Delayenhanced coherent chaotic oscillations in networks with large disorders.
ABSTRACT We study the effect of coupling delay in a regular network with a ring topology and in a more complex network with an alltoall (global) topology in the presence of impurities (disorder). We find that the coupling delay is capable of inducing phasecoherent chaotic oscillations in both types of networks, thereby enhancing the spatiotemporal complexity even in the presence of 50% of symmetric disorders of both fixed and random types. Furthermore, the coupling delay increases the robustness of the networks up to 70% of disorders, thereby preventing the network from acquiring periodic oscillations to foster disorderinduced spatiotemporal order. We also confirm the enhancement of coherent chaotic oscillations using snapshots of the phases and values of the associated Kuramoto order parameter. We also explain a possible mechanism for the phenomenon of delayinduced coherent chaotic oscillations despite the presence of large disorders and discuss its applications.
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 Springer Verlag, Berlin.
 Dynamics of Nonlinear TimeDelay Systems: , Springer Series in Synergetics. ISBN 9783642149375. SpringerVerlag Berlin Heidelberg, 2010. 01/2010;

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PHYSICAL REVIEW E 84, 066206 (2011)
Delayenhanced coherent chaotic oscillations in networks with large disorders
D. V. Senthilkumar,1R. Suresh,2Jane H. Sheeba,2M. Lakshmanan,2and J. Kurths1,3
1Potsdam Institute for Climate Impact Research, DE14473 Potsdam, Germany
2Centre for Nonlinear Dynamics, Department of Physics, Bharathidasan University, Tiruchirapalli 620 024, India
3Institute of Physics, Humboldt University, DE12489 Berlin, Germany
(Received 22 June 2011; revised manuscript received 24 October 2011; published 19 December 2011)
We study the effect of coupling delay in a regular network with a ring topology and in a more complex
network with an alltoall (global) topology in the presence of impurities (disorder). We find that the coupling
delay is capable of inducing phasecoherent chaotic oscillations in both types of networks, thereby enhancing
the spatiotemporal complexity even in the presence of 50% of symmetric disorders of both fixed and random
types. Furthermore, the coupling delay increases the robustness of the networks up to 70% of disorders, thereby
preventing the network from acquiring periodic oscillations to foster disorderinduced spatiotemporal order.
We also confirm the enhancement of coherent chaotic oscillations using snapshots of the phases and values
of the associated Kuramoto order parameter. We also explain a possible mechanism for the phenomenon of
delayinduced coherent chaotic oscillations despite the presence of large disorders and discuss its applications.
DOI: 10.1103/PhysRevE.84.066206PACS number(s): 05.45.Xt, 05.45.Pq, 05.45.Jn, 05.45.Gg
I. INTRODUCTION
Inrecenttimes,researchershavebeeninterestedinstudying
networks of oscillators with timedelayed coupling because
of their wide applications in science [1–4], engineering, and
technology [5–7]. Considering the fact that in most realistic
physical and biological systems [8–10] the interaction signal
propagates through media with limited speed, its finite
signal propagation time induces a timedelay in the received
signal [11–13]. For example, in biological neural networks, it
has been shown that the neural connections are full of variable
loops such that the propagation of signal through the loops
can result in a large timedelay (synaptic delay), and it is also
reported that the axons can generate timedelay up to 300 ms
[12]. A typical nonlinear timedelay system is a veritable
black box [13] and that the delay coupling itself gives rise to a
plethoraofnovelphenomena,suchasdelayinducedamplitude
death [7], phaseflip bifurcation [14], synchronizations of
differenttypes[15],multistability[16],chimerastates[17,18],
etc. in coupled nonlinear oscillator systems.
In this paper we consider a network of forced and damped
nonlinear pendula studied earlier by Braiman et al. [19],
commonly known as the forced FrenkalKontorova model. It
represents a straightforward physical realization of an array of
diffusively coupled Josephson junctions [20,21] in which the
applied current in each junction is modulated by a common
frequency.Thepossibilityofobtainingsynchronizedmotionin
one and twodimensional chaotic arrays of such systems has
been investigated in Refs. [22–25], where the complex chaotic
behaviorofthecollectivesystemswascompletelytamedwhen
acertainamountofimpurities(disorder)wasintroduced.Tobe
specific, disorder enhanced spatiotemporal regularity [19,26],
disorder enhanced synchronization [27,28], and taming chaos
with disorder [23–25] in such systems have received central
importance in recent research on complex systems and their
applications.
In our present studies we study a regular network with a
ring topology and a more complex network with an alltoall
(global) topology with different densities (sizes) of impurities
(disorder)andexaminetheeffectoftimedelayinthecoupling.
In particular, the oscillations of each pendulum affects the
oscillations of the pendula to which it is connected to, after
some finite timedelay τ. In such a configuration we are
interestedininvestigatingthepossibilityofachievingcoherent
chaotic dynamics in the network despite the presence of a
substantial amount of disorder and, thereby, enhancing the
spatiotemporal complexity, a counterintuitive result to the
expected (reported) outcome of taming chaos and enhancing
spatiotemporal order with even a negligible size of disorder
in the network (in the absence of coupling delay). Here
by coherent chaotic dynamics, we mean the emergence of
collective (phasecoherent or phasesynchronized) chaotic
oscillations (but not complete synchronization) in the entire
network despite the presence of disorder [29]. The delay
enhancedphasecoherentchaoticoscillationsarecharacterized
both qualitatively and quantitatively using snapshots of the
phases of the pendula in the networks and the Kuramoto
order parameter [30]. Recently, similar coherent states have
been observed in BoseEinstein condensates on tilted lattices
for strong field showing highly organized patterns, often
denoted as quantum carpets [31]. Enhancing spatiotemporal
complexity or at least preserving the original spatiotemporal
pattern in the midst of a noisy environment due to the
presence of disorder is crucial for applications, such as
spatiotemporal and/or secure communication [32] in spatially
extended systems, especially in biology and physiology [33],
in the state of art of modern computing, namely liquid state
machines (LSM), in which the degree of spatiotemporal
complexity of the network of dynamical systems determines
the highest degree of computational performance (i.e, mixing
property) [34], etc.
In particular, we will show that timedelay in the coupling
inducescoherentchaoticoscillationsofthenetworkofcoupled
systems, in both diffusively coupled pendula with periodic
boundary conditions and in a globally coupled network,
thereby enhancing the spatiotemporal complexity despite
the presence of a large number of disorders, even up to
half the size of the network. Furthermore, coupling delay
enhances the robustness of the network against disorders of
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D. V. SENTHILKUMAR et al.
