Transition to complete synchronization and global intermittent synchronization in an array of time-delay systems

Abstract

We report the nature of transitions from nonsynchronous to complete synchronization (CS) state in arrays of time-delay systems. We demonstrate that the transition to CS occurs distinctly for different coupling configurations. In particular, for unidirectional coupling, locally (microscopically) synchronization transition occurs in a very narrow range of coupling strength but for a global one (macroscopically) it occurs sequentially in a broad range of coupling strength preceded by an intermittent synchronization. On the other hand, in the case of mutual coupling a very large value of coupling strength is required for local synchronization and, consequently, all the local subsystems synchronize immediately for the same value of the coupling strength and hence globally synchronization also occurs in a narrow range of the coupling strength. In the transition regime, we observe a new type of synchronization transition where long intervals of high quality synchronization which are interrupted at irregular times by intermittent chaotic bursts simultaneously in all the systems, which we designate as global intermittent synchronization (GIS). We have also explained the mechanism behind this GIS transition by using unstable periodic orbit theory. The above studies are carried out in paradigmatic models such as piecewise linear and Mackey-Glass time-
delay systems.

Full text

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PHYSICAL REVIEW E 86, 016212 (2012)
Transition to complete synchronization and global intermittent synchronization in an array of
time-delay systems
R. Suresh,1D. V. Senthilkumar,2M. Lakshmanan,1and J. Kurths2,3,4
1Centre for Nonlinear Dynamics, School of Physics, Bharathidasan University, Tiruchirapalli 620 024, India
2Potsdam Institute for Climate Impact Research, 14473 Potsdam, Germany
3Institute of Physics, Humboldt University, 12489 Berlin, Germany
4Institute for Complex Systems and Mathematical Biology, University of Aberdeen, Aberdeen AB24 3UE, United Kingdom
(Received 29 February 2012; revised manuscript received 31 May 2012; published 13 July 2012)
We report the nature of transitions from the nonsynchronous to a complete synchronization (CS) state in arrays
of time-delay systems, where the systems are coupled with instantaneous diffusive coupling. We demonstrate
that the transition to CS occurs distinctly for different coupling configurations. In particular, for unidirectional
coupling, locally (microscopically) synchronization transition occurs in a very narrow range of coupling strength
but for a global one (macroscopically) it occurs sequentially in a broad range of coupling strength preceded
by an intermittent synchronization. On the other hand, in the case of mutual coupling, a very large value of
coupling strength is required for local synchronization and, consequently, all the local subsystems synchronize
immediately for the same value of the coupling strength and, hence, globally, synchronization also occurs in a
narrow range of the coupling strength. In the transition regime, we observe a type of synchronization transition
where long intervals of high-quality synchronization which are interrupted at irregular times by intermittent
chaotic bursts simultaneously in all the systems and which we designate as global intermittent synchronization.
We also relate our synchronization transition results to the above specific types using unstable periodic orbit
theory. The above studies are carried out in a well-known piecewise linear time-delay system.
DOI: 10.1103/PhysRevE.86.016212 PACS number(s): 05.45.Xt, 05.45.Pq
I. INTRODUCTION
Numerical and experimental investigations of chaotic syn-
chronizationincouplednonlinearsystemshavereceivedmuch
attention in recent years. This phenomena is omnipresent
and plays an important role in diverse areas of science and
technology[1,2].Inthesynchronizationprocess,twoidentical
chaotic systems do not always necessarily synchronize per-
fectly. Rather, long intervals of high-quality synchronization
are interrupted at irregular times by intermittent chaotic bursts
and such chaotic bursts along with the synchronization are
called on-off intermittency [3]. It has been shown that on-off
intermittency is a frequently occurring instability preced-
ing typical synchronization transitions in diverse dynamical
systems, mediated by unstable periodic orbits (UPOs) [4].
Further, as the coupling parameter is increased, a periodic
orbitembeddedintheattractorintheinvariantsynchronization
manifold can become unstable for perturbations (such as noise
and/orparametermismatches)transversetothemanifold.This
iscalledabubblingbifurcation,whichleadstotheformationof
riddled basins of attraction in the invariant manifold inducing
intermittent bursting (see Ref. [5] for more details). There
exists another type of bifurcation, called blowout bifurcation,
induced by changes in the transverse stability of an infinite
number of UPOs. Among these UPOs, some are transversely
stableandothersaretransverselyunstablenearthebifurcation.
