Article
Delay-enhanced coherent chaotic oscillations in networks with large disorders
Potsdam Institute for Climate Impact Research, DE-14473 Potsdam, Germany.
Physical Review E (Impact Factor: 2.29). 12/2011; 84(6 Pt 2):066206. DOI: 10.1103/PhysRevE.84.066206 Source: arXiv
Full-text
Available from: Suresh RamachandranPHYSICAL REVIEW E 84, 066206 (2011)
Delay-enhanced coherent chaotic oscillations in networks with large disorders
D. V. Senthilkumar,
1
R. Suresh,
2
Jane H. Sheeba,
2
M. Lakshmanan,
2
and J. Kurths
1,3
1
Potsdam Institute for Climate Impact Research, DE-14473 Potsdam, Germany
2
Centre for Nonlinear Dynamics, Department of Physics, Bharathidasan University, Tiruchirapalli 620 024, India
3
Institute of Physics, Humboldt University, DE-12489 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 all-to-all (global) topology in the presence of impurities (disorder). We ﬁnd that the coupling
delay is capable of inducing phase-coherent 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 disorder-induced 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
delay-induced coherent chaotic oscillations despite the presence of large disorders and discuss its applications.
DOI: 10.1103/PhysRevE.84.066206 PACS number(s): 05.45.Xt, 05.45.Pq, 05.45.Jn, 05.45.Gg
I. INTRODUCTION
In recent times, researchers have been interested in studying
networks of oscillators with time-delayed 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 time-delay 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 time-delay (synaptic delay), and it is also
reported that the axons can generate time-delay up to 300 ms
[12]. A typical nonlinear time-delay system is a veritable
black box [13] and that the delay coupling itself gives rise to a
plethora of novel phenomena, such as delay-induced amplitude
death [7], phase-ﬂip bifurcation [14], synchronizations of
different types [15], multistability [16], chimera states [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 Frenkal-Kontorova 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. The possibility of obtaining synchronized motion in
one- and two-dimensional chaotic arrays of such systems has
been investigated in Refs. [22–25], where the complex chaotic
behavior of the collective systems was completely tamed when
a certain amount of impurities (disorder) was introduced. To be
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 all-to-all
(global) topology with different densities (sizes) of impurities
(disorder) and examine the effect of time-delay in the coupling.
In particular, the oscillations of each pendulum affects the
oscillations of the pendula to which it is connected to, after
some ﬁnite time-delay τ. In such a conﬁguration we are
interested in investigating the possibility of achieving coherent
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 (phase-coherent or phase-synchronized) chaotic
oscillations (but not complete synchronization) in the entire
network despite the presence of disorder [29]. The delay-
enhanced phase-coherent chaotic oscillations are characterized
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 Bose-Einstein 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 s patially
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 time-delay in the coupling
induces coherent chaotic oscillations of the network of coupled
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
disorder-enhanced 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
very small periodic disorder itself is capable of suppressing the
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
globally connected heterogeneous neural network has revealed
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
will demonstrate our results on delay-induced coherent chaotic
oscillations despite the presence of large disorders, even up to
70%, in Sec. III. Similar results are presented in a network of
globally coupled pendula both with and without delay coupling
in Sec. IV. Finally, in Sec. V, we discuss our results and
conclusions.