PHYSICAL REVIEW E 84, 066206 (2011)
size greater than 50% of the network, thereby preserving
the original dynamical states of the network and preventing
disorderenhanced synchronous periodic oscillations of the
entire network leading to spatiotemporal order. It is to be
noted that in an array without delay even the presence of a
verysmallperiodicdisorderitselfiscapableofsuppressingthe
chaotic oscillations of the entire network and thereby inducing
spatiotemporal regularity as demonstrated in Refs. [23–25].
We will also explain an appropriate mechanism for delay
induced coherent chaotic oscillations leading to enhanced
spatiotemporal complexity based on a mechanism for taming
chaoticity (in the absence of delay), as reported in Ref. [19].
A relevant study focusing on macroscopic properties of the
globallyconnectedheterogeneousneuralnetworkhasrevealed
similar irregular collective behavior [35].
The paper is organized as follows. In Sec. II we will briefly
discuss the existing results on taming chaoticity leading to
spatiotemporal regularity without any delay coupling for a
linear array of nonlinear pendula with periodic boundary
conditions, which will be useful for a later comparison. We
willdemonstrateourresultsondelayinducedcoherentchaotic
oscillations despite the presence of large disorders, even up to
70%, in Sec. III. Similar results are presented in a network of
globallycoupledpendulabothwithandwithoutdelaycoupling
in Sec. IV. Finally, in Sec. V, we discuss our results and
conclusions.
II. LINEAR ARRAY OF PENDULA IN THE ABSENCE
OF COUPLING DELAY
WeconsiderachainofN forcedcouplednonlinearpendula
whose equation of motion can be written as [19,23–28]
ml2˙ xi= yi,
˙ yi= −γyi− mgl sinxi+ f + f?
+C[yi+1(t) − 2yi(t) + yi−1(t)],
where i = 1, 2,...,N. We choose the following periodic
boundaryconditions:x0= xNandxN+1= x1.Theparameters
are taken as follows: the mass of the bob m = 1.0, the
damping γ = 0.5, acceleration due to the gravity g = 1.0,
dc torque f = 0.5, angular frequency ω = 0.67, pendulum
length l = 1.0, f?
strength. The schematic diagram of the coupling configuration
is shown in Fig. 1, in which the first pendulum is connected
with the second and the Nth pendulum so that each pendulum
gets two inputs, without any delay, from its nearest pendula.
For the coupling strength C = 0.0, the pendula are uncoupled
and evolve independently.
In earlier studies [19,23–28], the authors have dealt with
an array of pendula with diffusive coupling but without delay
and have shown that the chaotic dynamics of the array is
controlled by the inclusion of impurities, which are disorders
in their natural frequencies and/or distributed initial phases
of the external forces. In particular, in Refs. [23–25], the
authorshaveconsideredachainofdiffusivelycoupledpendula
without delay and have shown that inclusion of even a single
periodic impurity is enough to tame chaos in a long chain of
length with N = 512. However, we would like to point out
that we are not able obtain the results with a single impurity
(1a)
isin(ωt)
(1b)
i= f?is the ac torque, and C is the coupling
1
2
3
4
5
N1
N
FIG. 1. (Color online) The schematic diagram of the array of
pendula with periodic boundary conditions.
as reported by these authors. Nevertheless, taming chaos and
achievingspatiotemporalregularitycanbeobtainedfor20%of
impurities for appropriate coupling strength for different sizes
of the array as reported by other authors [19,26–28]. In the
following, we will briefly illustrate the results of taming chaos
and achieving spatiotemporal regularity in an array of N = 50
coupled pendula Eq. (1) with ring configuration without any
coupling delay to appreciate the effect of delay coupling in the
following sections. The results have been confirmed for the
case of N = 512 too.
We introduce disorder in the network of chaotic pendula by
allowingoneormorependulatooscillateperiodicallyasinthe
earlierreports[19,26–28].Inordertofixtheparameters(ofthe
pendula) corresponding to the chaotic and periodic regions,
we start our analysis by plotting the bifurcation diagram
of a single pendulum as a function of the ac torque in the
range f?∈ (0,2) for fixed values of the other parameters. The
bifurcation diagram and its corresponding largest Lyapunov
exponent is depicted in Fig. 2(a), which exhibit a typical
bifurcationscenarioleadingtochaoticbehaviorforappropriate
values of the ac torque. To elucidate the dynamical behavior
of the ring of N coupled pendula as a function of a parameter,
we have explored an array of N = 3 pendula with periodic
boundary conditions in plotting the bifurcation diagram,
because each pendulum in an array of arbitrary length N > 2
iscoupledwithitsnearestneighborsandsoeachofthependula
effectively gets two inputs from its neighbors. Therefore
the basic configuration of N = 3 pendula is sufficient to
explain the bifurcation pattern of N coupled pendula in a ring
configuration for same values of the parameters. Indeed, we
have confirmed that the bifurcation diagram remains the same
irrespective of the value of N for the same set of parameter
values as in Fig. 2. The bifurcation diagram of a single
pendulum in a ring of N = 3 coupled pendula and the largest
Lyapunov exponent of the entire network for the value of the
coupling strength C = 0.5 in the same range of f?is depicted
in Fig. 2(b). The bifurcation scenario of each pendulum in
a ring of N = 3 coupled pendula is almost identical to that
of a single uncoupled pendulum [Fig. 2(a)] and the network
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1.5
0.5
0.5
1.5
0 0.5 1
f’
1.5 2
y
(a)
1.5
0.5
0.5
1.5
0 0.5 1
f’
1.5 2
y
(b)
1.5
0.5
0.5
1.5
0 0.5 1
f’
1.5 2
y
(c)
1.5
0.5
0.5
1.5
0 0.5 1
f’
1.5 2
y
(d)
1.6
FIG. 2. (Coloronline)Bifurcationdiagramsofasinglependulum
in a ring of three coupled pendula and the largest Lyapunov exponent
(of a single pendulum for C = 0.0 and that of the entire network
for C > 0.0) for different values of the coupling strength C and
the coupling delay τ. (a) C = 0.0 and τ = 0.0, (b) C = 0.5 and
τ = 0.0, (c) C = 0.5 and τ = 1.5, and (d) C = 0.6 and τ = 2.0
(inset shows that the pendulum exhibits chaotic oscillations for
f?
exponentandlightblue(lightgray)dotscorrespondtothebifurcation
diagram. Note that in all the cases (b)–(d) the three pendula are in
a completely synchronized state and, hence, the largest Lyapunov
exponent corresponds to the synchronization manifold.
d= 1.6). Red (dark gray) line corresponds to the largest Lyapunov
(ring of N = 3 coupled pendula) as a whole exhibits a positive
largest Lyapunov exponent as shown in Fig. 2(b).
