Document [original]

Chaos 29, 051103 (2019); https://doi.org/10.1063/1.5097570 29, 051103

Controlling chimera states via minimal

coupling modification

Cite as: Chaos 29, 051103 (2019); https://doi.org/10.1063/1.5097570

Submitted: 26 March 2019 . Accepted: 17 April 2019 . Published Online: 07 May 2019

Giulia Ruzzene, Iryna Omelchenko, Eckehard Schöll , Anna Zakharova , and Ralph G. Andrzejak

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Chaos ARTICLE scitation.org/journal/cha

Controlling chimera states via minimal coupling

modification

Cite as: Chaos 29, 051103 (2019); doi: 10.1063/1.5097570

Submitted: 26 March 2019 ·Accepted: 17 April 2019 ·

Published Online: 7 May 2019

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Giulia Ruzzene,1,a)Iryna Omelchenko,2Eckehard Schöll,2Anna Zakharova,2and Ralph G. Andrzejak1,3

AFFILIATIONS

1Department of Information and Communication Technologies, Universitat Pompeu Fabra, Carrer Roc Boronat 138,

08018 Barcelona, Catalonia, Spain

2Institut für Theoretische Physik, Technische Universität Berlin, Hardenbergstrasse 36, 10623 Berlin, Germany

3Institute for Bioengineering of Catalonia (IBEC), The Barcelona Institute of Science and Technology, Baldiri Reixac 10-12,

08028 Barcelona, Spain

a)Electronic mail: giulia.ruzzene@upf.edu

ABSTRACT

We propose a method to control chimera states in a ring-shaped network of nonlocally coupled phase oscillators. This method acts exclusively

on the network’s connectivity. Using the idea of a pacemaker oscillator, we investigate which is the minimal action needed to control chimeras.

We implement the pacemaker choosing one oscillator and making its links unidirectional. Our results show that a pacemaker induces chimeras

for parameters and initial conditions for which they do not form spontaneously. Furthermore, the pacemaker attracts the incoherent part of the

chimera state, thus controlling its position. Beyond that, we find that these control effects can be achieved with modifications of the network’s

connectivity that are less invasive than a pacemaker, namely, the minimal action of just modifying the strength of one connection allows one

to control chimeras.

Published under license by AIP Publishing. https://doi.org/10.1063/1.5097570

In networks of oscillators, chimera states are phenomena defined

as the coexistence of coherence and incoherence.1,2They were

first observed in 2002 by Kuramoto and Battogtokh who stud-

ied the dynamics of a ring-shaped network of nonlocally cou-

pled oscillators.1A chimera state is formed when the oscillators

spontaneously split into two complementary groups, one display-

ing an almost synchronous behavior and the other in which the

oscillators perform an erratic motion. Thus, the spatial symme-

try of the network’s equations is broken by the spatiotempo-

ral evolution of its dynamics. Since the initial discovery in ring

networks of nonlocally coupled oscillators, chimeras have been

observed for a variety of network node dynamics and network

coupling topologies.3–9The interest in the study of this fascinat-

ing phenomenon grew, thanks to the observation of chimeras

in experiments,5,10–16 and numerous conceptual links established

between chimera states on the one hand and natural and man-

made phenomena on the other.17–21 Many advances were made

in the understanding chimeras from a mathematical perspective

(see Ref. 22 and the references therein). For finite-size networks

of nonlocally coupled phase oscillators, chimera states are not

stable but can collapse to the synchronous state at any moment

in time.19,23,24 Another finite-size effect on chimera states is the

drifting of the groups within the network, which can be character-

ized as a Brownian motion.25 It has been shown that it is possible

to control these instabilities of chimera states.19,24,26–30 With the

expression controlling chimeras, we here mean the interactions

with the network aimed to influence the formation, the position,

and the collapse of the chimera state. Control methods include

closed feedback loops19,24,26,27 and open-loops.28–30

An open problem is to find the minimal action needed to

control chimera states. To address this problem, we choose to

act exclusively on the connectivity structure of the network. We

develop an open-loop mechanism based on the idea of a pace-

maker oscillator. We implement this mechanism selecting one

oscillator and gradually eliminating its incoming connections,

while maintaining its outgoing connections. We first show that

this modification can induce chimeras in cases in which they

do not form spontaneously. We then illustrate that the pace-

maker attracts the incoherent part of the chimera. Furthermore,

removing even a small fraction of the connections of one oscil-

lator or just lowering the strength of one connection is suffi-

cient to achieve control. These results also point to the fact that

Chaos 29, 051103 (2019); doi: 10.1063/1.5097570 29, 051103-1

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symmetry breaking is an essential part of chimera control. To fur-