II. LINEAR ARRAY OF PENDULA IN THE ABSENCE
OF COUPLING DELAY
We consider a chain of N forced coupled nonlinear pendula
whose equation of motion can be written as [19,23–28]
ml
2
˙
x
i
= y
i
, (1a)
˙
y
i
=−γy
i
− mgl sin x
i
+ f + f
i
sin(ωt)
+C[y
i+1
(t) − 2y
i
(t) + y
i−1
(t)], (1b)
where i = 1, 2,...,N. We choose the following periodic
boundary conditions: x
0
= x
N
and x
N+1
= x
1
. The parameters
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
i
= f
is the ac torque, and C is the coupling
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
authors have considered a chain of diffusively coupled pendula
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
1
2
3
4
5
N-1
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
achieving spatiotemporal regularity can be obtained for 20% 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
allowing one or more pendula to oscillate periodically as in the
earlier reports [19,26–28]. In order to ﬁx the parameters (of the
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
bifurcation scenario leading to chaotic behavior for appropriate
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
is coupled with its nearest neighbors and so each of the pendula
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
aringofN = 3 coupled pendula is almost identical to that
of a single uncoupled pendulum [Fig. 2(a)] and the network
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DELAY-ENHANCED COHERENT CHAOTIC OSCILLATIONS ... PHYSICAL REVIEW E 84, 066206 (2011)
-1.5
-0.5
0.5
1.5
0 0.5 1 1.5 2
y
f
’
(a)
-1.5
-0.5
0.5
1.5
0 0.5 1 1.5 2
y
f
’
(b)
-1.5
-0.5
0.5
1.5
0 0.5 1 1.5 2
y
f
’
(c)
-1.5
-0.5
0.5
1.5
0 0.5 1 1.5 2
y
f
’
(d)
1.6
FIG. 2. (Color online) Bifurcation diagrams of a single pendulum
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 = 0.0andτ = 0.0, (b) C = 0.5and
τ = 0.0, (c) C = 0.5andτ = 1.5, and (d) C = 0.6andτ = 2.0
(inset shows that the pendulum exhibits chaotic oscillations for
f
d
= 1.6). Red (dark gray) line corresponds to the largest Lyapunov
exponent and light blue (light gray) dots correspond to the bifurcation
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.
(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
angular velocities of the pendula; dark red (dark gray) indicates
negative velocities and green (light gray) positive velocities.
Narrow bands of red and green colors represent sudden rapid
motion of the pendula i n 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
spatiotemporal regularity as discussed in Ref. [19]. The density
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 50
y
N
(a)
-1.5
-1
-0.5
0
0.5
0 25 50
y
N
(b)
-1.5
-1
-0.5
0
0.5
0 25 50
y
N
(c)
FIG. 3. Poincar
´
e points corresponding to the network of pendula in a ring conﬁguration with N = 50 for the coupling strength C = 0.5in
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).
066206-3
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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
d
. The spatiotemporal
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 disorder-induced 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
d
of the disorders taming
can be achieved in a wide range of ac torque. A periodically
oscillating array of pendula for 20% of disorder is obtained for
a random distribution of f
d
∈ (0.1,0.5) as shown in Fig. 3(c)
along with their s patiotemporal 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.
III. LINEAR ARRAY WITH COUPLING DELAY
A. Effect of time-delay
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
ml
2
˙
x
i
= y
i
, (2a)
˙
y
i
=−γy
i
− mgl sin x
i
+ f + f
i
sin(ωt)
+C[y
i+1
(t − τ ) − 2y
i
(t) + y
i−1
(t − τ )], (2b)
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 N th 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
ring of N = 3 delay coupled pendula and the largest Lyapunov
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
diffusively coupled subsystems are synchronized to a common
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)]forτ = 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
d
= 0.5
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 s ufﬁciently large number of (1000 drive
cycles) transients in the presence of the coupling delay τ = 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
d
and the second one for a random distribution
of f
d
∈ (0.2,0.9). The corresponding spatiotemporal plot is
depicted in Fig. 6 for 10 drive cycles. Disorder of size 20%
-1.5
0
1.5
0 25 50
y
N
(a)
-1.5
0
1.5
0 25 50
y
N
(b)
-1.5
0
1.5
0 25 50
y
N
(c)
-1.5
0
1.5
0 25 50
y
N
(d)
-1.5
0
1.5
0 25 50
y
N
(e)
-1.5
0
1.5
0 25 50
y
N
(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
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
disorders.
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DELAY-ENHANCED COHERENT CHAOTIC OSCILLATIONS ... PHYSICAL REVIEW E 84, 066206 (2011)
FIG. 6. (Color online) Spatiotemporal representation of Fig. 5. Color bar is the same as in Fig. 4.
are uniformly distributed in t he array of N = 50 pendula, as
in Fig. 3(b) (where the array acquired synchronous periodic
oscillations) with ﬁxed and randomly distributed f
d
in the
periodic regime. The evolution of the array in this case is
illustrated in Figs. 5(a) and 5(d), respectively. The array
self-organizes to exhibit complex spatiotemporal patterns
[Figs. 6(a) and 6(d)] without any repetitive patterns, thereby
exhibiting delay-induced phase-coherent chaotic oscillations
(see Sec. III B below for conﬁrmation). It is to be noted that
the network of N = 50 delay coupled pendula does not exhibit
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
d
, respectively, along with the spatiotemporal representation
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 = 1.5ina
wide range of f
, which is indeed a s urprising result of delay
impact.