It is to be noted that the network of diffusively coupled
(N = 3) pendula is already in a synchronized state for the
chosen value of coupling strength C = 0.5. Consequently,
following a reasoning similar to what reported in Ref. [36] for
a system of two coupled chaotic oscillators, one gets that the
synchronization manifold has only a single positive Lyapunov
exponent for appropriate values of f?. The synchronization
manifold in this case is almost similar to the phase space of
a single system [Fig. 2(a)] as is evident from the bifurcation
diagram [Fig. 2(b)]. Hence the network as a whole exhibits a
single positive Lyapunov exponent for C = 0.5. More details
on synchronization manifold and its relation to the transition
of Lyapunov exponents of diffusively coupled systems can be
found in Ref. [36].
Poincar´ e (surface of section) points corresponding to the
network of N = 50 pendula in a ring configuration Eq. (1) for
the coupling strength C = 0.5 is depicted in Fig. 3. The entire
network of pendula exhibits coherent chaotic oscillations in
the absence of any periodically oscillating disorder as shown
FIG. 4. (Color online) Spatiotemporal representation of Fig. 3.
Here the horizontal axis corresponds to time t and the vertical axis to
the oscillator index N.
in Fig. 3(a) for the ac torque f?= 1.5. The spatiotemporal
representation of Fig. 3(a) is illustrated in Fig. 4(a), where the
horizontal axis corresponds to time t and the vertical axis to
the oscillator index N, which is plotted for 10 drive cycles
after leaving out sufficient transients (1000 drive cycles). It
is to be noted that the network of N = 50 coupled pendula
does not exhibit synchronous chaotic oscillations as is evident
from Fig. 3(a). Otherwise it would show identical color for
all the oscillators as a function of time. The colors code the
angularvelocitiesofthependula;darkred(darkgray)indicates
negative velocities and green (light gray) positive velocities.
Narrow bands of red and green colors represent sudden rapid
motion of the pendula in the array. The spatiotemporal plot
[Fig. 4(a)] shows that the evolution is not only nonperiodic
but is in fact chaotic without any repetitive patterns or regular
structures.
Next, impurities (disorders) with periodic oscillations are
symmetrically distributed in the array to investigate the effect
of disorder as in the earlier studies [19,26–28]. It is to be
noted that an asymmetric distribution of disorder does not
tame the array, thereby fostering synchronous evolution and
spatiotemporalregularityasdiscussedinRef.[19].Thedensity
of the disorder is increased from 1% along with the coupling
strength C. We find that for C = 0.5 the entire array gets
locked to a synchronous periodic evolution [Fig. 3(b)] for
20% impurities with their corresponding f?
d= 0.5 (so that the
1.5
0
1.5
0 25
N
50
y
(a)
1.5
1
0.5
0
0.5
0 25
N
50
y
(b)
1.5
1
0.5
0
0.5
0 25
N
50
y
(c)
FIG. 3. Poincar´ e points corresponding to the network of pendula in a ring configuration with N = 50 for the coupling strength C = 0.5 in
the absence of coupling delay. (a) Chaotically oscillating pendula for f?= 1.5 when no disorder is present, (b) periodically oscillating pendula
for 20% disorders with fixed f?= f?
d= 0.5, and (c) periodically oscillating pendula for 20% disorders with random f?
d∈ (0.1,0.5).
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D. V. SENTHILKUMAR et al.
PHYSICAL REVIEW E 84, 066206 (2011)
impurities oscillate periodically), leading to spatiotemporal
regularity [Fig. 4(b)]. Hereafter, we denote the ac torque
corresponding to chaotic states as f?and that corresponding
to (disordered) periodic states as f?
plot indicates repetitive patterns for every two drive cycles
confirming the periodic evolution of the array of pendula.
To be precise, for 20% disorder in the network of N = 50
coupled pendula, 10 disorders are placed at every fifth site in
the network. The disorderinduced spatial synchronized states
reported here are exactly for the same value of C and the
density of the disorders but for different sizes of the array
as reported in Refs. [19,26–28]. Furthermore, we find that
even for a random distribution of f?
can be achieved in a wide range of ac torque. A periodically
oscillatingarray of pendula for 20%of disorder isobtained for
a random distribution of f?
along with their spatiotemporal representation in Fig. 4(c).
In the following, we will demonstrate that the introduction
of coupling delay can sustain and enhance coherent chaotic
oscillations in the linear array with periodic boundary con
ditions with the density of disorder as large as 50% for the
same parameter values. For 50% disorder in the network of
N = 50 coupled pendula, 25 disorders are placed at every
alternate sites in the network. Furthermore, the coupling
delay increases the robustness of the network by preserving
the dynamical complexity of the given network for further
increase in the density of disorder to as large as 70% of the
size of the network. The array attains synchronous periodic
behavior for disorder greater than 70%. For 70% disorders
in the network of N = 50 coupled pendula, after placing 25
disorders at every alternate sites in the network, the remaining
10 disorders are placed anywhere either symmetrically or
asymmetrically.
d. The spatiotemporal
dof the disorders taming
d∈ (0.1,0.5) as shown in Fig. 3(c)
III. LINEAR ARRAY WITH COUPLING DELAY
A. Effect of timedelay
Now we consider a chain of N forced coupled nonlinear
pendula with periodic boundary conditions along with the
same parameter values as in Sec. II but with the introduction
of coupling delay. The dynamical equations then become
ml2˙ xi= yi,
˙ yi= −γyi− mgl sinxi+ f + f?
+C[yi+1(t − τ) − 2yi(t) + yi−1(t − τ)],
where i = 1, 2,...,N and τ is the coupling delay. Now, the
first pendulum (see Fig. 1) is connected with the second
and with the Nth pendulum with a delay τ, so that each
pendulum gets two delayed inputs from its nearest pendula.