ther strengthen this claim, we show that the opposite of the full

pacemaker, i.e., an oscillator with no outgoing links, produces

control effects qualitatively similar to what we obtain with the

pacemaker. The advantages of our method are its simple imple-

mentation and the possibility of controlling chimeras with min-

imal actions on the network coupling topology. We expect that

these aspects will make our method attractive for possible appli-

cations in which there is limited access to the system showing

chimera states.

I. INTRODUCTION

Methods to achieve control of chimera states have been imple-

mented in models featuring different types of oscillators. In 2014,

Sieber and co-workers published a closed-loop method for ring-

shaped networks of phase oscillators, which used a time-dependent

phase-lag parameter to prevent chimera states from collapsing to

the synchronized state.24 Another closed-loop method based on a

gradient dynamics that allows one to maintain the position of the

chimera state was proposed by Bick and Martens in 2015.26 In 2016,

Omelchenko et al.27 developed a feedback control mechanism, called

tweezers, to control chimera states in small networks of FitzHugh-

Nagumo and Van der Pol oscillators. This method uses two com-

ponents: a symmetric one to prevent the collapse of chimeras and

an asymmetric one to control their position.27 The tweezers mech-

anism was optimized in Ref. 31 allowing to control the size of the

domains forming the chimera state and the frequency difference

among the oscillators in each domain. Gambuzza and Frasca28 used

spatial pinning to control the position of chimera states in networks

of FitzHugh-Nagumo and phase oscillators. Isele et al. conducted

a study about control of the position of chimeras in networks of

oscillatory FitzHugh-Nagumo units.29 They introduced a barrier of

excitable units in the network, which attracts the incoherent region.29

In the work by Andrzejak et al.,19 closed-loop feedback control

schemes were used to suppress or promote the collapse of the chimera

to the synchronous state in networks of phase oscillators. Recently,

the possibility of controlling some features of chimera states in net-

works of Stuart-Landau oscillators acting on the initial conditions

and coupling scheme has been developed by Kalle et al.30 It was

also proven that it is possible to control not only classical phase

chimeras, but also amplitude chimeras which are observed in net-

works of Stuart-Landau oscillators.7Furthermore, in phase oscillator

networks with coupling functions involving higher order harmon-

ics, chimera states can be stabilized without external influence.32,33 All

these previous studies on control of chimeras rely on modifications

of parameters of the oscillators, and in some cases, these changes

are made according to information extracted from the system. In

real-world applications, however, it might be difficult to alter the indi-

vidual oscillators that form a network. Closed-loop feedback meth-

ods could also result unreliably, for example, when measurements of

system features are affected by noise.

With the goal of finding the minimal action needed to control

chimeras, we propose here an open-loop control mechanism that

avoids these issues and acts uniquely on the coupling topology of

the network, leaving unaltered the oscillators’ parameters. We con-

sider the Kuramoto-Sakaguchi model of nonlocally coupled phase

oscillators in a ring topology. Our control mechanism is based on

the idea of modifying the coupling topology so that a pacemaker

oscillator is present in the network. A pacemaker is an oscillator that

influences the other oscillators to which it is connected but is not

influenced by them. In other words, it is an oscillator whose links

are all unidirectional in the connectivity structure. Starting from this

extreme, we reduce the number of modified links, thus considering

modifications of the connectivity that are less and less invasive. We

finally push this mechanism to the limit and act only on one link. We

first remove it from the connectivity and then gradually increase its

strength until we restore the original connectivity. This allows us to

find the minimal intervention needed to control chimeras.

