained
fromEq.(3)witheitherk =1(hencealsok =N−1due
to degeneracy) or with k = N′ (hence also k = N′ +1
if N is odd), depending on whether |1−ε+εb(1)/η| is,
0
respectively, greater or smaller than |1−ε+εb(N′)/η|. 0 1 2 3 4 5 a 6
The condition λ¯⊥ < 0 leads to ε < ε < ε′ [9] (see also
c c
[13]), where
b(1) −1
ε (α,N,λ )=(1−e−λU) 1− and (5) FIG.1. Synchronizationdiagraminparameterspace(α,ε),
c U
(cid:18) η (cid:19) for different values of N and λU = ln2 (a), ln1.3 (b). Lines
b(N′) −1 correspond to analytical predictions; symbols to numerical
ε′c(α,N,λU)=(1+e−λU)(cid:18)1− η (cid:19) . (6) simulations using the Bernoulli (a) or triangular (b) maps.
Synchronization is transversally stable in the region between
InFig.1weshowavarietyofcriticalcurvesinparame- thecouple of curves for each set of values of theparameters.
terspace(α,ε)obtainedfordifferentlatticesizes. Stable
CSstates dwellinthe regionboundedby theα=0axis,
and two curve segments. The critical curves were ob-
that for w = 0.074 yields λ ≃ ln1.3 (notice that
tained analytically from Eqs. (5) (lower curve) and (6) U
λ = −wlnw − (1 − w)ln[1 − w]). Additional tests
(upper curve). The symbols shown stand for the numer- U
ical results determined from the condition R¯ = 1 with a (results not shown here) were performed for other maps
toleranceof10−6. Twodifferentvaluesofλ wereconsid- such as the logistic map f(x) = µx(1−x) with µ = 4
U
(hence λ = ln2) and µ = 3.6533 (hence λ ≃ ln1.30)
ered. Numerical results shown in Fig. 1 were computed U U
yieldingthesamedegreeofagreement. Fortheseinterval
for the piecewise linear (a) Bernoulli shift f(x) = 2x
maps, in principle, one must have 0 ≤ ε ≤ 1 in order to
(mod 1) (therefore λ = ln2) and (b) triangular map
[14] U guaranteethat the state variables x(nj) will remaininside
the interval [0,1]. But, reinjection into the interval can
x/w for 0≤x≤w be performed through an operation, for instance (mod
f (x)= (7)
w (cid:26)(1−x)/(1−w) for w<x≤1 1), such that it does not spoil the Lyapunov exponent
of the chaotic uncoupled map. If trajectory points were
2
not reinjected, one can still look at our results as valid ofmapssynchronize(becausethecriticalcurvesinFig.2
fortrajectoriesortrajectorysegmentsaslongasthestate do not intersect). Generically it is easier to synchronize
variablesremainconfinedwithinthegiveninterval. Any- a small number of maps. Consistently with this obser-
way,forothermapssuchasf(x)=exp{−[(x−0.5)/σ]2} vation, lattices of small size (e.g., N ≤ 5 for λ = ln2 )
U
[15], one may have any coupling ε ≥ 0 since the map is cansynchronizeforanyα, foracertainwindowofεthat
naturally defined in the full real axis. Tests performed narrowswithincreasingα. ForN ≤3thereis,naturally,
withthisGaussianmap(resultsnotshowninthisletter) no dependence on α and the lattice syncronizes for any
are also in good accord with theoretical predictions. λ >0.
U
In general terms, we observe that for weak coupling,
the maps do not synchronize. As the coupling strength 1.5
increases, synchronization can occur depending on the l = ln2
lattice parameters (α,N,λ ). However, a too high cou- e U a
U
plingintensityε>ε′ hasadestabilizinginfluenceonthe
c 0.0
CS state and the lattice no longer synchronizes.
0.5
Concerning lattice size, already Fig. 1 exhibits the in- 1.0 1.0
tuitivefactthatitismoredifficulttosynchronizealarger 2.0
latticethanashorterone,allotherparametersbeingkept 8
fixed. In the limit N →∞ we obtain
1−e−λU
ε (α,∞,λ ) = , (8)
c U 1−C(α) 0.5
where C(α)=limN→∞b(1)/η. This limit is equal to one
for α > 1, so that Eq. (8) yields a divergent result. For
α outside the domain of convergence of the series, i.e. 1 10 100 N
α<1, one gets
1−α π cos(x) FIG. 2. Synchronization critical lines in the plane (N,ε)
C(α) = dx. (9)
π1−α Z0 xα fordifferentvaluesofαandλU =ln2. Inthiscaseαc ≈0.77.
Symbols correspond to theoretical calculations. Dotted lines
In that same range of α one has areguidestotheeyes. Synchronizationistransversallystable
in theregion between thetwocurvesfor each set of valuesof
ε′(α,∞,λ ) = 1+e−λU, (10)
c U theparameters.
whichisindependentonα,thusityieldsastraightlinein
the plots ofFig.1. Fromthe intersectionofε (α,∞,λ )
c U
with ε′(α,∞,λ ) it results a critical value of the inter- Although Figs. 1(a) and 1(b) yield qualitative similar
c U
results, their comparison makes clear that, as expected,
action range α , such that, for α ≤ α < 1 (α < d in
c c c
the morechaoticthe uncoupledmapsare,themorediffi-
the d-dimensional case [9]), synchronization is possible
cult becomes to obtain their synchronization. The influ-
even in the thermodynamic limit N → ∞ for an appro-
ence of the Lyapunov exponent λ on the synchroniza-
priate window of ε. Observe the corresponding domains U
tion domains in the parameter space (α,ε) is displayed
in Fig. 1.