Similar delayed couplings are effective for the other pendula
in the array. For C = 0.0, the pendula are uncoupled and
evolve according to their own dynamics as before. As
the coupling delay will change the bifurcation scenario of
the coupled pendula as a function of the ac torque, we have
to look at the bifurcation diagrams to fix the values of the
strength of the ac torque f?in the periodic and chaotic
regimes. The bifurcation scenario of a single pendulum in a
ringofN = 3delaycoupledpendulaandthelargestLyapunov
exponent of the entire network for the value of the coupling
delay τ = 1.5 and for C = 0.5 is depicted in Fig. 2(c). This
network exhibits only a single positive Lyapunov exponent
for the chosen value of delay τ = 1.5, as the network of
diffusivelycoupledsubsystemsaresynchronizedtoacommon
synchronization manifold as discussed in Sec. II. Figure 2(d)
shows the bifurcation diagram and its corresponding largest
Lyapunov exponent for τ = 2.0 and C = 0.6.
The bifurcation diagram [Fig. 2(c)] for τ = 1.5 and C =
0.5 shares some common regimes of chaotic behavior in f?
with its corresponding undelayed case [Fig. 2(b)]. Therefore,
we fix f?= 1.5 for the chaotic pendula and f?= f?
for disorder characterized by periodic behavior as in Sec. II.
The Poincar´ e points as a function of the oscillator index (N)
after leaving out a sufficiently large number of (1000 drive
cycles)transientsinthepresenceofthecouplingdelayτ = 1.5
and for C = 0.5 are shown in Fig. 5 for different values
of density of disorder. The first row is plotted for disorders
with fixed f?
of f?
depicted in Fig. 6 for 10 drive cycles. Disorder of size 20%
(2a)
isin(ωt)
(2b)
d= 0.5
dand the second one for a random distribution
d∈ (0.2,0.9). The corresponding spatiotemporal plot is
1.5
0
1.5
0 25
N
50
y
(a)
1.5
0
1.5
0 25
N
50
y
(b)
1.5
0
1.5
0 25
N
50
y
(c)
1.5
0
1.5
0 25
N
50
y
(d)
1.5
0
1.5
0 25
N
50
y
(e)
1.5
0
1.5
0 25
N
50
y
(f)
FIG. 5. Same as in Fig. 3 but now in the presence of coupling delay τ = 1.5 and for different densities of disorders. First row is with
fixed value of the disorders f?
disorders.
d= 0.5 and the second row with random values of f?
d∈ (0.2,0.9). (a), (d) 20%, (b), (e) 50%, and (c), (f) 70% of
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FIG. 6. (Color online) Spatiotemporal representation of Fig. 5. Color bar is the same as in Fig. 4.
are uniformly distributed in the array of N = 50 pendula, as
in Fig. 3(b) (where the array acquired synchronous periodic
oscillations) with fixed and randomly distributed f?
periodic regime. The evolution of the array in this case is
illustrated in Figs. 5(a) and 5(d), respectively. The array
selforganizes to exhibit complex spatiotemporal patterns
[Figs. 6(a) and 6(d)] without any repetitive patterns, thereby
exhibiting delayinduced phasecoherent chaotic oscillations
(see Sec. IIIB below for confirmation). It is to be noted that
thenetworkofN = 50delaycoupledpenduladoesnotexhibit
synchronous oscillations as confirmed by the Figs. 6(a) and
6(d). We have increased the density of disorder up to 50% for
the same values of the parameters and the scenario is depicted
in Figs. 5(b) and 5(e) for fixed and random distributions of
f?
in Figs. 6(b) and 6(e). These figures show that the array of
pendula originally with 50% of periodic disorder evolves to
acquire collective coherent chaotic oscillations in the entire
array induced by the rather small coupling delay τ = 1.5 in a
wide range of f?, which is indeed a surprising result of delay
impact.
din the
d, respectively, along with the spatiotemporal representation
B. Delayenhanced phase coherence
Controlling of oscillator coherence by delayed feedback
has been observed both theoretically and experimentally in
Refs. [37,38]. In our investigation, we find that in addition
to the enrichment of the periodic disorder to (chaotic) higher
order oscillations, delay coupling also increases the coherence
of the collective chaotic oscillations of the whole network.
For a better understanding and confirmation of the delay
enhancedphasecoherentoscillationsoftheentirenetwork,we
investigate both qualitatively and quantitatively the coherence
property of the network macroscopically. For each of the
pendulum in system (2), one can define the phase as
θi= tan−1(yi/xi).
(3)
Here θirepresent the phases of the individual pendula in the
system. In order to visualize the effect of coupling delay on
phase coherence of the system, we plot the snapshot of the
phases of the pendula in Fig. 7. From Eq. (3) one can write
Xi= cosθi=
xi
??x2
??x2
i+ y2
yi
i
?,
?.
(4a)
Yi= sinθi=
i+ y2
i
(4b)
The Kuramoto order parameter r which quantifies the strength
of phase coherence is given by reiψ=1
r = 0 phase coherence is absent in the system and when
r ≈ 1 there is complete phase coherence in the system. Thus
r essentially quantifies the strength of phase coherence. To
be more quantitative one can use the time averaged order
parameter R =1
corresponds to phase incoherence, while a value near to unity
N
?N
j=1eiθj. When
T
?T
0rdt so that its low value (near to zero)
1
0.5
0
0.5
1
0 0.5
Xi
1
Yi
(d)
1
0.5
0
0.5
1
0 0.5
Xi
1
Yi
(b)
1
0.5
0
0.5
1
0 0.5
Xi
1
Yi
(c)
1
0.5
0
0.5
1
0 0.5
Xi
1
Yi
(a)
FIG. 7. Snapshotsofthephaseportraitsonthe(Xi,Yi)planeofthe
ringnetworkwithXi= cosθiandYi= sinθi.Here(a)correspondsto
C = 0.05and(b)–(d)correspondtoFigs.5(d)–5(f)withC = 0.5and
thestrengthoftheimpuritiesbeing20%, 50%,and70%,respectively.
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0.8
0.9
1
9300 9330
t
9360
r
(b)
0.24
0.28
0.32
9300 9330
t
9360
r
(a)
FIG. 8. Time evolution of the Kuramoto order parameter r for (a)
C = 0.05[Figs.7(a)and7(b)]C = 0.5with20%impurity[Fig.7(b)].
corresponds to phase coherence. Throughout the paper we
have estimated R for an average over 1200 time units.