In what follows, we first review the model of a ring-shaped net-

work of phase oscillators, then we introduce our procedure to modify

the coupling structure of the network with different intensities of the

control mechanism. We present results regarding the formation of

chimera states and the control of their position. Finally, we conclude

with a brief discussion of the results.

II. COUPLED OSCILLATOR MODEL

We use a ring-shaped network of Nnonlocally coupled phase

oscillators [see Figs. 1(a) and 1(c)]. This network is described by

the following system of differential equations for the time-dependent

FIG. 1. Implementation of the full pacemaker. In panel (a), we show a network of

12 oscillators that are nonlocally coupled with b=4. The links of one oscillator

are highlighted to better show the nonlocal coupling configuration. In panel (b), we

show how we change the connectivity to implement the pacemaker: we choose

one oscillator [the one with highlighted links in panel (a)] and we make its link

unidirectional. The corresponding coupling matrices G(i,j)are shown in panels

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phases φj(t)of the oscillators:23,25

φj(t)=ω−1

k=1

G(j,k)sin φj(t)−φk(t)+α, (1)

G(j,k)=(1 if |j−k| ≤ b,

0 otherwise, (2)

where i,j=1, ...,N. The oscillators’ natural frequency ωis set to

be zero without loss of generality. The connectivity matrix Gcorre-

sponds to a rectangular coupling kernel with broadness 2b+1.23,25

The phase-lag parameter is set to α=1.46.23 Reflecting the peri-

odic boundary conditions of the network’s ring shape, all sums and

differences of indexes are to be understood modulo N.

To solve the differential equations, we used the 4-th order

Runge-Kutta method, with fixed sampling time of dt =0.05. We inte-

grated Eq. (1) starting from initial conditions uniformly distributed

in the interval [0 , 2π). To detect chimera states, we adapted an

algorithm proposed by Isele et al.,29 which is based on the global

Kuramoto order parameter and the mean phase velocity profiles

that characterize chimera states (see Appendix A). Following the

terminology introduced in Ref. 19, we refer to the two complemen-

tary groups forming the chimera states as the high coherence group

(HCG) and the low coherence group (LCG). In Figs. 2(a)–2(c), we

show three independent realizations of Eq. (1). Panels (d)–(f) are

their corresponding representations in terms of the HCG and LCG.

Panel (a) shows a typical chimera state and illustrates the drift of the

LCG and HCG. Panel (b) shows another chimera state that collapses

after a short time. Finally, for the realization in panel (c), no chimera

is formed and the oscillators synchronize after a short transient.

III. MODIFYING NETWORK CONNECTIVITY TO

CONTROL CHIMERAS

Our control mechanism acts on the connectivity matrix G

defined in Eq. (2). We implement the idea of a pacemaker oscillator

in the model in the following way. We decide to have the pacemaker

in position i, which corresponds to setting to zero all the elements of

the i-th row of G, except for the diagonal entry G(i,i). Accordingly,

the i-th oscillator does not receive any input and as a consequence

it oscillates at a constant angular frequency ˙

φi(t)= − sin(α). How-

ever, since the i-th column of Gis maintained, this constant frequency

is received by all oscillators within the coupling range bof oscilla-

tor i[see Fig. 1, panels (b) and (d)]. Subsequently, we implement

gradually less invasive modifications of the coupling matrix G. Like

we just described, in the pacemaker configuration, only the diagonal

element is maintained at G(i,i)=1. Starting from this most inva-

sive control, we then restore the pair of first off-diagonal elements

G(i,i−1)=G(i,i+1)=1, then the pair of second off-diagonal

elements G(i,i−2)=G(i,i+2)=1, etc. This process is contin-

ued until we set the elements G(i,i−b+1)=G(i,i+b−1)=1.

Therefore, at this stage only the elements G(i,i−b),G(i,i+

b)remain modified to zero. We refer to the case in which

FIG. 2. Uncontrolled chimera states drift along the network over time and may collapse to the synchronous state. The pacemaker stabilizes chimera states. In panels

(a)–(c), we display instantaneous phase velocities for three different realizations of Eq. (1) for N=35, b=12, α=1.46, ω=0. Panels (d)–(f) illustrate the division of the

corresponding solutions into the high coherence group (HCG) and the low coherence group (LCG). In panels (g)–(i), we display the effects of the presence of a pacemaker

in position 18 on the solutions shown in panels (a)–(c), respectively. The pacemaker was activated at the beginning of the simulations. Panels (j)–(l) are analogous to panels

(g)–(i), but here the pacemaker was activated after 150 dimensionless time units.

