Chaos 27, 114320 (2017); https://doi.org/10.1063/1.5008385 27, 114320
© 2017 Author(s).
Time-delayed feedback control of coherence
resonance chimeras
Cite as: Chaos 27, 114320 (2017); https://doi.org/10.1063/1.5008385
Submitted: 25 April 2017 . Accepted: 13 September 2017 . Published Online: 26 October 2017
Anna Zakharova, Nadezhda Semenova , Vadim Anishchenko, and Eckehard Schöll
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Time-delayed feedback control of coherence resonance chimeras
Anna Zakharova,
1,a)
Nadezhda Semenova,
2
Vadim Anishchenko,
2
and Eckehard Sch€
oll
1
1
Institut f€ur Theoretische Physik, Technische Universit€at Berlin, Hardenbergstr. 36, 10623 Berlin, Germany
2
Department of Physics, Saratov State University, Astrakhanskaya str. 83, 410012 Saratov, Russia
(Received 25 April 2017; accepted 13 September 2017; published online 26 October 2017)
Using the model of a FitzHugh-Nagumo system in the excitable regime, we investigate the influence
of time-delayed feedback on noise-induced chimera states in a network with nonlocal coupling, i.e.,
coherence resonance chimeras. It is shown that time-delayed feedback allows for the control of the
range of parameter values where these chimera states occur. Moreover, for the feedback delay close
to the intrinsic period of the system, we find a novel regime which we call period-two coherence
resonance chimera. Published by AIP Publishing. https://doi.org/10.1063/1.5008385
Coherence resonance chimeras in nonlocally coupled net-
works of excitable elements represent partial synchroni-
zation patterns composed of spatially separated domains
of coherent and incoherent spiking behavior, which are
induced by noise. These patterns are different from clas-
sical chimera states occurring in deterministic oscillatory
systems and combine properties of the counter-intuitive
phenomenon of coherence resonance, i.e., a constructive
role of noise, and chimera states, i.e., the coexistence of
spatially synchronized and desynchronized domains in a
network of identical elements. Another distinctive feature
of the particular type of chimera we study here is its
alternating behavior, i.e., periodic switching of the loca-
tion of coherent and incoherent domains. Applying
time-delayed feedback, we demonstrate how to control
coherence resonance chimeras by adjusting delay time
and feedback strength. In particular, we show that feed-
back increases the parameter intervals of existence of
chimera states and has a significant impact on their alter-
nating dynamics leading to the appearance of novel pat-
terns, which we call period-two coherence resonance
chimera. Since the dynamics of every individual network
element in our study is given by the FitzHugh-Nagumo
(FHN) system, which is a paradigmatic model for
neurons in the excitable regime, we expect wide-range
applications of our results to neural networks.
I. INTRODUCTION
The processes occurring in nature are inevitably affected
by internal and external random fluctuations, i.e., noise.
Even at a relatively low intensity, noise can significantly
influence the behavior of a dynamical system. Noise can
play a constructive role and give rise to new dynamic behav-
ior, e.g., stochastic bifurcations, stochastic synchronization,
or coherence resonance.
1–7
The counter-intuitive effect of
coherence resonance describes a non-monotonic behavior of
the regularity of noise-induced oscillations in the excitable
regime, leading to an optimum response in terms of regular-
ity of the excited oscillations for an intermediate noise
strength. It has been previously shown that coherence reso-
nance can be modulated by applying time-delayed feedback in
excitable
8,9
as well as in non-excitable systems.
10,11
In particu-
lar, for appropriate choices of time delay, either suppression or
enhancement of coherence resonance can be achieved.
Recently, a new type of coherence resonance, coherence
resonance chimeras, has been discovered.
12,13
It combines
temporal features of coherence resonance, and spatial prop-
erties of chimera states,
14,15
i.e., coexistence of spatially
coherent and incoherent domains in a network of identical
elements. This phenomenon is distinct from classical chime-
ras, which occur in deterministic oscillatory elements.
16,17
It
is well known that in the presence of time delay, simple
dynamical systems can exhibit complex behavior, such as
delay-induced bifurcations,
18
delay-induced multistability,
19
stabilization of unstable periodic orbits,
20
or stationary
states,
21
to name just a few examples. Chimera states have
been investigated for noisy systems
22
and delayed systems as
well. In general, chimera patterns tend to form clusters in the
presence of time delay.