Using Eqs. (4) (where the xiare wrapped to be between 0
and 2π) we have plotted the distribution of phases associated
with Eq. (2) in the (Xi,Yi) plane in Fig. 7. We present the
results for two specific values of the coupling strength C for
the same value of delay parameter τ = 1.5, that is for a low
value of coupling C = 0.05 with 20% impurity [Fig. 7(a)] and
forC = 0.5with20%, 50%,and70%impuritiesinFigs.7(b)–
7(d), respectively. The phases of the pendula are distributed
apart on the unit circle for C = 0.05, as illustrated in Fig. 7(a),
indicating a poor or a very low coherence of the pendula,
whichisalsoconfirmedbythelowvalueofthetimeaverageof
the Kuramoto order parameter R = 0.316. The time evolution
of the corresponding order parameter r itself is depicted in
Fig. 8(a). The phases of the pendula in the entire network
corresponding to Figs. 5(d)–5(f), that is for C = 0.5 with
20%, 50%,and70%impurities,aredepictedinFigs.7(d)–7(f),
respectively. The phases are now confined to a much smaller
region on the unit circle for C = 0.5 in the presence of the
delay coupling confirming thedelayenhanced phasecoherent
oscillations of the entire network. This is indeed confirmed
by much higher values of the time averaged order parameter
R = 0.964, 0.973, and 0.982 for Figs. 7(b)–7(d), respectively.
Also, the time evolution of the order parameter corresponding
to Fig. 7(b) is shown in Fig. 8(b). Thus we have confirmed
the existence of delayenhanced phasecoherent oscillations
in the entire network of delay coupled pendula for appropriate
coupling strength C.
C. Possible mechanism
Twosimplemechanismweresuggestedfortamingofchaos
by disorder and fostering of periodic patterns in the array
without delay in Ref. [19]. Indeed, we find the manifestations
of both these mechanism in the delay coupled networks as
well under appropriate conditions. It is essential to understand
the first of these two mechanism to understand the mechanism
behind the delay induced coherent chaotic oscillations. The
first mechanism depends on the topological features of the
attractors. The periodic disorder needed to stabilize a chaotic
array depends on both the distance and the direction in the
parameter space to the nearest periodic attractor, which is
controlled by the magnitude and distribution of the disorder
[19]. For this mechanism to work it is not essential to
introduce disorder since uncoupled chaotic oscillators can
becomeperiodicwhencoupled.Thisphenomenonisexplicitly
observed from the bifurcation diagrams shown in Fig. 2. The
chaotic regimes in the range of f?∈ (0.5,0.97) in Fig. 2(a)
for the uncoupled system becomes periodic for the same
parameters when a coupling delay is introduced [seeFigs. 2(c)
and 2(d)]. The second mechanism deals with the locking of
the chaotic pendula by the periodic ones to the external ac
drive [19], which is observed for disorder greater than 70% in
our case.
In this paper we are interested in delayinduced coherent
chaotic oscillations in the network, the mechanism of them
is a simple extension of the first, as will be explained in the
following. The presence of delay in the coupling extends the
phase space dimension as a timedelay system is essentially
an infinitedimensional system [13]. Therefore, the dimension
and the phase space (characterized by multiple unstable direc
tions corresponding to multiple positive Lyapunov exponents
of the delaycoupled network) of the chaotic attractors of the
delaycoupled network also increases. This in turn increases
the robustness of the chaotic attractors against even nearby
periodicorbits(disorders)intheparameterspaceandhencethe
presenceofalargepercentageofperiodicdisorder(whichdoes
not extend over multidimensional phase space) is not capable
of taming the chaoticity of the pendula. Furthermore, delay
1
0.5
0
0.5
1
1.5
0 25
N
50
y
(a)
0.5
0
0.5
1
1.5
0 25
N
50
y
(b)
1
0.5
0
0.5
1
1.5
0 25
N
50
y
(c)
1
0.5
0
0.5
1
1.5
0 25
N
50
y
(d)
1
0.5
0
0.5
1
1.5
0 25
N
50
y
(e)
1.5
0
1.5
0 25
N
50
y
(f)
FIG. 9. Poincar´ e points corresponding to the network of pendula, with f?= 1.6 for chaotic pendula, in a ring configuration with N = 50
for the coupling strength C = 0.6 with coupling delay τ = 2.0 and for different values of density of disorders. First row is with fixed value of
the disorders f?
d= 1.1 and the second row with random values of f?
d∈ (0.5,1.2). (a), (d) 20%, (b), (e) 50%, and (c), (f) 70% of disorders.
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FIG. 10. (Color online) Spatiotemporal representation of Fig. 9. Color bar is the same as in Fig. 4.
being a source of instability, by inducing chaotic oscillations
[13,40–43], the periodic disorder acquires chaotic oscillations
for suitable values of the delay resulting in coherent chaotic
oscillations of the entire array. It is to be noted that increasing
the delay alone gives rise to a rich variety of behavior, such as
periodic, higher order oscillations, chaotic and hyperchaotic
attractors with a large number of positive Lyapunov exponents
as observed in several bifurcation diagrams presented earlier
as a function of the delay even in scalar timedelay systems
[13,39]. Furthermore, periodic orbits of very large periods are
also created due to the delay which are not present in the
undelayed systems and these higher order oscillation manifest
in the array in place of disorder when the size of disorder is
larger than 50%.
D. Effect of increased disorder and coupling delay
We have also increased the amount of disorder to more
than half the size of the network to investigate the effect of
delay coupling. Indeed this scenario may also be considered
as the one in which chaotic impurities coexist in a sea of
a periodically oscillating network and one may expect the
suppressionofchaoticoscillationstoachievecoherentperiodic
oscillations so as to enhance spatiotemporal order of the
network. Nevertheless, the presence of delay in the coupling
prohibits suppression of any chaotic pendula up to 70% of
disorder and induces chaoticity in the periodic impurities
adjacent to a chaotic pendulum, while the impurities away
from it acquire higher order oscillations. Furthermore, it is to
be noted that for a density of disorder larger than 50%, the
distribution of disorder becomes nonuniform (asymmetric).
For instance, we have distributed 70% of disorders, while
retaining chaoticity only in the remaining 30% pendula for
the same values of τ and C. The dynamical organization
of the array with 70% of disorder is shown in Figs. 5(c)
and 5(f), which again depicts the delayinduced chaoticity
in the periodic pendula adjacent to the chaotic pendulum,
while the other periodic pendula away from the chaotic
pendulumacquirehigherorderoscillations.Thecorresponding
selforganized complex spatiotemporal behavior is shown in
Figs. 6(c) and 6(f). We have also confirmed the higher order
oscillations of the pendula from their corresponding phase
space plots. We can conclude that the complexity of the
network as a whole is increased in the presence of delay
coupling even if the impurities exceed half the size of the
network, thereby confirming the robustness of the network
against large disorderinduced by the coupling delay.