B. Delay-enhanced 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-
enhanced phase-coherent oscillations of the entire network, 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
θ
i
= tan
−1
(y
i
/x
i
). (3)
Here θ
i
represent 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.FromEq.(3) one can write
X
i
= cos θ
i
=
x
i
x
2
i
+ y
2
i
, (4a)
Y
i
= sin θ
i
=
y
i
x
2
i
+ y
2
i
. (4b)
The Kuramoto order parameter r which quantiﬁes the strength
of phase coherence is given by re
iψ
=
1
N
N
j=1
e
iθ
j
. When
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
T
T
0
rdt so that its low value (near to zero)
corresponds to phase incoherence, while a value near to unity
-1
-0.5
0
0.5
1
0 0.5 1
Y
i
X
i
(d)
-1
-0.5
0
0.5
1
0 0.5 1
Y
i
X
i
(b)
-1
-0.5
0
0.5
1
0 0.5 1
Y
i
X
i
(c)
-1
-0.5
0
0.5
1
0 0.5 1
Y
i
X
i
(a)
FIG. 7. Snapshots of the phase portraits on the (X
i
,Y
i
) plane of the
ring network with X
i
= cos θ
i
and Y
i
= sin θ
i
. Here (a) corresponds to
C = 0.05 and (b)–(d) correspond to Figs. 5(d)–5(f) with C = 0.5and
the strength of the impurities being 20%, 50%, and 70%, respectively.
066206-5
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D. V. SENTHILKUMAR et al. PHYSICAL REVIEW E 84, 066206 (2011)
0.8
0.9
1
9300 9330 9360
r
t
(b)
0.24
0.28
0.32
9300 9330 9360
r
t
(a)
FIG. 8. Time evolution of the Kuramoto order parameter r for (a)
C = 0.05 [Figs. 7(a) and 7(b)] C = 0.5 with 20% 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 x
i
are wrapped to be between 0
and 2π ) we have plotted the distribution of phases associated
with Eq. (2)inthe(X
i
,Y
i
) 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
for C = 0.5 with 20%, 50%, and 70% impurities in Figs. 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,
which is also conﬁrmed by the low value of the time average of
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%, and 70% impurities, are depicted in Figs. 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 the delay-enhanced phase-coherent
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 delay-enhanced phase-coherent oscillations
in the entire network of delay coupled pendula for appropriate
coupling strength C.
C. Possible mechanism
Two simple mechanism were suggested for taming of chaos
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
become periodic when coupled. This phenomenon is explicitly
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 [see Figs. 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 delay-induced 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 time-delay system is essentially
an inﬁnite-dimensional system [13]. Therefore, the dimension
and the phase space (characterized by multiple unstable direc-
tions corresponding to multiple positive Lyapunov exponents
of the delay-coupled network) of the chaotic attractors of the
delay-coupled network also increases. This in turn increases
the robustness of the chaotic attractors against even nearby
periodic orbits (disorders) in the parameter space and hence the
presence of a large percentage of periodic disorder (which does
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 50
y
N
(a)
-0.5
0
0.5
1
1.5
0 25 50
y
N
(b)
-1
-0.5
0
0.5
1
1.5
0 25 50
y
N
(c)
-1
-0.5
0
0.5
1
1.5
0 25 50
y
N
(d)
-1
-0.5
0
0.5
1
1.5
0 25 50
y
N
(e)
-1.5
0
1.5
0 25 50
y
N
(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.