23,24
The role of time-delayed cou-
pling has been previously investigated in two-population net-
works of oscillators.
25
In particular, it has been reported that
coupling delay induces globally clustered chimera states in
which the coherent and incoherent regions span both popula-
tions.
26,27
Experimental evidence for chimera states in sys-
tems with time delay has been provided for chemical
oscillators
28
and electronic or optoelectronic systems.
29,30
Internal delayed feedback has been shown to induce chimeras
in systems of globally coupled phase oscillators
31
and laser
networks.
32
Chimera states in the presence of both delayed
feedback and noise have been investigated in Ref. 33.
Here we investigate the interplay of noise and time-
delayed feedback in a network of nonlocally coupled excit-
able elements and mainly focus on the role of feedback for
coherence resonance chimeras. A distinctive feature of this
type of chimera is that it is induced by noise and occurs in a
certain restricted interval of noise intensity and systems
parameters. The question we address here is whether these
intervals can be increased by introducing time-delayed feed-
back. By exploring the impact of time delay, we uncover the
mechanisms to control coherence resonance chimeras by
time-delayed feedback. Our results show that applying
a)
1054-1500/2017/27(11)/114320/8/$30.00 Published by AIP Publishing.27, 114320-1
CHAOS 27, 114320 (2017)
time-delayed feedback promotes the occurrence of coher-
ence resonance chimeras and induces new regimes.
II. MODEL
We consider a ring of Nidentical nonlocally coupled
FitzHugh-Nagumo (FHN) systems with time-delayed feed-
back in the presence of Gaussian white noise
edui
dt ¼uiu3
i
3viþr
2RX
iþR
j¼iR
buuðujuiÞþbuvðvjviÞ
þceðuiðtÞuiðtsÞÞ;
dvi
dt ¼uiþaiþr
2RX
iþR
j¼iR
bvuðujuiÞþbvv ðvjviÞ
þffiffiffiffiffiffi
2D
pniðtÞ;(1)
where u
i
and v
i
are the activator and inhibitor variables,
respectively, i¼1,…, Nand all indices are modulo N,ris
the coupling strength, Ris the number of nearest neighbours
in each direction on a ring. We also introduce a coupling
range which is the normalized number of nearest neighbours
r¼R/N, where Nis the total number of elements in the
network. Further, niðtÞ2Ris Gaussian white noise, i.e.,
hniðtÞi¼0 and hniðtÞnjðt0Þi¼dijdðtt0Þ;8i;j, and Dis the
noise intensity. The feedback term is characterized by time
delay sand strength c. A small parameter responsible for the
time scale separation of fast activator and slow inhibitor is
given by e>0 and a
i
defines the excitability threshold. For
an individual FHN element, it determines whether the system
is excitable, or oscillatory. For jaij<1, the system is in the
oscillatory regime where the steady state is unstable and self-
sustained oscillations are observed. For jaij>1, the system is
in the excitable regime and characterized by a locally stable
steady state. It is important to note that the FHN system can
also exhibit excitability in the oscillatory regime prior to
canard explosion (0.995 <a<1), where a sufficiently large
perturbation can trigger a spike on top of small-amplitude sub-
threshold oscillations emerging from the supercritical Hopf
limit cycle bifurcation at a¼1. In the present study, we
assume that all elements are in the excitable regime close to
the threshold (aia¼1:001 except for Figs. 8–13). Equation
(1) contains not only direct but also cross couplings between
activator (u) and inhibitor (v) variables, which is modeled by a
rotational coupling matrix
34
B¼buu buv
bvu bvv
¼cos /sin /
sin /cos /
;(2)
where /2½p;pÞ. Here we fix the parameter /¼p=2
0:1. In the absence of time delay s¼0, chimera states have
been found for this value of /in both the deterministic oscil-
latory
34
and the noisy excitable regime.
12,13
Moreover, it has
been shown that chimera states occurring in the excitable
regime
12,13
are different from those detected in the oscilla-
tory regime.