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all coefficients of the i-th row of Gare set to zero as “full

pacemaker” [Figs. 1(b) and 1(d)], and to the intermediate modi-

fications of Gdescribed above as “partial pacemaker”. The “pace-

maker intensity” ψis defined as the ratio between the number

of removed links and the initial number of bidirectional con-

nections of the pacemaker. The lowest possible nonzero value of

ψis 1/b, which corresponds to just two unidirectional links of

oscillator i. Finally, we set G(i,i−b)=1 and G(i,i+b)=ξ, where

ξis varied from 0 to 1. That means, for ξ=1, the unchanged

connectivity matrix Gis restored [see Eq. (2)].

The two rightmost columns of Fig. 2 show the effects of the full

pacemaker: attracting the LCG and preventing the collapse to the

synchronized state. In panels (g)–(i), we start the system with the

same initial conditions as in panels (a)–(c), respectively, but now a

pacemaker is present in position i=18. We see how the pacemaker

attracts the LCG. In panel (i), the collapse to the synchronous state

is avoided. Panels (j)–(l) show effects analogous to the ones in panels

(g)–(i) but now the pacemaker is activated only after 150 time units.

IV. TRIGGERING CHIMERA STATES

First, we use a pacemaker to induce chimera states for parame-

ters and initial conditions for which they do not form spontaneously

[see again Fig. 2, panels (c),(f),(i), and (l)]. We compare the per-

centage of chimeras obtained with different pacemaker intensities

ψand for every pacemaker intensity, we use the same set of initial

conditions. It is known that the lifetime of chimeras increases with

the number of oscillators N(Ref. 23) and the drifting increases with

decreasing N.25 Since our control aims to counteract these instabil-

ities, we focus on small networks of up to N=50, and we insert a

pacemaker in position i=1 of the network. To detect chimeras, we

used the algorithm described in Appendix A. For each value of the

network size, we consider all the possible values of coupling range b

varying from local coupling b=1 to global coupling b=N−1

2when

Nis odd, or from b=1 to the maximum possible value b=N−2

when Nis even. For this section, integration was performed over

4·105sampling times, corresponding to 2 ·104dimensionless time

units, and all analyses were performed over an evaluation interval

of 2500 dimensionless time units I1=[17 500, 20 000]. We consid-

ered 100 independent realizations for all pairs of values of network

size Nand coupling range b. The results are displayed in Fig. 3.

We clearly see that the region of the parameter space in which

chimeras are detected is broader when a pacemaker is present in the

network [see Figs. 3(b)–3(d)]. When no control is applied to the net-

work [Fig. 3(a)], no chimeras are found for N<32 and for relative

coupling range b/Noutside the interval [0.25, 0.4]. This is due to the

presence of chimera states whose lifetime is shorter than the inte-

gration time and to initial conditions that collapse immediately to

the synchronous state without ever forming a chimera state. In the

region where chimera states are present for the unchanged connec-

tivity [Fig. 3(a)], we observe an increase in their percentage when the

pacemaker is present [Figs. 3(b)–3(d)]. In particular, a low intensity

pacemaker with, obtained cutting only two incoming links, already

induces chimeras for small values of N<32 [see Fig. 3(b)]. For pace-

maker intensity ψ≈0.5 [panel (c)], we obtain results that are close

to the case of the full pacemaker [panel (d)].

V. CONTROLLING THE POSITION OF CHIMERA STATES

Secondly, we study the control of the position of chimera

states. In Fig. 2(a), we see that the two complementary groups

LCG and HCG drift along the network.23 This drifting is par-

ticularly pronounced for small networks and it was character-

ized as a Brownian motion.23 Figures 2(g)–2(l) show how the full

pacemaker attracts the LCG, thus preventing its chaotic motion

along the network. We study how different pacemaker intensi-

ties ψaffect the chimera’s position. To do this, we set N=

50 and b=18. For these parameters, the occurrence of chimera

states is more likely in comparison with smaller sizes N, while

the drifting of the LCG and HCG is still substantial. The pace-

maker is in position i=25. For this section, integration was per-

formed over 2 ·105sampling times, corresponding to 104dimen-

sionless time units, and all analyses were performed over an evalu-

ation interval of 2500 dimensionless time units I2=[7500, 10 000].