It is a well-accepted fact that on-off intermittency is a
common phenomenon which occurs in a wide variety of
natural systems, including neural networks [6,7], biological
systems [8], laser systems [9,10], electronic circuits [11,12],
complex networks [13], coupled chaotic systems [14], earth-
quake occurrence [15], and other physical systems such as
Hamiltonian systems and self-driven particle systems [16,17].
Specifically, it has been reported that the dynamics of clusters
inanetworkcanexhibitanextremeformofintermittency[18]:
A substantial percentage of synchronized nodes forms a giant
cluster most of the time, while many small clusters can also
occur at other times. Thus, the cluster sizes can vary in a
highly intermittent fashion as a function of time. Recently, it
has been shown [19] that the transition to intermittent chaotic
synchronization [in the case of complete synchronization
(CS)] for phase-coherent attractors (R¨ ossler attractors) occurs
immediately as soon as the coupling parameter is increased
from zero and, for non-phase-coherent attractors (Lorenz
attractors), the transition occurs slowly in the sense that it
occurs only when the coupling is sufficiently strong; this is
known as delayed transition.
It has been already shown that the transition from non-
synchronization to any type of synchronization is preceded
by intermittent synchronization in coupled chaotic systems.
For example, intermittent lag synchronization (ILS) [20],
intermittentphasesynchronization(IPS)[21,22],andintermit-
tent generalized synchronization (IGS) [23] are some of the
synchronization transitions characterized by the intermittent
behavior as a function of a coupling parameter. Recently, IGS
has been numerically observed in unidirectionally coupled
time-delay systems [24]. It has been found that the onset of
generalized synchronization is preceded by on-off intermit-
tency and the transition behavior differs for different coupling
schemes. In particular, the intermittent transition occurs in a
broad range of coupling strength for error feedback coupling
configuration and in a narrow range of coupling strength
for direct feedback coupling configuration beyond certain
threshold values of the coupling strength.
The transition between various types of synchronization
and their mechanism are not yet well understood, especially
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SURESH, SENTHILKUMAR, LAKSHMANAN, AND KURTHSPHYSICAL REVIEW E 86, 016212 (2012)
in time-delay systems. Further, the dynamics of a large
ensemble of coupled time-delay systems such as regular and
complex networks are not yet well studied and only very
few studies are available in the literature [25,26]. The study
of synchronization in ensembles of time-delay systems has
received attention recently in view of the infinite dimensional
nature and feasibility of the experimental realization of time-
delay systems. Particularly, considerable attention is being
paid to time-delay systems with instantaneous coupling due
to their extensive applications in different fields such as signal
and image processing, pattern recognition, chaotic neural
networks, secure communication, and cryptography [27–32].
In particular, In Refs. [29,30], it was demonstrated that, in
chaotic communication experiments, the time-delayed optical
fiber ring laser system is capable of transmitting encoded
signals with a speed of 1 Gb/s over a long distance fiber-optic
channel (≈120 Km).
Motivated by the above, we will investigate the synchro-
nization transitions in an array of coupled time-delay sys-
tems with different (instantaneous) coupling configurations.
Particularly, in this paper, we demonstrate that the transition
to CS occurs distinctly for different coupling configurations
in a regular array of coupled time-delay systems. In a
unidirectional array, the transition from nonsynchronization
to CS occurs locally (microscopically) in a narrow range
of coupling strength and, globally (macroscopically), the
systems synchronize one by one with the drive system as
a function of the coupling strength, which is known as
sequential synchronization. But in a mutually coupled array,
everyindividualsystemsynchronizesimmediatelyinanarrow
range (after a large threshold value) of the coupling strength
and so, globally, the synchronization transition is immediate
as a function of the coupling strength in contrast to sequential
synchronization. It is also to be noted that in the transition
regime we observe a type of synchronization behavior called
globalintermittentsynchronization(GIS)wherelongintervals
ofhigh-qualitysynchronizationareinterruptedbylargedesyn-
chronized chaotic bursts simultaneously in all the systems in
the array.