34
In the presence of Gaussian white noise, a
special type of chimera state called coherence resonance
chimera appears in a ring of Nnonlocally coupled excitable
FHN systems (Fig. 1).
In the present work, to control these patterns, we intro-
duce time-delayed feedback to the activator variable in Eq.
(1). For that purpose, we fix all the parameters of the system
in the regime of coherence resonance chimera and vary those
characterizing the feedback term: cand s. For c¼0, Eq. (1)
demonstrates coherence-resonance chimeras with the period
T4:76. This regime can also be observed in the presence
of time-delayed feedback for c¼0.2, s¼1.0 and is shown as
a space-time plot color-coded by the variable u
i
in Figs. 1(a)
and 1(b). One can clearly distinguish the regions of coherent
and incoherent spiking.
To characterize spatial coherence and incoherence of
chimera states, one can use the local order parameter
35,36
Zk¼1
2dZX
jjkjdZ
eiHj
;k¼1;…N;(3)
where the geometric phase of the j-th element is defined by
Hj¼arc tanðvj=ujÞ
34
and Z
k
¼1 and Zk<1 indicate coher-
ence and incoherence, respectively. Figure 1(c) represents a
space-time plot color-coded by Z
i
and illustrates coexistence
of coherent and incoherent domains with the latter character-
ized by values of Z
i
noticeably below unity (dark regions).
One of the main features of these noise-induced chimera
states is their alternating behavior which is absent in the
oscillatory regime without noise. In more detail, the incoher-
ent domain of the chimera pattern switches periodically its
position on the ring, although its width remains fixed [Figs.
1(b) and 1(c)]. This property has been previously described
in Ref. 12 and the explanation based on the time evolution of
the coupling term has been provided in Ref. 13. Taking into
account that the system (1) involves both direct and cross-
couplings between activator uand inhibitor vvariables, in
FIG. 1. (a), (b) Space-time plots and (c) local order parameter for the
coherence-resonance chimera. Initial conditions: randomly distributed on
the circle u2þv2¼4. Parameters: N¼500, e¼0:05;/¼p=20:1,
a¼1.001, r¼0:4, r¼0.2, D¼0.0002, c¼0:2;s¼1:0.
114320-2 Zakharova et al. Chaos 27, 114320 (2017)
total we have four coupling terms. It turns out that coupling
terms form patterns shown as space-time plots in Figs.
2(a)–2(d).
The crucial point is that the coupling acts as an addi-
tional term and shifts the nullclines of every individual ele-
ment of the network. The coupling term with the strongest
impact corresponds to cross-coupling for the variable v[Fig.
2(c)]. It means that the coupling significantly influences the
_
u¼0 nullcline and shifts the threshold parameter awhich is
responsible for the excitation. As a result for a certain group
of nodes, the threshold becomes lower due to coupling, and
the probability of being excited by noise increases.
Therefore, the elements of this group are the first to start the
large excursion in the phase space and experience random
spiking. The elements constituting the rest of the network
spike coherently since they are pulled by already excited
nodes and are, therefore, excited by coupling and not by
noise. This scenario can also be obtained for the system Eq.
(1) in the presence of time-delayed feedback (Fig. 2). Due to
the feedback an additional term appears in Eq. (1) and should
be taken into account. Its evolution in time for all nodes of
the network is shown in Fig. 2(e). The color-code bar clearly
indicates that the values of the feedback term are larger than
those of the coupling terms. However, for the chosen value
of delay time s¼1.0 the feedback does not have any
essential impact on the behavior of coherence resonance chi-
meras since it is less than the intrinsic period of oscillations
T¼4.76 (Figs. 1and 2).
For the better understanding of this alternating dynamics
in the presence of time-delayed feedback, we study the
impact of the coupling on activator and inhibitor nullclines
for selected nodes of the system Eq. (1). In particular, we
consider a sequence of phase portraits for the nodes i¼241
(red dot) and i¼1 (blue dot) which belong to the incoherent
and coherent domains, respectively (Fig. 3). First, all the ele-
ments are located near the steady state [Fig. 3(a)]. After a
while, the vertical nullcline of the node i¼241 is shifted to
the left of the value u¼–a¼–1.001 due to positive coupling
term [panel (b)]. Consequently, this node can be more easily
excited by noise [panel (c)]. Due to nonlocal coupling the
excited node pulls its neighbours and they also start spiking.