Following Ref. 34, if at some point the system synchronized, we

started over with new initial conditions. For every time step twe

define the position of the center of the LCG denoted by l(t)which

varies in the set Lof numbers from 0.5 to N=50 in steps of 0.5

(see Appendix B). Furthermore, we calculate the size s(t)of the LCG

and the distance d(t)=l(t)−25 of its center from the pacemaker

position i=25 (see Appendix B). In Fig. 4(a), we show the temporal

FIG. 3. A pacemaker triggers chimeras for initial conditions for which they do not form spontaneously. Comparison of the number of chimera states observed for different

values of the network size Nand coupling broadness b(the other network parameters are α=1.46, ω=0). For each pair of values, we solved the model 100 times without

control [panel (a), ψ=0], with low control intensity ψ=1

b[panel (b)], intermediate control intensity ψ≈0.5 [panel (c)] and with a full pacemaker corresponding to ψ=1

[panel (d)].

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FIG. 4. The center of the LCG is attracted by pacemakers of different intensities.

Panel (a) is the temporal evolution of the center of the LCG for the same realization

of Eq. (1) with different pacemaker intensities, starting with no control up to the

full pacemaker. In panel (b), we show four independent realizations of Eq. (1) with

a low pacemaker intensity.

evolution of the position of the center of the LCG for four solutions

of Eq. (1) corresponding to four different pacemaker intensities ψ=

0, 0.06, 0.5, 1, where 0.06 =1

b. The initial conditions were the same

in every realization. In panel (b), we show four different realiza-

tions of Eq. (1) with a low pacemaker intensity ψ=0.06. This is

the lowest possible value in our setting, as it corresponds to only two

unidirectional links. In both panels, one can appreciate the attracting

effect of the pacemaker on the center of the LCG. As soon as we

switch on a pacemaker, even with a low intensity, the center of the

LCG is attracted by the pacemaker, as it becomes evident from the dif-

ference in the characteristics of the blue to the red curve in Fig. 4(a).

The control effect becomes stronger for increasing pacemaker inten-

sity [purple and black curves in Fig. 4(a)]. In Fig. 4(b), we see how

the weakest possible pacemaker with ψ=0.06 attracts the center of

the LCG for different initial conditions, but the motion of center is

more pronounced in these curves than in the black curve in panel

(a), which corresponds to the full pacemaker.

Next, we study the position of the center of the LCG throughout

100 independent realizations for each pacemaker intensity. For every

time step tand every control intensity ψ, we thus have a distribution

C(l(t),ψ) of the position of the LCG center. For the uncontrolled

system, there is no preferred position for the LCG of the chimera

state over time and across different realizations. The distribution

C(l(t), 0), corresponding to the uncontrolled system, is shown in

Fig. 5(a). As we can see from the blue curve in Fig. 5(e), the dis-

tribution C[l(t), 0] is uniform on Lduring the interval I2. As soon

as we break the symmetry of the coupling topology of the oscilla-

tor network, the distribution of the position of the center changes

and we see how the center position is attracted by the partial or full

pacemaker. Figures 5(b)–5(d) shows the effect of increasing the pace-

maker intensity ψin position 25. In Fig. 5(b), only 2 incoming links

of oscillator 25 were cut, while 18 links were removed in panel (c)

(corresponding to ψ=0.06 and ψ=0.5, respectively). Figure 5(d)

corresponds to 36 links removed, i.e., the full pacemaker (ψ=1).

FIG. 5. Cutting incoming links of one oscillator allows one to control the position of the chimera state. We show the effects on chimera states of the presence of a partial/full

pacemaker in position 25, in a network with N=50, b=18, α=1.46. The color scale in panels (a)–(d) represents values of the distributions of the LCG center C[l(t),ψ]

over 100 independent realizations. Panel (a) shows how the center of the low coherence group (LCG) is positioned without any control, that is with an unmodified matrix G.