Next, the value of the coupling delay is further increased
to examine whether the delay enhances the coherent chaotic
oscillations and increases the robustness of the array against
more than 70% disorder. We find that increase in the coupling
delay also leads to the same results for appropriate value
of the coupling strength and the network of pendula attains
synchronous periodic oscillations leading to spatiotemporal
order for disorder of size more than 70%. To be specific, we
fix the coupling delay as τ = 2.0 and C = 0.6. For impurities
of periodic type the ac torque is fixed as f?
chaotic pendula it is chosen as f?= 1.6 using the bifurcation
diagram shown in Fig. 2(d) (as seen in the inset). The first row
in Fig. 9 is plotted for disorders with fixed f?
d= 1.1 and for
dand the second
1
2
3
4
5
y(tτ)
FIG. 11. (Color online) Schematic diagram of a network of
five oscillators with alltoall (global) coupling. In this figure each
oscillator gets four delayed input from the remaining oscillators in
the network, and also a delayed feedback from itself when j = i.
0662067
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D. V. SENTHILKUMAR et al.
PHYSICAL REVIEW E 84, 066206 (2011)
1.5
0.5
0.5
1.5
0 0.5 1
f’
1.5 2
y
(b)
0.9
1.5
0.5
0.5
1.5
0 0.5 1
f’
1.5 2
y
(a)
1.4
1.5
0.5
0.5
1.5
0 0.5 1
f’
1.5 2
y
(c)
1
FIG. 12. (Color online) The largest Lyapunov exponents of the network of N = 20 globally coupled pendula and the bifurcation diagram
of a single pendulum in the network for different values of the coupling strength C and the coupling delay τ. The inset shows that the network
exhibits chaotic oscillations for the specific values of f?chosen in the text. (a) C = 0.2 and τ = 0.0, (b) C = 0.2 and τ = 1.5, and (c) C = 0.3
and τ = 2.0. Red and dark blue (dark gray) lines correspond to the largest Lyapunov exponents and light blue (light gray) dots correspond to
the bifurcation diagram.
row for random distribution of f?
coherent chaotic oscillations of the whole network in the
presenceof20%disorderareshowninFigs.9(a)and9(d).The
corresponding spatiotemporal representation is illustrated in
Figs.10(a)and10(d).ThenetworkofN = 50coupledpendula
oscillates chaotically even in the presence of 50% disorder
as depicted in Figs. 9(b) and 9(e) along with their complex
spatiotemporal patterns in Figs. 10(b) and 10(e), respectively.
Further increase in the density of disorder to 70% continues to
resultinchaoticoscillationsofdisordersadjacenttothechaotic
pendulaandhigherorderoscillationsindisordersfurtheraway
from it, as shown in Figs. 9(c) and 9(f). The corresponding
dynamical organization of the network of pendula with 70%
disorder to selforganized complex spatiotemporal structures
is illustrated in Figs. 10(c) and 10(f).
d∈ (0.2,0.9). Delayinduced
E. Summary
Thus we have shown that the infected sites are healed or in
otherwordsthedisordersintheringnetworkacquirescoherent
chaotic oscillating behavior induced by timedelay in the
coupling,therebyenhancingthespatiotemporalcomplexityfor
a uniform (symmetric) distribution of the impurities as large
as 50% of the array. Note that in the absence of delay in the
coupling, the whole network will become infected (ordered)
even for 20% of disorder. Furthermore, for the density of
disorder larger than 50%, the distribution of disorder becomes
nonuniform (asymmetric). In this case, the impurities adjacent
tothechaoticelementacquireschaoticity,whiletheimpurities
located away from the chaotic ones acquire higher order
oscillationsresultinginenhancedcomplexityofthenetwork.It
is also to be appreciated that the delay in the coupling not only
enhances the coherent chaotic oscillations, but also increases
the robustness of the network against any infection (disorder)
of even more than half the size of the network.
In the next section we will extend our investigation to
a network of globally coupled pendula and show that we
essentially obtain similar results. In particular, coupling delay
can enhance the dynamical complexity of disordered pendula
leading to delayinduced coherent chaotic oscillations up to
50% of symmetric disorder. For asymmetric disorder of size
greater than 50%, the coupling delay can induce chaotic
oscillations in disordered pendula adjacent to chaotic pendula
andthoseawayfromitwillacquirehigherorderoscillationsup
to 65% disorder resulting in the enhanced complex behavior
of the existing network.
IV. GLOBAL DELAYED COUPLING
Most natural systems involve complicated coupling be
tween them and that the individual oscillators are not only
coupled with their nearest neighbors but also with all other
elements in the network. Such a global coupling plays an
important role in a large number of dynamical systems
ranging from the physical [44], chemical [45], and bio
logical [46] to social and economic [47,48] networks and
electronic systems [49]. Global coupling is also being studied
in reactiondiffusion systems, for example, as a reaction
diffusion with global coupling (RDGC) model, to understand
the mechanism behind the electromechanic dynamics of the
heart and generation of successive ectopic beats [50] and also
to understand the mechanism behind the oscillatory regime
in the NashPanfilov model [51]. In addition, delayed global
couplinghasbeenshowntoinduceinphasesynchronizationin
an array of semiconductor lasers [3]. It has been demonstrated
that global coupling is more efficient than local coupling to
1.5
0.5
0.5
1.5
0 5 10 15 20
y
N
(a)
1.5
0.5
0.5
0 5 10 15 20
y
N
(b)
1.5
0.5
0.5
0 5 10 15 20
y
N
(c)
FIG. 13. Poincar´ e points of globally coupled network of N = 20 pendula for the coupling strength C = 0.2 in the absence of coupling
delay τ = 0.0. (a) Chaotically oscillating pendula for f?= 1.4, (b) periodically oscillating pendula for 20% of disorders with fixed f?
and (c) periodically oscillating pendula for 20% of disorders with random f?
d= 0.3,
d∈ (0,0.3).
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FIG. 14. (Color online) Spatiotemporal representation of Fig. 13.
Color bar is the same as in Fig. 4.
achieve nonstationary and stationary inphase operations with
and without delay, respectively, in Refs. [52,53].
In this section we will investigate the effect of delay
coupling in the presence of disorder in a globally coupled
network of pendula, where every pendulum is connected to all
the other (N − 1) pendula in the network with a delay τ and
it gets a selfdelayed feedback only when j = i. To explain
the coupling configuration, a schematic diagram of a network
of five oscillators is shown in Fig. 11 (the dotted lines show
the selfdelayed feedback only when j = i). The model is
represented as
ml2˙ xi= yi,
˙ yi= −γyi− mgl sinxi+ f + f?