The coupling can also shift the vertical nullcline to the right
of the value u¼–a¼–1.001 [panel (d)].
Similar results have been previously obtained for the
case without time-delayed feedback.
13
Therefore, for the
strength c¼0.2 and time delay s¼1.0 the feedback does not
have an impact on the nullclines. Consequently, coherence
resonance chimeras observed for small time delay of the
feedback are the same as in the case without feedback.
III. DYNAMIC REGIMES IN THE PRESENCE OF
TIME-DELAYED FEEDBACK
Since our main goal is to study the impact of time-
delayed feedback, we now choose the parameters of the sys-
tem in the regime of coherence resonance chimera and vary
FIG. 2. Space-time plots of coupling terms for u
i
and v
i
variables in the
coherence-resonance chimera regime: (a) direct coupling for the u
i
variable,
(b) cross-coupling for the u
i
variable, (c) cross-coupling for the v
i
variable,
(d) direct coupling for the v
i
variable, and (e) space-time plot of the delay
term. Parameters: N¼500, e¼0:05;/¼p=20:1, a¼1.001, r¼0:4,
r¼0.2, D¼0.0002, and c¼0:2;s¼1:0.
FIG. 3. Activator and inhibitor nullclines _
uiand _
vi, respectively, for the
selected nodes i¼241 (left column) and i¼1 (right column) of the system
Eq. (1) in the coherence-resonance chimera regime for (a) t¼997.4, (b)
t¼997.6, (c) t¼997.7, and (d) t¼998.4. Parameters: N¼500, e¼0:05,
a¼1.001, r¼0:4, r¼0.2, D¼0.0002, c¼0:2;and s¼1:0.
114320-3 Zakharova et al. Chaos 27, 114320 (2017)
only the feedback parameters cand s. For fixed feedback
strength c¼0.4, we observe the change of dynamic regimes
by tuning the delay time s. For s¼3.6, all the nodes of the
network spike coherently, i.e., in-phase synchronization
occurs [Fig. 4(a)]. The feedback with s¼2.2 shifts the sys-
tem into the regime which is incoherent in space and peri-
odic in time: all the nodes demonstrate spiking behavior, but
the spiking events of the neighboring nodes are not corre-
lated [Fig. 4(b)].
To gain a general view of the dynamics in the network
of nonlocally coupled noisy excitable elements in the pres-
ence of time-delayed feedback, we construct the map of
regimes of the system Eq. (1) in the (c,s) parameter plane
(Fig. 5). For visualization reasons, we have divided the map
into two panels: panel (a) corresponds to the sinterval from
0 to 7 and includes the values sT, where Tis the period of
the dynamics without delay (T4.76); panel (b) corresponds
to larger values of sincluding s2T9.52.
Note that the other parameters of the network are cho-
sen in the coherence resonance chimera state which now
occurs only for certain intervals of delay time s. We detect
three main regions [yellow (light-grey) in Fig. 5] separated
by in-phase synchronization domains [red (dark-grey)
regions in Fig. 5] and regimes of spatially incoherent spik-
ing (hatched regions Fig. 5). Although the map of regimes
is dominated by various oscillatory patterns, for relatively
small feedback strength c<0.2 and time delay 3.2 <s<4,
we also observe a small regime of steady state [white
region in Fig. 5(a)].
Moreover, the diagram is characterized by multistabil-
ity since spatially incoherent spiking can coexist with
chimera states or in-phase synchronization. The overall
structure of the map resembles a sequence of synchroniza-
tion tongues although there are no clear resonances for
delay times equal to the multiples of the intrinsic period
T4.76. Nevertheless, applying the feedback with delay
time s2T9.52 does not change the dynamics dramati-
cally, and the regime of coherence resonance chimera
is still observed for a wide range of feedback strength
(Fig. 5).