066206-6
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DELAY-ENHANCED COHERENT CHAOTIC OSCILLATIONS ... PHYSICAL REVIEW E 84, 066206 (2011)
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 f unction of the delay even in scalar time-delay 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
suppression of chaotic oscillations to achieve coherent periodic
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 delay-induced chaoticity
in the periodic pendula adjacent to the chaotic pendulum,
while the other periodic pendula away from the chaotic
pendulum acquire higher order oscillations. The corresponding
self-organized 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 disorder-induced 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
d
= 1.1 and for
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
and the second
1
2
3
4
5
y(t-τ)
FIG. 11. (Color online) Schematic diagram of a network of
ﬁve oscillators with all-to-all (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.
066206-7
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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 1.5 2
y
f
’
(b)
0.9
-1.5
-0.5
0.5
1.5
0 0.5 1 1.5 2
y
f
’
(a)
1.4
-1.5
-0.5
0.5
1.5
0 0.5 1 1.5 2
y
f
’
(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 = 0.2andτ = 0.0, (b) C = 0.2andτ = 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
d
∈ (0.2,0.9). Delay-induced
coherent chaotic oscillations of the whole network in the
presence of 20% disorder are shown in Figs. 9(a) and 9(d).The
corresponding spatiotemporal representation is illustrated in
Figs. 10(a) and 10(d). The network of N = 50 coupled pendula
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
result in chaotic oscillations of disorders adjacent to the chaotic
pendula and higher order oscillations in disorders further away
from it, as shown in Figs. 9(c) and 9(f). The corresponding
dynamical organization of the network of pendula with 70%
disorder to self-organized complex spatiotemporal structures
is illustrated in Figs. 10(c) and 10(f).
E. Summary
Thus we have shown that the infected sites are healed or in
other words the disorders in the ring network acquires coherent
chaotic oscillating behavior induced by time-delay in the
coupling, thereby enhancing the spatiotemporal complexity for
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
to the chaotic element acquires chaoticity, while the impurities
located away from the chaotic ones acquire higher order
oscillations resulting in enhanced complexity of the network. 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 delay-induced 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
and those away from it will acquire higher order oscillations up
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 reaction-diffusion systems, for example, as a reaction
diffusion with global coupling (RDGC) model, to understand
the mechanism behind the electromechanic dynamics of t he
heart and generation of successive ectopic beats [50] and also
to understand the mechanism behind the oscillatory regime
in the Nash-Panﬁlov model [51]. In addition, delayed global
coupling has been shown to induce in-phase synchronization in
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
d
= 0.3,
and (c) periodically oscillating pendula for 20% of disorders with random f
d
∈ (0,0.3).
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DELAY-ENHANCED COHERENT CHAOTIC OSCILLATIONS ... PHYSICAL REVIEW E 84, 066206 (2011)
FIG. 14. (Color online) Spatiotemporal representation of Fig. 13.
Color bar is the same as in Fig. 4.
achieve nonstationary and stationary in-phase 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 self-delayed 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 self-delayed feedback only when j = i). The model is
represented as
ml
2
˙
x
i
= y
i
, (5a)
˙
y
i
=−γy
i
− mgl sin x
i
+ f + f
i
sin(ωt)
+
C
N
N
j=1
[y
j
(t − τ ) − y
i
(t)], (5b)
where i = 1, 2,...,N. All the parameters have been chosen to
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
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
the case of linear coupling (Secs. II and III), it is not meaningful
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
network itself and present the ﬁrst one or two largest Lyapunov
exponents of the entire network and the bifurcation diagram
of a s ingle 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 delay-enhanced 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).Weﬁxf
= 1.4
for chaotic pendula [as conﬁrmed from the positive Lyapunov
exponent shown in the inset of Fig. 12(a)] and f
d
= 0.3for
-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
d
= 1.5 and the second row with random values of f
d
∈ (1.0,1.7). (a), (d)
20%, (b), (e) 50%, and (c), (f) 65% of disorders.
066206-9
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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
any disorder is depicted in Fig. 13(a) for C = 0.2 along with its
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
d
= 0.3, as illustrated in Fig. 13(b) for the same value of
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).Wehave
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
d
∈ (0,0.3). It is also to be
noted that we have also got similar results for random values
of f
d
∈ (1.7,2.0).
In the next section we will demonstrate the existence of
delay-induced coherent chaotic oscillations leading to en-
hanced spatiotemporal complexity of the network for disorder
of size as large as 65%.