N
?
wherei = 1, 2,...,N.Alltheparametershavebeenchosento
be the same as in the previous section. We restrict ourselves to
N = 20 oscillators for computational convenience; however,
similar results have also been obtained for larger number
(5a)
isin(ωt)
+C
N
j=1
[yj(t − τ) − yi(t)],
(5b)
of oscillators for appropriate coupling delay and coupling
strength.
Furthermore, we wish to add that in order to fix the system
parameters pertaining to chaotic and periodic regimes, unlike
thecaseoflinearcoupling(Secs.IIandIII),itisnotmeaningful
to consider the bifurcation scenario with low numbers of
pendula, like N = 3 or 4, in the case of global coupling as the
bifurcation diagrams will change appreciably when the value
of N changes. So in our following study of the bifurcation
scenario and the Lyapunov spectrum, we analyze the full
networkitselfandpresentthefirstoneortwolargestLyapunov
exponents of the entire network and the bifurcation diagram
of a single pendulum in the network.
A. Globally coupled pendula without delay
We will start our investigation by plotting the bifurcation
diagrams and the Lyapunov exponents for delineating the
periodic and chaotic regimes in the case of N = 20 globally
coupled pendula. Enhancement of spatiotemporal regularity
and taming chaoticity in globally coupled network has not
yet been reported to the best of our knowledge. Hence the
comparison of delayenhanced coherent chaotic oscillations
leading to spatiotemporal complexity will be meaningful
only when the globally coupled chaotic network in the
presence of a few periodic disorder is tamed when there is
no delay. Therefore, in this section we will show that the
globally coupled chaotic network is indeed tamed leading to
spatiotemporal order in the absence of coupling delay.
The largest Lyapunov exponent of the full network and
the bifurcation diagram of a pendulum in the network of
the globally coupled (N = 20) case are plotted in Fig. 12(a)
for C = 0.2 in the range of f?∈ (0,2) when no delay is
present. We find that all the pendula in this network exhibit
an almost similar bifurcation scenario, and that the network
as a whole exhibits multiple positive Lyapunov exponents.
However, these values are close to each other and so we
present only the largest one in Fig. 12(a). We fix f?= 1.4
for chaotic pendula [as confirmed from the positive Lyapunov
exponent shown in the inset of Fig. 12(a)] and f?
d= 0.3 for
1.5
0.5
0.5
1.5
0 5 10 15 20
y
N
(a)
0.5
0.5
1.5
0 5 10 15 20
y
N
(b)
0.5
0.0
0.5
1.0
0 5 10 15 20
y
N
(c)
1.5
0.5
0.5
1.5
0 5 10 15 20
y
N
(d)
4
2
0
2
0 5 10 15 20
y
N
(e)
4
2
0
2
0 5 10 15 20
y
N
(f)
FIG. 15. Poincar´ e points of globally coupled network of N = 20 pendula, chaotic for f?= 0.92, for the coupling strength C = 0.2 and the
coupling delay τ = 1.5. First row with fixed value of the disorders f?
20%, (b), (e) 50%, and (c), (f) 65% of disorders.
d= 1.5 and the second row with random values of f?
d∈ (1.0,1.7). (a), (d)
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D. V. SENTHILKUMAR et al.
PHYSICAL REVIEW E 84, 066206 (2011)
FIG. 16. (Color online) Spatiotemporal representation of Fig. 15. Color bar is the same as in Fig. 4.
periodic disorder from the bifurcation diagram. Chaotically
oscillating pendula in the globally coupled network without
anydisorderisdepictedinFig.13(a)forC = 0.2alongwithits
spatiotemporal representation in Fig. 14(a) for 10 drive cycles.
As discussed in the case of diffusive coupling in Sec. II, the
globally coupled network is also tamed exhibiting periodic
oscillations for 20% periodic uniform disorder with fixed
f?
C in the absence of delay. The corresponding spatiotemporal
plot shows spatiotemporal regularity with repetitive patterns
for every two drive cycles as depicted in Fig. 14(b). We have
obtained similar results of taming chaoticity [Fig. 13(c)] from
the network leading to spatiotemporal order [Fig. 14(c)] for
random values of the ac torque f?
noted that we have also got similar results for random values
of f?
In the next section we will demonstrate the existence of
delayinduced coherent chaotic oscillations leading to en
hanced spatiotemporal complexity of the network for disorder
of size as large as 65%.
d= 0.3, as illustrated in Fig. 13(b) for the same value of
d∈ (0,0.3). It is also to be
d∈ (1.7,2.0).
B. Globally coupled pendula with coupling delay
The largest two Lyapunov exponents of the network of
globally delay coupled pendula and the bifurcation diagram
of a single pendulum in the network are plotted in Fig. 12(b)
for the same value of coupling delay and coupling strength
as in the nondelay case reported in the previous section
(Sec. IVA) for comparison. Again the system as a whole
exhibits 20 positive Lyapunov exponents and only the first
two largest positive Lyapunov exponents differ appreciably
from the other almost identical positive Lyapunov exponents.
Now, we fix f?= 0.92 for chaotic pendula [as confirmed
from the positive Lyapunov exponent shown in the inset
of Fig. 12(b)] and f?
bifurcation diagram. Poincar´ e points representing the delay
induced coherent chaotic oscillations of N = 20 pendula
in the network with 20% symmetric disorder with fixed
f?
spatiotemporal patterns in Fig. 16(a). A random distribution
of f?
in delayinduced coherent chaotic oscillations [Fig. 15(d)] and
enhanced spatiotemporal complexity [Fig. 16(d)]. The density
of disorder is increased further up to half the size of the
network with both fixed f?
f?
shows increased complexity of the entire network as depicted
in their corresponding spatiotemporal plots Figs. 16(b) and
16(e). The globally delaycoupled network remains robust
d= 1.5 for periodic disorder from the
d= 1.5 are illustrated in Fig. 15(a) with its complex
d∈ (1.0,1.7) corresponding to 20% disorder also results
d= 1.5 and random distribution of
d∈ (1.0,1.7) as in Figs. 15(b) and 15(e), respectively, which
1.5
0.5
0.5
1.5
0 5 10 15 20
y
N
(a)
1.5
0.5
0.5
1.5
0 5 10 15 20
y
N
(b)
1.5
0.5
0.5
1.5
0 5 10 15 20
y
N
(c)
1.5
0.5
0.5
1.5
0 5 10 15 20
y
N
(d)
1.5
0.5
0.5
1.5
0 5 10 15 20
y
N
(e)
1.5
0.5
0.5
1.5
0 5 10 15 20
y
N
(f)
FIG. 17. Poincar´ e points of globally coupled network of N = 20 pendula, chaotic for f?= 0.96, for the coupling strength C = 0.3 and the
coupling delay τ = 2.0. First row with fixed value of the disorders f?