IV. IMPACT OF THE FEEDBACK ON COHERENCE
RESONANCE CHIMERA EXISTENCE: NOISE
INTENSITY RANGE
Without feedback, as previously reported,
12,13
coher-
ence resonance chimeras are observed for a certain restricted
interval of noise intensity 0:000062 D0:000325 for the
following parameters of the system: N¼500, e¼0:05,
a¼1.001, /¼p=20:1, r¼0.2, r¼0.4 (this set of param-
eters is fixed throughout this section). Time-delayed feed-
back modifies this interval. To illustrate this effect, we
consider two cases: c<0.5 and c>0.5 which allows for a
better understanding of the impact of feedback strength on
this interval. Also for the two values of parameter c,we
choose different delay times sfrom all three regions of coher-
ence resonance chimeras shown in the (s,c) plane in Figs.
5(a) and 5(b). Time-delayed feedback slightly changes the
range of noise intensity values where chimera states occur in
the system (1) for both considered values of feedback
strength: c¼0.2 (Fig. 6) and c¼0.6 (Fig. 7).
For rather weak feedback strength c¼0.2, the interval
of existence of chimera patterns is enlarged for all the con-
sidered delay times. Interestingly, the right boundary of this
interval can be shifted in the direction of stronger noise [Fig.
6(b)] as well as in the direction of lower noise intensities
[Figs. 6(a),6(d), and 6(e)] and remains almost unchanged for
delay time s¼4.76 T[Fig. 6(c)]. Therefore, by appropri-
ately choosing the feedback delay time, one can adjust the
value of noise intensity for which spatially incoherent spik-
ing replaces coherence resonance chimeras within the
FIG. 4. Space-time plots for the variable u
i
(left panels) and local order
parameter Z
i
in the regime of (a) complete in-phase synchronization for c¼
0:4;s¼3:6 and (b) spatial incoherence for c¼0:4;s¼2:2. Other parame-
ters: N¼500, e¼0:05, a¼1.001, /¼p=20:1, D¼0.0002, r¼0.2, and
r¼0:4.
FIG. 5. Dynamic regimes in the (s;c) parameter plane. Red (dark-gray)
regions: in-phase synchronization [see space-time plot in Fig. 4(a)]; hatched
regions: spatial incoherence [see space-time plot in Fig. 4(b)]; white region:
steady state; yellow (light-gray) regions: coherence resonance (CR) chimeras
[see Fig. 4(d)]. Parameters: N¼500, e¼0:05, a¼1.001, /¼p=20:1,
D¼0.0002, r¼0.2, and r¼0:4.
114320-4 Zakharova et al. Chaos 27, 114320 (2017)
interval 0:00030 D0:00035 (Fig. 6). The transition
from the steady state to coherence resonance chimeras for
increasing noise occurs at the left boundary [Fig. 6(f)] which
is shifted by the feedback to smaller noise intensities [Fig.
6(e)]. Furthermore, due to feedback, chimera states appear
even at zero noise intensity [Figs. 6(a)–6(d)]. Therefore,
time-delayed feedback promotes coherence resonance chi-
meras not only by increasing the noise range where they
exist but also by inducing these patterns in the absence of
noise. The largest range of Dcorresponds to s¼6.0 [Fig.
6(b)]. It is important to note that on the borders of the inter-
vals, the multistability is observed. Chimera states can coex-
ist with the steady state on the left border and with the
regime of spatially incoherent spiking on the right border.
Large feedback strength c¼0.6 can also shift the left
boundary of the chimera interval to lower [Figs. 7(a) and
7(b)] and even zero [Fig. 7(c)] noise values. The multistabil-
ity on the borders also occurs. Interestingly, for c¼0.6 the
chimera state overlaps with the complete synchronization
regime on the left boundary and not with the steady state as
in the case of weak feedback strength. The right boundary
strongly depends on sand shifts into the direction of lower
noise intensities [Figs. 7(a),7(b), and 7(c)]. The largest
detected interval for c¼0.6 corresponds to s¼4.76 T
[Fig. 6(b)] and for s¼9.52 2Twe even observe shrinking
of the interval [Fig. 7(a)].
If we compare the interval of chimera existence without
time-delayed feedback 0:000062 D0:000325 [Figs.
6(f) and 7(d)] with the interval the most enlarged by the
feedback 0:000001 D0:00035, it turns out that we
achieve 33% improvement rate.