Tounderstandthetwodistincttransitionscenarios,wefocus
on the theory of unstable periodic orbits, which are the basic
building blocks of chaotic and hyperchaotic attractors. The
sequential and immediate synchronization transitions to CS
are characterized by calculating the probability of synchro-
nization and the average probability of synchronization as a
functionofthecouplingstrength.Theexistenceofintermittent
synchronization is corroborated by using a spatiotemporal
difference and a power-law behavior of the laminar phase
distributions.
The remaining paper is organized as follows: In Sec. II,
we will explain the occurrence of sequential synchronization
preceded by intermittent synchronization in an array of
unidirectionally coupled piecewise linear time-delay systems
and, in Sec. III, we consider a mutual coupling configuration
and explain the occurrence of instantaneous synchroniza-
tion transition in the array. Further, we demonstrate the
existence of GIS and provide a possible mechanism for
the occurrence of this synchronization transition to CS in
the array. Finally, we discuss our results and conclusion in
Sec. IV.
II. SYNCHRONIZATION IN A PIECEWISE LINEAR
TIME-DELAY SYSTEMS: LINEAR ARRAY WITH
UNIDIRECTIONAL COUPLING
We consider the following unidirectionally coupled time-
delay systems of the form
˙ x1= −αx1(t) + βf[x1(t − τ)],
˙ xi= −αxi(t) + βf[xi(t − τ)] + ε[xi−1(t) − xi(t)],
where i = 2,3,...,N. We choose an open end boundary
condition. α, β are system parameters, τ is the time-delay,
and ε is the strength of the coupling between the systems.
The nonlinear function f(x) is chosen to be a piecewise linear
functionwithathresholdnonlinearity,whichhasbeen studied
recently [33],
f(x) = AF∗− Bx.
Here
⎧
⎩
The system parameters for the piecewise linear system (1)–
(3) are fixed as follows: α = 1.0, β = 1.2, τ = 6.0, A = 5.2,
B = 3.5, and x∗is the threshold value fixed at x∗= 0.7. Note
that for this set of parameter values a single uncoupled system
exhibits a hyperchaotic attractor with three positive Lyapunov
exponents (LEs) (see Ref. [34]).
To demonstrate the nature of the dynamical transition to a
completesynchronizationregime,weconsideranarrayofN =
30 unidirectionally coupled identical piecewise linear time-
delay systems (1)–(3) (each system having different initial
conditions). Here, x1(t) acts as the drive and the remaining
systems [xi(t),i = 2,3,...,30] as the response systems. In
the absence of coupling [ε = 0.0 in Eq. (1)], all the systems
evolve independently according to their own dynamics. On
increasing the coupling strength, the system x1(t) starts to
drive the system x2(t). Consequentially, the system x3(t) starts
tofollowthedrivesystemx1(t)forlargervaluesofε andthisis
continueduptotheNthsystem.Hence,globalsynchronization
isachievedviasequentialsynchronizationofthesystemsinthe
arrayasafunctionofcouplingstrength.Tobemorespecific,on
(1a)
(1b)
(2)
F∗=
⎨
−x∗,
x,
x∗,
x < −x∗
−x∗? x ? x∗
x > x∗
.
(3)
FIG. 1. (Color online) The probability of synchronization ?i(ε)
as a function of the coupling strength ε. The system index i
illustrates the occurrence of sequential synchronization transition to
CS in unidirectionally coupled piecewise linear time-delay systems
[Eq. (1)]. Black indicates the absence of synchronization [?i(ε) =
0.0], whereas yellow (light gray) represents the occurrence of CS
[?i(ε) = 1.0].
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5
10
15
20
25
30
5 10 15 20 25 30
i
j
(c)
5
10
15
20
25
30
5 10 15 20 25 30
i
j
(a)
5
10
15
20
25
30
5 10 15 20 25 30
i
j
5
10
15
20
25
30
5 10 15 20 25 30
i
j
(b)
(d)
FIG. 2. (Color online) Snap shots of node vs. node plots indicat-
ing sequential synchronization in unidirectionally coupled piecewise
linear systems for different values of coupling strength. (a) ε = 0.4,
(b) ε = 0.7, (c) ε = 0.87, and (d) ε = 1.1.
increasing the coupling strength, ε, from zero, nearby systems
to the drive in the array synchronize sequentially with it, while
the faraway systems are still in their transition state. The
other desynchronized systems will synchronize sequentially
for further larger values of ε. The occurrence of sequential
phase synchronization in an array of unidirectionally coupled
time-delaysystemshasbeenshowninRef.[26]andsequential
desynchronization in a network of spiking neurons is reported
in Ref. [35] as a function of coupling strength, ε. We may also
note here a somewhat analogous situation occurs but now as
a function of time for a fixed coupling strength in an array
of unidirectionally coupled chaotically evolving systems in
Refs. [36,37].