In (b), the pacemaker intensity is ψ=0.06, while in (c), we have ψ=0.5. In panel (d), the configuration corresponding to the full pacemaker, i.e., ψ=1 is displayed. In

panel (e), we show the corresponding time averages of the spatial distributions C(l(t),ψ) of the LCG center position over the interval I2.

Chaos 29, 051103 (2019); doi: 10.1063/1.5097570 29, 051103-5

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FIG. 6. Partial pacemaker is sufficient

to control the chimera’s position. We show

effects of different control intensities on

the position of chimera states. Panel (a)

shows a plot of the order parameters

0(ψ). In panel (b), we display values of

the distance d(ψ) of the LCG center from

the pacemaker position i=25. The aver-

age size of the LCG is shown in panel (c),

depending on the pacemaker intensity ψ.

Panels (d)–(f) are analogous to (a)–(c),

respectively, but here 0,d,sare calcu-

lated for varying ξ=G(25, 7). For ξ=

1 we have the uncontrolled system, while

ξ=0 corresponds to one unidirectional

link. The network size is N=50, the cou-

pling range is b=18 and the phase lag is

α=1.46. All time averages were calcu-

lated over the evaluation interval I2. The

error bars display the standard deviation

of the averages over the 20 sets of 100

independent initial conditions.

The control effect is clearly visible already in panel (c). Looking at the

time averaged spatial distributions in panel (e), one can also observe

how these become narrower with a pronounced peak around posi-

tion 25 as we approach the case of the full pacemaker [see Fig. 5(e),

black curve].

To further quantify the effects of our control mechanism, we

define the following order parameter:

0(t,ψ) =

2NX

l∈L

C(l(t),ψ)eiθl

, (3)

where θl=2πl

Nfor l∈L,tis in the evaluation interval I2, and

|·| is the modulus of complex numbers. We calculate the order

parameter 0for 20 distributions of the LCG center position which

were obtained from 20 sets of 100 independent initial conditions.

For every pacemaker intensity ψ, we obtained order parameters

01(t,ψ),...,020(t,ψ), average distances d1(t,ψ),...,d20(t,ψ), and

average LCG sizes s1(t,ψ),...,s20(t,ψ). We then calculated the

mean of their temporal averages over the interval I2, thus obtain-

ing functions of the pacemaker intensity 0(ψ),d(ψ),s(ψ). These

values are shown in Figs. 6(a)–6(c). In panel (a), we see how the

value of 0(ψ) sharply increases when passing from pacemaker inten-

sity ψ=0 to ψ=0.06 (the lowest possible intensity in this setting),

and then increases more slowly toward the value corresponding to

the full pacemaker. The results in Fig. 6 confirm that the pacemaker

attracts the LCG, in the sense that the distance d(ψ) of the LCG cen-

ter from the pacemaker position decreases as the pacemaker intensity

ψincreases. In panel (c), another effect of our control mechanism is

shown: an increase in the size s(ψ) of the incoherent group LCG.

The last step of our analysis is to modify only one value of

G. We repeated the analysis described before, setting the value

of the coefficient G(25, 7)=ξ, where ξgoes from 1 (unchanged

matrix G) to 0 (one unidirectional link). The results are repre-

sented in Figs. 6(d)–6(f). In panel (d), we observe an increase of the

order parameter 0(ξ) when the modification of the coupling matrix

becomes stronger. In particular we see that, as we decrease the value

of G(25, 7)=ξ(which corresponds to increasing the control inten-

sity), the distribution of the position of the LCG center becomes more

and more similar to the one obtained in Fig. 5(b), where two links

were made unidirectional, as it is reflected in the increasing values of

0(ξ) [Fig. 6(d)] and the decreasing values of the distance d(ξ) of the

LCG center from the pacemaker [Fig. 6(e)]. The effect on the size of

the LCG shown in Fig. 6(f) is not as pronounced as it was in the case

of the transition from no control to the full pacemaker.