V. IMPACT OF THE FEEDBACK ON COHERENCE
RESONANCE CHIMERA EXISTENCE: THRESHOLD
PARAMETER RANGE
It has been previously shown that coherence resonance
chimera can be obtained only in a small interval of a
(0:995 a1:004). To analyze the impact of time-delayed
feedback, we again consider two cases: c¼0:2andc¼0:6
and different values of delay time. Figure 8corresponds to the
case of small feedback strength c¼0:2 and Fig. 9illustrates
the results for the case of larger feedback strength c¼0:6.
For the two considered values of c, time-delayed feed-
back significantly changes the range of the threshold parame-
ter awhere coherence resonance chimeras exist. Moreover,
in both cases, this interval is increased the most when the
delay time is equal to the intrinsic period of the system s
¼4:76 T[Figs. 8(c) and 9(b)]. However, smaller feedback
strength allows for the stronger enlargement of the interval:
for c¼0:2 and s¼4:76, it is 0:993 a1:017 and is
more than doubled compared to the case without feedback
0:995 a1:004 [Fig. 8(c)].
While tuning the threshold parameter a, we observe
multistability on the boundaries of the coherence resonance
FIG. 6. Dynamic regimes depending on the noise intensity Dfor feedback
strength c¼0:2 and different values of delay time: (a) s¼9:52, (b)
s¼6:0, (c) s¼4:76, (d) s¼1:8, (e) s¼0:8, and (f) s¼0. Dynamic
regimes: steady state (yellow/light grey); spatially incoherent spiking (pink/
dark grey); coherence resonance chimeras (hatching). Other parameters:
N¼500, e¼0:05, a¼1.001, /¼p=20:1, r¼0.2, and r¼0:4.
FIG. 7. Dynamic regimes depending on the noise intensity Dfor feedback
strength c¼0:6 and different values of delay time: (a) s¼9:52, (b)
s¼4:76, (c) s¼0:8, and (d) s¼0. Dynamic regimes: steady state (yellow/
light grey); spatially incoherent spiking (pink/dark-grey); synchronization
(green/grey); coherence resonance chimeras (hatching). Other parameters:
N¼500, e¼0:05, a¼1.001, /¼p=20:1, r¼0.2, and r¼0:4.
FIG. 8. Dynamic regimes depending on the threshold parameter afor feed-
back strength c¼0:2 and different values of delay time: (a) s¼9:52, (b)
s¼6:0, (c) s¼4:76, (d) s¼1:8, (e) s¼0:8, and (f) s¼0. Dynamic
regimes: steady state (yellow/light grey); spatially incoherent spiking (pink/
dark grey); coherence resonance chimeras (hatching). Other parameters:
N¼500, e¼0:05, D¼0.0002, /¼p=20:1, r¼0.2, and r¼0:4.
114320-5 Zakharova et al. Chaos 27, 114320 (2017)
chimera regime where this pattern coexists with spatially
incoherent spiking [Figs. 8(a)–8(d) and 9(a)–9(c)]. For
increasing parameter a, the coherence resonance chimeras
disappear in the absence of feedback, and a steady state is
observed [Figs. 8(f) and 9(d)]. However, for c¼0:2;sT
[Figs. 8(a)–8(c)] and for all considered values of time delay
in the case of strong feedback c¼0:6 [Figs. 9(a)–9(c)], the
steady state is replaced by spatially incoherent spiking, i.e.,
the feedback induces oscillatory behaviour of the network.
As it can be seen from Fig. 8, for decreasing delay time
sfrom 9.52 to 0, we observe a nonlinear modulation of the
size of the a-interval of existence of chimera states. To gain
more insight into this effect, we define the parameter range
for which this pattern exists in the (a,s) plane (Fig. 10). We
detect isolated regions occurring for certain disconnected
intervals of s. The region centered at the time delay value
close to the intrinsic period of the system s¼4:76 T
clearly indicates the enlargement of the a-interval.
Interestingly, at the top of this region for sTand
a>1.01, we find a novel chimera regime (hatching in Fig.
10) which is induced by time-delayed feedback and has not
been previously shown for the system (1) without delay. The
space-time plot for the variable u
i
and the local order param-
eter indicate the coexistence in space of coherent and inco-
herent spiking as well as alternating behavior, typical
features of coherence resonance chimeras (Fig. 11).