Here, we find that, locally, the synchronization in the
array occurs immediately in a very narrow range of coupling
strength; globally, it occurs in a broader range of ε due to
sequential synchronization. To characterize these local and
global synchronization transitions, we have calculated the
probability of synchronization ?i(ε) (which is defined as
the fraction of time during which |x1(t) − xi(t)| < δ occurs,
where δ is a small but an arbitrary threshold) and the aver-
0
0.2
0.4
0.6
0.8
1
0 0.2 0.4 0.6 0.81 1.2
Φi(ε), Ψ(ε)
ε
i=15
i=20
i=30
Ψ(ε)
FIG. 3. (Color online) The probability of synchronization ?i(ε)
of selected systems (i = 15,20,30) and the average probability of
synchronization [?(ε)] in a unidirectionally coupled array [Eq. (1)]
as a function of the coupling strength ε.
age probability of synchronization [?(ε) =
Here the asynchronized state is characterized by ?i(ε) = 0,
CS by ?i(ε) = 1, and the transition region by intermediate
values less than unity.
To understand the dynamical organization of sequential
synchronization in the array [Eq. (1)], we have calculated the
probabilityofsynchronizationasafunctionofε andthesystem
index i, which is depicted in Fig. 1. In this figure, the black
indicatestheasynchronizedstate[?i(ε) = 0.0]andtheyellow
(light gray) corresponds to the complete synchronization
state [?i(ε) = 1.0], while intermediate colors represent the
transition region. From this figure one can clearly see the
occurrence of sequential synchronization as a function of ε
where the nearby systems to the drive get synchronized first
forlowervaluesofε,whereasthefarsystemsaresynchronized
at larger ε.
Sequential synchronization can also be visualized using
snapshots of the oscillators in the node versus node plots.
We regard the oscillators in the array as synchronized when
the probability of synchronization ?i(ε) > 0.96, which are
indicated by solid circles. Figure 2 shows node versus node
diagrams for various values of the coupling strength. For
ε = 0.4 none of the oscillators are synchronized with the
drive system [see Fig. 2(a)]. Figure 2(b) indicates that the
first seven oscillators are synchronized with the drive for
ε = 0.7. Further increase in the coupling strength results in
increase in the size of the synchronized cluster,resulting in the
formation of sequential synchronization. Figures 2(c) and 2(d)
are depicted for ε = 0.87 and 1.1, respectively, illustrating
sequential synchronization.
To discuss the nature of the synchronization transition
locally, we have calculated the probability of synchronization
1
N−1
?N
i=2?i(ε)].
0
1
2
10000
2
15000 20000
Δ x1,15
t
0
1
2
10000
2
15000 20000
Δ x1,20
t
0
1
2
10000
2
15000 20000
Δ x1,25
t
0
1
10000 15000 20000
Δ x5,20
t
0
1
10000 15000 20000
Δ x5,25
t
0
1
10000 15000 20000
Δ x5,30
t
(a)(b)(c)
(d)(e)(f)
FIG. 4. The difference between some selected piecewise linear time-delay systems [Eq. (1)] shows intermittent synchronization. (a) ?x1,15
for ε = 0.76, (b) ?x1,20for ε = 0.81, (c) ?x1,25for ε = 0.86, (d) ?x5,20for ε = 0.76, (e) ?x5,25for ε = 0.81, and (f) ?x5,30for ε = 0.86.
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0.001
0.01
0.001 0.01 0.1
Λ(t)
t
i=15
i=20
i=30
fit
FIG. 5. (Color online) The statistical distribution of the laminar
phase for the systems i = 15 for ε = 0.76, i = 20 for ε = 0.81,
and the system i = 30 for ε = 0.93, all satisfying −3
scaling.