VI. ALTERNATIVE SYMMETRY BREAKING MECHANISM

To further understand which are the important aspects of the

control mechanism introduced in this paper, we analyze here what

happens when we reverse the pacemaker idea. That is, we select

one oscillator with index iand we cut all of its outgoing links. In

terms of the coupling matrix Gof Eq. (2), this corresponds to setting

G(j,i)=0 for a fixed column iand for all j6= i. This new configura-

tion is equivalent to isolating oscillator ifrom the rest of the network,

but we continue to show its dynamics in our results. Figure 7 shows

what happens when we repeat the simulations of Fig. 2 substituting

the pacemaker with the new symmetry breaking configuration which

consists in cutting the outgoing link of oscillator 18. Panels (a)–(c) of

Fig. 7 are replicas of (a)–(c) of Fig. 2 (uncontrolled chimeras), and we

can see how the remaining panels of Fig. 7 are qualitatively similar to

the corresponding panels obtained in Fig. 2 using the full pacemaker.

These findings provide further evidence that the essential element for

chimera control is the disruption of the spatial symmetry of the ring

network.

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FIG. 7. Cutting the outgoing links of one oscillator acts like a full pacemaker. In

panels (a)–(c), we display instantaneous phase velocities for three different real-

izations of Eq. (1) for N=35, b=12, α=1.46, ω=0 (same initial conditions

as in Fig. 2). In panels (g)–(i), we display the effects of the symmetry breaking

described in Sec. VI with i=18 on the solutions shown in panels (a)–(c), respec-

tively. The symmetry breaking was activated at the beginning of the simulations.

Panels (j)–(l) are analogous to panels (g)–(i), but here the symmetry breaking was

activated after 150 dimensionless time units.

VII. DISCUSSION

We introduced a method based on the idea of a pacemaker

oscillator which allows one to control chimera states in small net-

works of phase oscillators. By varying the control intensity, we were

able to investigate which is the minimal action needed to control

chimera states. We found that modifying only one coefficient in

the connectivity matrix is enough to control the chimera’s position.

Appealing features of our method are the simplicity of its imple-

mentation, which lies in the fact that no feedback from the system

is needed and that it does not intervene on the oscillators’ param-

eters. Interestingly, there are strong analogies between our results

and the ones elaborated by Isele et al.29 Although they use a dif-

ferent model and a completely different control mechanism, they

also observed that the symmetry breaking element in the network

attracts the incoherent group and stabilizes the chimera state. The

effects of symmetry breaking in the evolution of chimera states also

emerge in the recent work by Yao et al. in Ref. 35. They perturbed

the dynamics of a ring-shaped network of phase oscillators by select-

ing a target oscillator and forcing it to have a fixed phase difference

with respect to the local mean field of its neighbors. This perturbation

induces the incoherent group to be centered around the target oscil-

lator. Our results confirm the occurrence of this self-adaptation35 of

the chimera position and generalize the findings in Ref. 35 showing

that weaker changes in the network are sufficient not only to control

the chimera’s position, but also to trigger chimeras for parameters

and initial conditions for which they do not form spontaneously.

Moreover, the full pacemaker can be used to generate a chimera state

after the system has collapsed to synchronous solution, as we show

in Fig. 2.

It is worth to point out that the idea of a pacemaker was already

introduced in the study of synchronization of the Kuramoto model.

In Refs. 36 and 37, a pacemaker is used to synchronize random net-

works of phase oscillators. We showed that the same mechanism

produces the opposite effect for the Kuramoto-Sakaguchi model. In

fact, it promotes the existence of chimera states when the oscillators

are nonlocally coupled. This comparison underlines the importance

of the interplay of nonlocal coupling, the phase lag, and the control

mechanism in the control of chimera states. Given that our method

acts exclusively on the connectivity of the network and not on the

intrinsic dynamics of the oscillators, we conjecture that it may work

also for networks made of different types of oscillators and more

complex topologies.

ACKNOWLEDGMENTS

This project was supported by the Volkswagen Foundation,

the Spanish Ministry of Economy and Competitiveness (Grant

No. FIS2014-54177-R), the CERCA Program of the Generalitat de

Catalunya (G.R. and R.G.A.), and the European Union’s Horizon

2020 program under the Marie Sklodowska-Curie Grant Agreement

No. 642563 (R.G.A). We acknowledge support from the Deutsche

Forschungsgemeinschaft (DFG) in the framework of the SFB 910,

Projektnummer 163436311 (I.O., E.S., and A.Z.).