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. IV A) 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
d
= 1.5 for periodic disorder from the
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
d
= 1.5 are illustrated in Fig. 15(a) with its complex
spatiotemporal patterns in Fig. 16(a). A random distribution
of f
d
∈ (1.0,1.7) corresponding to 20% disorder also results
in delay-induced 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
d
= 1.5 and random distribution of
f
d
∈ (1.0,1.7) as in Figs. 15(b) and 15(e), respectively, which
shows increased complexity of the entire network as depicted
in their corresponding spatiotemporal plots Figs. 16(b) and
16(e). The globally delay-coupled network remains robust
-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 s trength C = 0.3 and the
coupling delay τ = 2.0. First row with ﬁxed value of the disorders f
d
= 1.5 and the second row with random values of f
d
∈ (1.2,2.0). (a), (d)
20%, (b), (e) 50%, and (c), (f) 65% of disorders.
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DELAY-ENHANCED COHERENT CHAOTIC OSCILLATIONS ... PHYSICAL REVIEW E 84, 066206 (2011)
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 period-1 disorder acquires
higher order oscillations resulting in a self-organized complex
spatiotemporal representation [Figs. 16(c) and 16(f)]. For some
distributions of f
d
, the network remains robust even up to 70%
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 l arger 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 t he network of
globally coupled pendula and the bifurcation diagram of a
pendulum in the network are illustrated in Fig. 12(c).We
chose f
= 0.96 for a chaotic pendulum [as conﬁrmed from the
positive Lyapunov exponent shown in the inset of Fig. 12(c)]
and f
d
= 1.5 for disorders from the bifurcation diagram.
Poincar
´
e points shown in Figs. 17(a) and 17(d) indicate the
-1
-0.5
0
0.5
1
0 0.5 1
Y
i
X
i
(b)
-1
-0.5
0
0.5
1
0 0.5 1
Y
i
X
i
(c)
-1
-0.5
0
0.5
1
0 0.5 1
Y
i
X
i
(d)
-1
-0.5
0
0.5
1
0 0.5 1
Y
i
X
i
(a)
FIG. 19. Snapshots of the phase portraits on the (X
i
,Y
i
) 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 = 0.2and
the strength of the impurities being 20%, 50%, and 65%, respectively.
chaotically oscillating pendula for 20% uniform disorder for
both ﬁxed and random f
d
∈ (1.2,2.0), respectively. Their
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
d
isshowninFig.17(b)
[with its spatiotemporal plot in Fig. 18(b)] and for random
f
d
∈ (1.2,2.0) in Fig. 17(e) [with its spatiotemporal plot in
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) and 18(f), respectively. The chaotic pendula remain
unaltered, while the period-1 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. III B, we conﬁrm the existence
of the delay-enhanced phase-coherent oscillations in the
globally connected network of pendula by looking at the
distribution of phases in the (X
i
,Y
i
)=(
x
i
√
x
2
i
+y
2
i
,
y
i
√
x
2
i
+y
2
i
)
plane. This is indeed shown in Fig. 19.ForC = 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%
impurities the phases are conﬁned to a narrow region of the unit
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).
0.8
0.9
1
7530 7560 7590
r
t
(b)
0.2
0.25
7530 7560 7590
r
t
(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 all-to-all (global) topology and studied the effect of
the s ize of disorder. We mainly ﬁnd that the coupling delay
can induce phase-coherent 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
discussed a mechanism for the delay-induced coherent chaotic
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
of the network against any disorder, for example, in examining
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 DST-IRPHA research
project. J.H.S. is supported by a DST–FAST TRACK Young
Scientist r esearch 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(FP7-ICT-2009-C).
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- [Show abstract] [Hide abstract] ABSTRACT: In this paper, the nonlinear dynamical characteristics of the Duffing–van der Pol oscillator subject to both external and parametric excitations with time delayed feedback control are analyzed. Using the multiple scale method, the primary and principal parametric resonances are discussed. Both the periodic and chaos dynamical behaviors are presented by numerical solutions. From the results, it can be seen that the frequency and amplitude responses for the primary and principal parametric resonances can be affected by the time delayed control. Moreover, it is not only the external and parametric excitations that have an influence on the periodic and chaotic motions, but also the control parameters.