2power-law
forsomeselectedsystems(i = 15,20,and30)inthearrayasa
function of ε (see Fig. 3). ?i(ε) of the system i = 15 is plotted
as a function of ε (represented by the solid squares). In the
range of ε ∈ (0,0.76), there is an absence of any entrainment
between the systems, resulting in an asynchronous behavior,
and ?15(ε) is practically zero in this region. However, starting
from the value ε = 0.76 and above, there appear some finite
values less than unity attributing to the transition regime.
Beyond ε = 0.78, ?15(ε) attains unit value indicating CS. We
have also plotted the probability of synchronization in Fig. 3
for two more selected systems, i = 20 and 30, represented by
theasterisksymbolandsolidtriangles,respectively,indicating
the immediate transition to CS locally. The system i = 20
attains CS at ε = 0.84 and the system i = 30 reaches the
CS state at ε = 0.98. From this figure, one can understand
the occurrence of sequential synchronization of the individual
systems (locally) in the array as a function of the coupling
strength. To explain the global (macroscopic) synchronization
phenomenon, we have calculated the average probability of
synchronization [?(ε)] of the N = 30 systems as a function
of ε and depicted it in Fig. 3 (represented by the solid circles).
It confirms sequential synchronization by gradual increase in
?(ε) as a function of ε (which indeed exactly matches with
Fig. 1).
Next, in the transition regime, we observe intermittent
synchronization in every individual system and this can be
0
0.5
10000
0.4
15000 20000
Δ X
t
0
0.2
10000 15000 20000
Δ X
t
(a)
(b)
FIG. 6. The average difference (?X) of all (N − 1) piecewise
linear time-delay systems in the array [Eq. (1)] with the drive x1
shows an intermittent synchronization transition. (a) ε = 0.85 and
(b) ε = 0.89.
FIG. 7. (Color online) The probability of synchronization ?i(ε)
as a function of ε and the system index i illustrating the occurrence
of instantaneous synchronization transition to CS both locally and
globally in mutually coupled piecewise linear time-delay systems
[Eq. (4)].
characterized qualitatively by a difference in the magnitudes
of the states between the systems (?x1,i= |x1− xi|), for
selected ones (i = 15, 20, and 25). Figures 4(a)–4(c) shows
intermittent synchronization in the above-mentioned systems
forε = 0.76,0.81,and0.86,respectively.Wealsofindthatthe
synchronization quality in the transition region depends on the
respective positions of the response systems from the drive, as
well as on the distance between the two units in the system
and the coupling strength. We have additionally plotted the
difference between the systems ?5,20, ?5,25, and ?5,30for the
above set of values of coupling strength [Figs. 4(d)–4(f)] to
demonstrate the above features.
The statistical features associated with the intermittent
dynamics is also analyzed by the distribution of the laminar
phases ?(t) with amplitudes less than a threshold value of ?
(here we have choosen ? = |x1(t) − xi(t)| = |0.001|,
15,20,and25).Auniversalasymptoticpower-lawdistribution
?(t) ∝ tαis observed for the above threshold value of ?
with the exponent α = −1.5. Figure 5 shows the laminar
phase distribution of the above selected systems. The solid
circles represent a laminar distribution of the system i = 15
for ε = 0.76, the solid triangles correspond to the laminar
distribution of the system i = 20 for ε = 0.81, and the solid
squares represent a laminar distribution of the system i = 30
for ε = 0.93 which clearly display the −3
a typical characterization of on-off intermittency. It should be
noted that this result does not change for a large range of ?.
i =
2power-law scaling,
5
10
15
20
25
30
5 10 15 20 25 30
i
j
(a)
5
10
15
20
25
30
5 10 15 20 25 30
i
j
(b)
FIG. 8. (Color online) Snapshots of node vs. node plots of
mutually coupled piecewise linear systems indicating instantaneous
synchronization. (a) ε = 58.0; (b) ε = 61.0.
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PHYSICAL REVIEW E 86, 016212 (2012)
Tounderstandthephenomenonofintermittentsynchroniza-
tion transition globally in the whole array, we have calculated
the average difference (?X =
(N − 1) systems in the array with the drive x1. Figures 6(a)
and6(b)showtheaveragedifferenceforthecouplingstrengths
ε = 0.85 and 0.89, respectively.
