# Delay-enhanced 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 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.

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**ABSTRACT:**The book is well organized and presents the most important classical and modern essentials of chaotic time delay systems and synchronization. It is suitable for senior undergraduate and graduate students as well as practical engineers and researchers interested in dynamics of nonlinear time-delay systems and synchronization. The book is organized as follows: The first four chapters (‘Delay differential equations (DDEs)’; ‘Linear stability and bifurcation analysis’; ‘Bifurcation and chaos in time-delayed piecewise linear dynamical system’; ‘A few other interesting chaotic delay differential equations’) introduce the basics concerning time-delay systems. Chapters 5 and 6 deal with amplitude death caused by delay, with time-delay induced bifurcations, chimera states and networks with time-delay feedback/coupling. “Different types of synchronizations (complete, generalized, lag, anticipatory and phase) and their transitions in coupled time-delay systems with constant delay are the topics of the Chapters 7-10.” Chapter 11 deals with ‘DTM induced oscillating synchronization’, and Chapter 12 addresses the subject ‘Exact solution of certain time delay systems: the car-following models’. In the appendices the following topics are discussed: A: ‘Computing Lyapunov exponents for time-delay systems’; B: ‘A brief introduction to synchronization in chaotic dynamical system’; C: ‘Recurrence analysis’; D: ‘Some more examples of DDEs’. References are given for each appendix.Dynamics of Nonlinear Time-Delay Systems: , Springer Series in Synergetics. ISBN 978-3-642-14937-5. Springer-Verlag Berlin Heidelberg, 2010. 01/2010; - [Show abstract] [Hide abstract]

**ABSTRACT:**Although the study of functional differential equations has received a great deal of attention for the last thirty years, as yet there have been yet few expository books on the topic, and still fewer books not necessarily restricted to specialists of this field. The monograph under review fulfils this aim, and in our opinion represents a successful and motivating introduction to the subject, whilst still being of interest to specialists. The book is divided into five chapters: all of them except chapter 2, which is a short one, devoted to the Hopf bifurcation theorem, are of comparable size. Each chapter is almost self-contained, with very few cross-references, which facilitates reading. Non classical mathematical notions (e.g. a fixed point theorem for sums of operators by Nashed and Wong) are stated wherever necessary. The book considers three main aspects: stability, oscillations and population dynamics. Stability is the primary topic, while “oscillations [are studied] in order to make use of the knowledge of oscillatory solutions in stability investigations”; and population dynamics are treated as a ground for applying methods and results presented elsewhere in the book. This latter choice is well-justified both by the importance that FDE’s have gained over the last few years in population dynamics and the personal production of the author, who is a recognized specialist on the subject. Throughout the book, the author points out ideas and methods, voluntarily avoiding excessive generalizations, even restricting such or such a proof to a particular case. Thus, only discrete delays or distributed delays with an integrable kernel are considered. In the same way, when nonlinear systems are studied, only autonomous ones are considered, and only systems with one positive equilibrium. Most statements are formulated in a way accessible even to non specialists. As a result, the book should be a useful tool for applied research. Amongst the methods provided, the most complete treatment is for the Lyapunov method. The author presents many examples, which cover known models, delay and neutral equations, predator-prey type equations, etc., where Lyapunov functions are built and used. Other methods are presented too, including Routh-Hurwitz criteria for stability, the Hopf bifurcation theorem. Special features such as cooperative systems, which lead to monotonically increasing dynamical systems, are dealt with in detail. Oscillation theory for FDE’s is the second important issue of the book. The latest developments on this topic, some of which are contributions by the author, are included. A comparison method which has been extended recently to various classes of FDE’s (vectorial, neutral, etc.) is presented in detail. The book gives an account of the developments on the subject over the last few years: FDE’s with piecewise constant arguments, impulses, stability switches, neutral equations, etc. Lastly, each chapter is followed by numerous exercises which either apply the notions treated in the chapter, or aim at widening the scope of the text.

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PHYSICAL REVIEW E 84, 066206 (2011)

Delay-enhanced 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, DE-14473 Potsdam, Germany

2Centre for Nonlinear Dynamics, Department of Physics, Bharathidasan University, Tiruchirapalli 620 024, India

3Institute 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.84.066206PACS number(s): 05.45.Xt, 05.45.Pq, 05.45.Jn, 05.45.Gg

I. INTRODUCTION

Inrecenttimes,researchershavebeeninterestedinstudying

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

plethoraofnovelphenomena,suchasdelay-inducedamplitude

death [7], phase-flip 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 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.Thepossibilityofobtainingsynchronizedmotionin

one- and two-dimensional 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 all-to-all

(global) topology with different densities (sizes) of impurities

(disorder)andexaminetheeffectoftime-delayinthecoupling.

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

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 (phase-coherent or phase-synchronized) chaotic

oscillations (but not complete synchronization) in the entire

network despite the presence of disorder [29]. The delay-

enhancedphase-coherentchaoticoscillationsarecharacterized

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 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 time-delay 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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1539-3755/2011/84(6)/066206(13)©2011 American Physical Society

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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

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

willdemonstrateourresultsondelay-inducedcoherentchaotic

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

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

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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DELAY-ENHANCED COHERENT CHAOTIC OSCILLATIONS ...

PHYSICAL REVIEW E 84, 066206 (2011)

-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).

066206-3

Page 4

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 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?

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 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

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

066206-4

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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 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

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. 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. 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-

enhancedphase-coherentoscillationsoftheentirenetwork,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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PHYSICAL REVIEW E 84, 066206 (2011)

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 thedelay-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

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 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

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.

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 function 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

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 delay-induced chaoticity

in the periodic pendula adjacent to the chaotic pendulum,

while the other periodic pendula away from the chaotic

pendulumacquirehigherorderoscillations.Thecorresponding

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?

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(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

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 self-organized complex spatiotemporal structures

is illustrated in Figs. 10(c) and 10(f).

d∈ (0.2,0.9). Delay-induced

E. Summary

Thus we have shown that the infected sites are healed or in

otherwordsthedisordersintheringnetworkacquirescoherent

chaotic oscillating behavior induced by time-delay 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 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

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 reaction-diffusion 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 Nash-Panfilov model [51]. In addition, delayed global

couplinghasbeenshowntoinducein-phasesynchronizationin

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).

066206-8

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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

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 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). 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)

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

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

delay-induced 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 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?

f?

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

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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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

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 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. IIIB, 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 (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.

066206-11

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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 size 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

discussedamechanismforthedelay-induced 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 DST-IRPHA 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(FP7-ICT-2009-C).

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