Furthermore, the alternation takes place periodically and the
incoherent domain switches its position on the ring.
However, the switching events occur not for every spiking
cycle as in the coherence resonance chimera state [Figs. 1(b)
and 1(c)], but for every second spiking event [Figs. 11(a)
and 11(b)]. Due to this distinguishing feature, we call this
pattern period-two coherence resonance chimera.
To understand the mechanism of this alternation, we
consider a temporal sequence of the phase portraits of the
system Eq. (1) with the nullclines indicated for four selected
nodes of the network. As it can be seen from Fig. 11, the
incoherent domain alternates between two regions: the first
region corresponds to the nodes i2½0;125;½450;500and
the second region is i2½200;375. For this reason, we
choose nodes i¼69 and i¼60 from the first region and
nodes i¼231 and i¼284 from the second region. Next we
analyze their dynamics during one period (Fig. 12).
We begin our observation when the incoherent spiking
occurs in the second region i2½200;375. The node i¼231
(red in Fig. 12 right column) starts the excursion in the phase
space first since its nullclines are shifted and it can, there-
fore, be excited more easily by noise [right panel in Fig.
12(a)]. At the same time, the elements i¼60 and i¼69 from
the coherent domain (first region) rest in the steady state
since their nullclines are unchanged [left panel in Fig. 12(a)].
Next, the nullclines of the other nodes from the second
region are modified [see node i¼284 (blue) in the right
panel of Figs. 12(b) and 12(c)]. Consequently, they are now
also excited by noise and, therefore, incoherently [right panel
in Fig. 12(c)], while the elements from the coherent domain
FIG. 9. Dynamic regimes depending on the threshold parameter afor feed-
back strength c¼0:6 and different values of delay time: (a) s¼9:52, (b)
s¼4:76, (c) s¼0:8, and (d) s¼0. Dynamic regimes: steady state (yellow/
light grey); spatially incoherent spiking (pink/dark grey); coherence reso-
nance chimeras (hatching). Other parameters: N¼500, e¼0:05,
D¼0.0002, /¼p=20:1, r¼0.2, and r¼0:4.
FIG. 10. Coherence resonance chimera states in (a;s)-plane (yellow/grey
regions). The hatched region corresponds to the period-two coherence reso-
nance chimera. Parameters: N¼500, e¼0:05, D¼0.0002, /¼p=20:1,
r¼0.2, r¼0:4;and c¼0:2.
FIG. 11. Space-time plot for the vari-
able u
i
(a) and local order parameter
Z
i
(b) in the regime of period-two
coherence-resonance chimera. Initial
conditions: randomly distributed on
the circle u2þv2¼4. Incoherent
domains are marked by rectangles
in panel (a). Parameters: N¼500,
e¼0:05, a¼1.012, r¼0:4, r¼0.2,
D¼0.0002, c¼0:2;and s¼4:76.
114320-6 Zakharova et al. Chaos 27, 114320 (2017)
still stay in the vicinity of the steady state with the nullclines
unchanged [left panel in Fig. 12(c)]. Further, when the nodes
from the incoherent domain are well on the way [right panel
in Fig. 12(c)], they pull the nodes from the first region that,
therefore, also start spiking [left panel in Fig. 12(c)]. Since
they are excited not by noise but due to the pulling of the
neighbours, their spiking is coherent. After performing a
spike all the nodes return to the steady state [Fig. 12(d)].
This scenario is typical for coherence resonance chimeras
(see Fig. 3) for which the interchange of coherent and inco-
herent domains takes place during each subsequent excita-
tion. However, this is not the case for period-two coherence
resonance chimera as we see below.
Next we consider the second excitation for the elements
of the network Eq. (1). At the next moment in time a small
group of nodes including i¼60 and i¼69 from the first
region starts an excursion in the phase space due to noise
[left panel in Fig. 12(e)], while the elements from the second
region remain in the steady state [right panel in Fig. 12(e)].
However, the spiking of this small group is weak since time-
delayed feedback significantly shifts the nullclines (the cubic
ones down and the vertical ones to the right) and does not
allow the elements to make the full cycle in the phase space
before going back to the steady state [left panel in Fig.