Thereasonbehindthesequentialsynchronizationtransition
is in accordance with the sequential stabilization of all the
unstable periodic orbits of the response systems in the array
as a function of the coupling strength. It is a well-established
fact that a chaotic/hyperchaotic attractor contains an infinite
number of UPOs of all periods. Synchronization between the
coupled systems is said to be stable if all the UPOs of the
response systems are stabilized in the transverse direction to
the synchronization manifold. Consequently, all the trajecto-
ries transverse to the synchronization manifold converge to
it for suitable values of ε. For sequential synchronization, the
UPOsinthecomplexsynchronizationmanifoldoftheresponse
systems near to the drive are stabilized, first, for appropriate
threshold values of the coupling strength ε as it is increased,
while the UPOs of the far systems remain unstable for these
values of ε. Once the coupling is increased further, the UPOs
of the far systems are gradually stabilized as a function of the
coupling strength. Unfortunately, methods for locating UPOs
have not been well established for time-delay systems, which
has hampered a qualitative proof for the gradual stabilization
of UPOs by locating them.
1
N−1
?N
j=2|x1− xj|) of the
III. SYNCHRONIZATION IN A PIECEWISE LINEAR
TIME-DELAY SYSTEMS: LINEAR ARRAY WITH
BIDIRECTIONAL COUPLING
In this section, we consider an array of mutually coupled
(bidirectional coupling) piecewise linear time-delay systems
with identical subunits. The dynamical equation then becomes
˙ xi= −αxi(t) + βf[xi(t − τ)]
+ε[xi+1(t) − 2xi(t) + xi−1(t)],
where i = 1,2,...,N. We choose open end boundary con-
ditions. The parameter values are the same as those in
Sec. II. The nonlinear function f(x) is chosen as in Eqs. (2)
and (3). In the mutual coupling case there are no drive and/or
response systems where each and every oscillator shares the
signals mutually with its two nearest neighbors. Therefore, the
synchronization transition is instantaneous due to the mutual
sharing of the signals and one needs a very large value of
(4)
0
0.2
0.4
0.6
0.8
1
0 20 40 60 80
Φi(ε), Ψ(ε)
ε
i=15
i=20
i=30
Ψ(ε)
FIG. 9. (Color online) The probability of synchronization ?i(ε)
of selected systems (i = 15, 20, and 30) and the average probability
of synchronization [?(ε)] in the mutually coupled array [Eq. (4)] as
a function of ε.
ε to attain CS. In the transition regime, we have observed
an intermittent synchronization transition in all the systems
simultaneously in the array.
Wehavecalculatedtheprobabilityofsynchronizationofall
the N = 30 systems in the array as a function of the coupling
strength ε and the system index i (see Fig. 7). In this figure the
black represents the desynchronized state [?(ε) = 0.0] and
CS is represented by the yellow (light gray) [?(ε) = 1.0]. The
transitionregimeisindicatedbyintermediatecolors.Fromthis
figure one can clearly see that locally every individual system
requires large values of ε to attain CS and globally all the
systems synchronize immediately for the same value of ε.
In the mutually coupled array, all the systems get syn-
chronized immediately in a narrow range of the coupling
strength, in contrast to sequential synchronization. Figure 8(a)
is plotted for ε = 58.0, where none of the oscillators in the
array are synchronized, whereas for ε = 61.0, all the systems
are completely synchronized as depicted in Fig. 8(b).
To characterize the nature of synchronization transitions to
CS both locally and globally, we again use the probability of
synchronization ?(ε) and the average probability of synchro-
nization ?i(ε), respectively. In Fig. 9, we have plotted ?i(ε)
for some selected piecewise linear systems (i = 15,20,30) as
a function of ε. For instance, we have illustrated ?i(ε) for
the system i = 15 in Fig. 9 (represented by the solid squares).
From this figure, one can observe that in the range of ε ∈
(0,50) there is an absence of any entertainment between the
systems resulting in asynchronous behavior and ?15(ε) is low
[?15(ε) < 0.4]. However, for ε > 50 there appear oscillations
in ?15(ε) in the range of ε ∈ (50,60) exhibiting intermittent
transition.Beyondε = 60.0,?15(ε) = 1indicatingperfectCS
of the system i = 15. We have also calculated ?(ε) for the
FIG. 10. (Color online) The spatiotemporal difference [?x(t)] of the mutually coupled piecewise linear systems for various values of
coupling strengths. (a) ε = 0.0, (b) ε = 50.2, (c) ε = 54.6, (d) ε = 61.0. Here the black indicates that the difference is zero and the red and
green (dark and light gray) indicate the bursting amplitudes.