20%, (b), (e) 50%, and (c), (f) 65% of disorders.
d= 1.5 and the second row with random values of f?
d∈ (1.2,2.0). (a), (d)
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FIG. 18. (Color online) Spatiotemporal representation of Fig. 17. Color bar is the same as in Fig. 4.
against disorder of a size as large as 65%, as shown in
Figs. 15(c) and 15(f) for both fixed and random values
of ac torque, in which case the period1 disorder acquires
higher order oscillations resulting in a selforganized complex
spatiotemporalrepresentation[Figs.16(c)and16(f)].Forsome
distributionsoff?
of disorder.
As in the case of the ring network, we further increased the
value of delay in the coupling to examine the change in the
robustness of the network against disorder and we obtained
the same results even for larger values of delay and for
appropriate values of coupling strength. For instance, we
present our results for τ = 2.0 and C = 0.3 in the following.
The two largest Lyapunov exponents of the network of
globally coupled pendula and the bifurcation diagram of a
pendulum in the network are illustrated in Fig. 12(c). We
chosef?= 0.96forachaoticpendulum[asconfirmedfromthe
positive Lyapunov exponent shown in the inset of Fig. 12(c)]
and f?
Poincar´ e points shown in Figs. 17(a) and 17(d) indicate the
d,thenetworkremainsrobustevenupto70%
d= 1.5 for disorders from the bifurcation diagram.
1
0.5
0
0.5
1
0 0.5
Xi
1
Yi
(b)
1
0.5
0
0.5
1
0 0.5
Xi
1
Yi
(c)
1
0.5
0
0.5
1
0 0.5
Xi
1
Yi
(d)
1
0.5
0
0.5
1
0 0.5
Xi
1
Yi
(a)
FIG. 19. Snapshots of the phase portraits on the (Xi,Yi) plane for
a globally coupled network of pendula. Here (a) corresponds to C =
0.05, and (b)–(d) corresponds to Figs. 15(d)–15(f) with C = 0.2 and
thestrengthoftheimpuritiesbeing20%, 50%,and65%,respectively.
chaotically oscillating pendula for 20% uniform disorder for
both fixed and random f?
spatiotemporal representation is depicted in Figs. 18(a) and
18(d). The evolution of the pendula in the network in the
presence of 50% disorder for fixed f?
[with its spatiotemporal plot in Fig. 18(b)] and for random
f?
Fig. 18(e)]. Figures 17(c) and 17(f) exemplify the dynamical
nature of the globally coupled network in the presence of
65% disorder with both fixed and random values of ac torque.
The corresponding spatiotemporal dynamics is depicted in
Figs.18(c)and18(f),respectively.Thechaoticpendularemain
unaltered, while the period1 disorders acquire higher order
oscillations for sizes larger than 50% resulting in increased
spatiotemporal complexity of the original network, indicating
the robustness of the delay coupled network against disorder
induced synchronous periodic oscillations.
Finally, as discussed in Sec. IIIB, we confirm the existence
of the delayenhanced phasecoherent oscillations in the
globally connected network of pendula by looking at the
distribution of phases in the (Xi,Yi) = (
plane. This is indeed shown in Fig. 19. For C = 0.05
(with 20% impurity) the phases are distributed on a large
part of the unit circle [Fig. 19(a)] and this reveals a poor
coherence of the pendula as confirmed by the low value of the
time averaged order parameter R = 0.267. The evolution of
the corresponding order parameter is depicted in Fig. 20(a).
On the other hand, for C = 0.20 with 20%, 50%, and 65%
impuritiesthephasesareconfinedtoanarrowregionoftheunit
circle as shown in Figs. 19(b)–19(d), which is also confirmed
by the corresponding time averaged order parameters R =
0.986, 0.988, and 0.992, respectively. Also, the evolution of
the order parameter r corresponding to Fig. 19(d) is shown in
Fig. 20(b).
d∈ (1.2,2.0), respectively. Their
dis shown in Fig. 17(b)
d∈ (1.2,2.0) in Fig. 17(e) [with its spatiotemporal plot in
xi
x2
√
i+y2
i,
yi
x2
√
i+y2
i)
0.8
0.9
1
7530 7560
t
7590
r
(b)
0.2
0.25
7530 7560
t
7590
r
(a)
FIG. 20. Time evolution of the Kuramoto order parameter r for
(a) C = 0.05 and (b) C = 0.2 with 20% impurity.
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D. V. SENTHILKUMAR et al.
PHYSICAL REVIEW E 84, 066206 (2011)
V. SUMMARY AND CONCLUSION
In this paper we have analyzed the dynamics of a regular
network with a ring topology, and a more complex network
with alltoall (global) topology and studied the effect of
the size of disorder. We mainly find that the coupling delay
can induce phasecoherent chaotic oscillations in the entire
network, thereby enhancing the spatiotemporal complexity
even in the presence of large disorder of a size as large as 50%
in contrast to the undelayed case, where even a 20% disorder
can render the whole network to be periodic and thereby
taming chaos. Furthermore, the delay coupling is also capable
of increasing the robustness of the network against a large size
of the disorder up to 70% of the size of the original network,
thereby increasing the dynamical complexity of the network
for suitable values of the coupling strength. We have also
discussedamechanismforthedelayinduced coherentchaotic
oscillations leading to spatiotemporal complexity in the pres
ence of large disorders. We have also confirmed the delay
enhanced coherent chaotic oscillations both qualitatively and
quantitatively. We note here that the results are also robust
against the size of the network and the size of the impurities
(disorders) have to be fixed proportional to the size of the
network. We expect that one can use the results of our analysis
to more realistic complex networks to increase the robustness
ofthenetworkagainstanydisorder,forexample,inexamining
the cascading failures of complex networks, specifically in
power grids and in controlling disease spreading in epidemics,
spatiotemporal, and secure communication and to increase the
robustness and complexity of reservoir computing or liquid
state machines.
ACKNOWLEDGMENTS
The work of R.S. and M.L. is supported by the Department
of Science and Technology (DST), Government of India
Ramanna program, and also by a DSTIRPHA research
project. J.H.S. is supported by a DST–FAST TRACK Young
Scientist research project. M.L. is also supported by a
Department of Atomic Energy Raja Ramanna program. D.V.S
and J.K. acknowledges the support from EU under Project
No. 240763 PHOCUS(FP7ICT2009C).
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