As N diminishes, the upper curve in Fig. 1, which is in Fig. 3. The exhibited numerical results were acquired
for N =21 either Bernoulli or triangular maps.
a straightline for infinite N,gains a negative inclination
andextendsforlargeαtovaluesofεlessthanunity,indi- If the positive Lyapunov exponent of the uncoupled
map increases, the synchronization domain shrinks, col-
catingthatthedestabilizingeffectofverystrongcoupling
lapsing in the limit λ → ∞. In the opposite limit of
is more easily attained. The lower curve segment has a U
λ →0+ one gets
positive inclination, connecting to the upper curve at a U
point that forms a cusp for small N. For still smaller N
ε →0 and (11)
(e.g., N ≤ 5 for λ = ln2 and N ≤ 8 for λ = ln1.3) c
U U
the two critical curves do not intersect each other even b(N′) −1
′
ε →2 1− . (12)
in the limiting case of first neighbors (α→∞). c (cid:18) η (cid:19)
The effect of lattice size can also be observedin Fig. 2
that exhibits the synchronization domains in the plane Thislimitvalueofε′ dependsonαandN. IfN →∞and
c
(N,ε) for different values of α. Similar plots have been α → 0(∞), ε′ goes to 2.0 (1.0) in the limit of vanishing
c
observed in scale-free networks [16], as if there were an chaos. (Although N = 21, all these extreme behaviors
average or effective α in such cases. While for α > α are already insinuated in Fig. 3.) As a consequence, the
c
thereisunupperboundN (α,λ )ofthenumberofmaps critical value α < 1 below which the lattice synchro-
b U c
forwhichsynchronizationoccurs;forα<α anynumber nizes in the thermodynamic limit depends on the degree
c
3
of chaoticity of the uncoupled maps. This dependence When α > α , it must be N ≤ N (α,λ ) for the lat-
c b U
can be explicitly obtained by inversion of tice to synchronize, where N decreases with increasing
b
α−α (as shown in Fig 2). In the limit α → ∞, it is
c
2 easy to obtain, from the condition ε′ > ε , an approxi-
λ =ln −1 (13) c c
U (cid:20)C(α ) (cid:21) mateexpressionforthemaximalsize,validwhenonehas
c
sufficiently small λ and large N :
U b
extracted from Eqs. (8) and (10). The critical value α
c 2
as a function of λU is displayed in Fig. 4. In the limit of Nmax ≡Nb(∞,λU) ≤ πrλ . (14)
vanishing (infinite) λ , α goes 1.0 (0.0). U
U c
In Fig. 5, we exhibit the maximal size N for which
max
2 synchronization can be achieved in the limit of nearest-
N = 21 exp( l U) neighborcouplings(henceforanyα)asafunctionofλU,
e 3 together with the approximationgiven by Eq. (14).
2
1.3
1.1
1.05 100
1.01
1 N
max
10
0
0 2 4 6 a
1
0.01 0.10 1.00 l
FIG.3. Synchronizationdomainsinparameterplane(α,ε)
for 21 coupled chaotic maps and various values of λU. Nu- U
mericalsimulationswereperformedforBernoulliortriangular
maps. CriticalcurveswereobtainedanalyticallyfromEqs.(5) FIG.5. Maximal lattice size Nmax ≡Nb(α,λU), for which
and (6).
synchronizationcanbeachievedforanyα,asafunctionofthe
chaoticity indicator λU (circles), determined from the condi-
tion ε′c > εc for α = ∞. The solid line corresponds to the
approximation given by Eq. (14).
1.0
a
Summarizing, we have presented numerical and ana-
c
lytical results for the CS states in lattices of coupled
identical chaotic maps with interactions that decay with
distance as a power law. We have scrutinized the role
0.5 of the system parameters in the ability of the lattice
to attain complete synchronizationusing various chaotic
maps. We observed, in the coupling parameters plane,
anoveralldecrease of the area of the synchronizationre-
gions,asthenumberofcoupledmapsisincreased. More-
over, the shape of these regions is bounded by critical
curves which vary with the lattice size in a fashion we
0.0
0 1 2 3 4 5 were able to predict analytically based on the Lyapunov
l
spectrumofthe synchronizedstate ofthe lattice, forvir-
U tually any chaotic map, with excellent agreement with
numerical results. Similar findings have been given for
FIG. 4. Critical value αc, below which synchronization is thedependenceofthesynchronizationregionsonthede-
stable even in the thermodynamic limit, as a function of λU gree of chaoticity of uncoupled maps, as well as for the
(symbols), determined from Eq. (13). The dotted line is a maximal lattice size for which complete synchronization
guide to theeyes. of chaos can be achieved.
4
ACKNOWLEDGMENTS: [7] S.E.deS.PintoandR.L.Viana,Phys.Rev.E61,5154
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We thank Sandro E. de S. Pinto for interesting dis- [8] R.L.VianaandA.M.Batista,Chaos,SolitonsandFrac-
cussions. This work was partially supported by Brazil- tals 9, 1931 (1998).
[9] C. Anteneodo, S.E. de S. Pinto, A.M. Batista and R.L.
ian agencies CNPq, FAPERJ, Fundac¸a˜o Arauc´aria and
Viana, Phys. Rev. E. 68, 045202(R) (2003) [erratum
PRONEX.
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P.M. Gade, H.A. Cerdeira and R. Ramaswamy, Phys.
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