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0
0.5
1
10000 15000 20000
Δ x1,20
t
0
0.5
10000
1
15000 20000
Δ x1,20
t
0
0.5
1
10000 15000 20000
Δ x1,20
t
0
0.5
1
10000 15000 20000
Δ x1,30
t
0
0.5
10000 15000 20000
Δ x1,30
t
0
0.5
1
10000 15000 20000
Δ x1,30
t
0
0.5
1
10000
1
15000 20000
Δ x1,20
t
0
0.5
10000 15000 20000
Δ x1,30
t
(a)
(b)
(c) (d)
(e)
(g)
(h)
(f)
FIG. 11. [(a)–(d)] The difference between systems 1 and 20 [?x1,20= |x1(t) − x20(t)|] for ε = 50.2, 54.6, 58.1, and 61.0. [(e)–(h)] The
difference between systems 1 and 30 [?x1,30= |x1(t) − x30(t)|] is plotted for the same set of coupling strength values given above.
systems i = 20 and i = 30, represented by asterisk symbols
andsolidtriangles,respectively,whichshowsimilartransitions
to CS almost at the same value of ε. We have also confirmed
a similar immediate transition to CS in all the systems in the
array (see Fig. 7).
To examplify the global synchronization phenomenon, we
have calculated the average probability of synchronization
[?(?)] of N = 30 systems as a function of the coupling
strength as shown in Fig. 9 by the solid circles. In the range of
ε ∈ (0,54), there is an absence of any synchronization and so
?(?)ishavingzeroorlowvalues[?(?) < 0.3].Intherangeof
ε ∈ (54,59), ?(?) is characterized by some finite values less
than unity, and beyond ε > 59.0 there is a sudden jump to the
valueof?(?) = 1.0,corroboratingthefactthatallthesystems
synchronize immediately at the same value of the coupling
strength attributed to the occurrence of global CS. Further,
in the transition region we have found long time intervals of
high-quality synchronization, which is interrupted at irregular
time intervals by intermittent chaotic bursts simultaneously in
all the systems in the array we call GIS.
To demonstrate the existence of GIS, we have calculated
the spatiotemporal difference [?x(t) = |x1(t) − xi(t)|,
2,3,...,30] of the array as a function of time and the
oscillator index i as in Fig. 10 for different values of ε.
Here the black indicates zero difference [?x(t) = 0.0] and
the red and green (dark and light gray) indicate bursting
amplitudes. In the absence of coupling (ε = 0.0), the systems
are evolving independently and so there is no correlation
between the systems as shown in Fig. 10(a). If we increase
thecoupling,weobserveseveralintermittentburstsalongwith
the synchronization as depicted in Figs. 10(b) and 10(c) for
ε = 50.2 and 54.6, respectively. From these figures, one can
clearly see the occurrence of aperiodic intermittent chaotic
bursts along with the synchronized regions simultaneously in
allthesystemsinthearray.Beyondε > 60onecanobserveCS
i =
0.0001
0.001
0.01
0.1
0.001 0.01 0.1
Λ(t)
t
(a)
ε=54.6
ε=58.1
fit
0.0001
0.001
0.01
0.1
0.001 0.01 0.1
Λ(t)
t
(b)
ε=54.6
ε=58.1
FIG. 12. (Color online) The statistical distribution of the laminar
phase for selected piecewise linear systems [Eq. (4)] (a) i = 20 and
(b)i = 30satisfyinga−3
ε = 54.6 and 58.1.
2power-lawscalingforthecouplingstrength
as illustratedin Fig. 10(d) where the spatiotemporal difference
of the systems is exactly zero for ε = 61.0.