12(f)]. At the same time for the nodes from the second
region, the feedback shifts the cubic nullclines up and the
vertical nullclines to the left making them more easily excit-
able by noise [right panel in Fig. 12(f)]. Therefore, the inco-
herent spiking is again induced in the second region, while
the nodes from the first region are pulled coherently due to
coupling [Fig. 12(g)]. After that all nodes return again to the
steady state. Next during the third excitation, the nullclines
for the elements from the first region are shifted in a way
making the excitation threshold lower and, therefore, the
spiking starts from the first region due to noise: the node
i¼60 is the first to spike [left panel in Fig. 12(h)]. Hence,
finally the coherent and incoherent domains are interchanged
and further the steps described above repeat with the only
difference that the first region is now incoherent, while the
second corresponds to coherent spiking.
Thus, it is the time-delayed feedback that prevents the
alternation for every spiking cycle. As it can be seen from Fig.
2, the largest coupling term corresponds to cross-coupling in
the v-equation of the system (1). However, the contribution of
the feedback term is significantly stronger than that of the cou-
pling terms. For this reason, alternating behaviour can only
occur when the time-delayed feedback term is close to zero.
That can be illustrated by considering the impact of coupling
terms and feedback upon the first and the second equation in
system (1) in the regime of period-two coherence resonance
chimeras (Fig. 13). This figure clearly indicates that the inter-
change of coherent and incoherent domains in the chimera
pattern occurs when the feedback term is close to zero (line A
in Fig. 13). On the other hand, the alternation fails when the
feedback term is non-zero and the coupling term almost
vanishes (line B in Fig. 13).
VI. CONCLUSIONS
In conclusion, we have investigated the impact of time-
delayed feedback on the dynamics of a network of nonlo-
cally coupled FitzHugh-Nagumo elements in the excitable
regime in the presence of noise. Our special attention is
given to a recently discovered chimera state, i.e., coherence
resonance chimeras. We demonstrate that time-delayed feed-
back promotes this pattern: it allows for control of the range
FIG. 12. Activator and inhibitor nullclines _
uand _
v, respectively, for the
selected nodes i¼69—black, i¼60—pink (left column) and i¼231—red,
i¼284—blue (right column) of the system Eq. (1) in period-two coherence-
resonance chimera regime for (a) t¼969.9, (b) t¼970.15, (c) t¼970.9, (d)
t¼974.5, (e) t¼975.0, (f) t¼975.9, (g) t¼976.65, and (h) t¼980.55.
Parameters: N¼500, e¼0:05, a¼1.012, r¼0:4, r¼0.2, D¼0.0002,
c¼0:2;and s¼4:76.
114320-7 Zakharova et al. Chaos 27, 114320 (2017)
of parameter values where noise-induced chimera exists and
in most cases increases this range. Moreover, the feedback
induces coherence resonance chimeras for vanishing noise
intensities. Additionally, we show that the threshold parame-
ter interval of coherence resonance chimeras can be more
than doubled by applying feedback with delay time close to
the intrinsic period of the system. Compared to the case
without feedback, this provides an essential improvement
which could be relevant for the experimental realization of
coherence resonance chimeras. Furthermore, when the feed-
back delay coincides with the intrinsic period of the network,
we find a novel feedback-induced regime which we call
period-two coherence resonance chimera. We explain the
alternating behavior of this novel pattern by analyzing the
evolution of the nullclines due to the coupling and feedback
terms of the network.
ACKNOWLEDGMENTS
This work was supported by DFG in the framework of
SFB 910 and by the Russian Ministry of Education and
Science (Project Code 3.8616.2017/8.9) and the Russian
Foundation for Basic Research (Grant No. 15-02-02288).
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FIG. 13. Space-time plots of coupling
terms for uand vvariables in the
period-two coherence resonance chi-
mera regime: (a) time-delayed feed-
back for the uvariable, (b) coupling
for the vvariable, and (c) space-time
plot of u
i
variable. Parameters: N¼500,
e¼0:05, a¼1.012, r¼0:4, r¼0.2,
D¼0.0002, c¼0:2;and s¼4:76.
114320-8 Zakharova et al. Chaos 27, 114320 (2017)