APPENDIX A: ALGORITHM FOR CHIMERA DETECTION

We used the following algorithm for the detection of chimera

states.29 We integrate Eq. (1) to obtain a solution φj(t)for 400 000

time steps of width dt =0.05. In the absence of control, we calculate

the Kuramoto global order parameter

R(t)=1

k=1

eiφk(t),

and we compute its temporal average R=hR(t)iI1] over the evalua-

tion interval I1=[17 500, 20 000] described in Sec. IV. We compute

the mean phase velocities

j=dφj(t)

dt I1

for j=1, ...,N,

and we determine the range of the mean phase velocity profile:2

=max

j=1,...,Nj−min

j=1,...,Nj.

In the case an oscillator acts as a pacemaker, we exclude this oscillator

from the averaging.

If we find that R∈[0.65, 0.8]and ∈[0.1, 1], then the solu-

tion φj(t)is classified as a chimera state. These threshold values are

based on preanalysis results. The other possible scenarios for Eq. (1)

are solutions which are completely incoherent and solutions in which

the oscillators are all synchronized or almost all synchronized. The

former are discarded by the lower bound on the order parameter.

The latter are also ruled out because R=1 if all oscillators are syn-

chronized. The condition on the mean phase velocity comes into play

when we have values of Rclose to 0.65. In this case, in the solu-

tion there is no clear distinction between coherent and incoherent

group and  < 0.1. The upper-bound for discards situations that

Chaos 29, 051103 (2019); doi: 10.1063/1.5097570 29, 051103-7

Published under license by AIP Publishing.

Chaos ARTICLE scitation.org/journal/cha

are rarely observed in the presence of high coupling and a pacemaker,

in which a chimera state is not formed but the synchronized state is

disturbed by few oscillators that have a different frequency from the

synchronized block.

APPENDIX B: HCG AND LCG GROUPS

In the analysis presented in Sec. V, we used the concepts of

high coherence group (HCG) and low coherence group (LCG) that

form a chimera state. These two groups were defined following the

algorithm presented in Ref. 19. In what follows, all indexes and sums

of indexes are to be understood modulo N. For the j-th oscillator,

we consider its two nearest neighbors on each side, that is oscillators

j−2, j−1, j+1, j+2. For every time instant t, we calculate

the pairwise local order parameters Rj+2,j+1(t),Rj+1,j(t),Rj,j−1(t),

Rj−1,j−2(t), where

Ra,b(t)=

2eiφa(t)+eiφb(t)

We define the following function:

χ(j,t)=









1 if Rj+2,j+1(t),Rj+1,j(t),

Rj,j−1(t)and Rj−1,j−2(t) > 0.995,

0 otherwise.

(B1)

At time t, the HCG is formed by all oscillators with indexes jsuch that

χ(j,t)=1, the LCG is formed by the remaining oscillators. Once we

defined the HCG and LCG, we can define the border of the LCG and

its center. For every time t∈I2, we look for indexes ib,jbwhich satisfy

the following conditions:

χ(ib−1, t)=1 and χ(ib,t)=χ(ib+1, t)=0,

χ(jb−1, t)=χ(jb,t)=0 and χ(jb+1, t)=1. (B2)

If such indices exist, we say that the border of the LCG is

B(t)= {ib,jb}. Apart from the main LCH, it may happen that there

are small islands of incoherent oscillators inside the HCG. In this

case, we find multiple pairs of indexes i1

b,j1

b,i2

b,j2

b,...,in

b,jn

bsatisfying

the conditions above. We choose index kcorresponding to the biggest

incoherent group and the border is B(t)= {ik

b,jk

b}. The position l(t)of

the center of the LCG at time tis defined according to the following

rule:

• if ib<jb, then l(t)=ib+jb

• if ib>jb, then l(t)=ib+jb+N

2mod 50.

The center position l(t)defined above can be an integer or half-

integer between 0.5 and the network size N. The size of the LCG at

time tis s(t)=50 −P50

k=1χ(i,t).

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