ToelaboratetheoccurrenceofGISinthearraymoreclearly,
wehavecalculatedthedifferencebetweenthesystemscoupled
in the array and plotted for some selected systems (i = 20 and
30)fordifferentvaluesofthecouplingstrength.Thedifference
between the systems 1 and 20, ?x1,20(t) = |x1(t) − x20(t)|,
is plotted for ε = 50.2, 54.6, and 58.1 in Figs. 11(a)–11(c),
respectively, which clearly displays the existence of aperiodic
intermittent bursts along with the synchronized regions. We
havealsoplottedthedifference?x1,30(t) = |x1(t) − x30(t)|for
the same values of ε as shown in Figs. 11(e)–11(g). It is to be
noted that in both systems (i = 20,30) the intermittent bursts
simultaneouslyoccuratthesametimeandthisoccursinallthe
other systems connected in the array, confirming the existence
of GIS. In Figs. 11(d) and 11(h) the difference between
the systems completely vanishes for ε = 61.0, indicating the
occurrence of CS. We have plotted the above figures with 104
time units after leaving a sufficient number of transients.
Further statistical features associated with the intermittent
dynamics of the entire array are also analyzed by calculating
the distribution of the laminar phases ?(t), which is shown
in Figs. 12(a) and 12(b) for selected systems i = 20 and 30,
respectively, for ε = 54.6 and 58.1 which clearly display the
−3
The reason for the occurrence of GIS can be explained as
follows: As we have already explained, a chaotic attractor can
be considered as a pool of infinitely many UPOs of allperiods.
Synchronization between the systems are asymptotically sta-
bleifalltheUPOsofthesystemsarestabilizedinthetransverse
direction to the synchronization manifold. Consequently, all
the trajectories transverse to the synchronization manifold
converge to it for suitable values of the coupling strength
and this is reflected in the stabilization of the UPOs on
synchronization. From our results, we find that the UPOs
of the systems are stabilized in the complex synchronization
manifold only for a very large value of coupling strength
after a certain threshold value. It is also to be noted that the
intermittency transition in the case of a bidirectional coupling
configurationisduetothefactthatthestrengthofthecoupling
ε contributes only less significantly to stabilize the UPOs as
the error in the coupling term in Eq. (4) gradually becomes
smaller from the transition regime after a certain threshold
value of the coupling strength.
2power-law scaling to confirm the on-off intermittency.
IV. CONCLUSION
In conclusion, we have shown the existence of sequential
and instantaneous synchronization transitions in an array of
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TRANSITION TO COMPLETE SYNCHRONIZATION AND ...
PHYSICAL REVIEW E 86, 016212 (2012)
time-delay systems with different coupling configurations. If
the systems are coupled with unidirectional configuration,
we have observed an immediate synchronization transition
to CS microscopically and, if we consider the macroscopic
synchronization behavior of the entire array, we find that
the transition region is gradually increasing as a function
of ε due to sequential synchronization, which is verified by
the probability of synchronization and average probability of
synchronization. In the transition regime we have observed
the existence of intermittent synchronization. On the other
hand, if we consider an array of mutually coupled time-
delay systems, every individual system (microscopically)
synchronizes immediately for a very large value of ε and,
globally (macroscopically), the synchronization transition
occurs immediately in the whole array. In the transition
region, a type of synchronization, called GIS, occurs, which is
characterizedbylongintervalsofhigh-qualitysynchronization
interrupted at irregular times by intermittent chaotic bursts
simultaneously in all the systems.
The reason (mechanism) for these two distinct transition
scenarios is explained based on unstable periodic orbit theory.
The GIS is confirmed using the spatiotemporal difference
and a power-law behavior of the laminar length distributions
with −3
carried out in a well-known piecewise linear time-delay
system. We have also confirmed the occurrence of the above
resultsforanotherwellknowntime-delaysystems,namelythe
Mackey-Glass system [38], with an array length of N = 50
and we observe the same kind of sequential and instantaneous
synchronizationtransitionsprecededbyGISforunidirectional
and bidirectional coupling configurations, respectively.
2power-law scaling. The above studies have been
ACKNOWLEDGMENTS
The work of R.S. and M.L. is supported by a Department
of Science and Technology (DST), IRHPA, research project.
M.L.isalsosupportedbyaDepartmentofAtomicEnergyRaja
Ramanna fellowship and a DST Ramanna program. D.V.S.
and J.K. acknowledge support from the EU under Project No.
240763 PHOCUS(FP7-ICT-2009-C) and J.K. acknowledges
support from IRTG 1740(DFG).
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