ORIGINAL RESEARCH
published: 11 February 2019
doi: 10.3389/fams.2019.00008
Frontiers in Applied Mathematics and Statistics | www .frontiersin.org 1 February 2019 | V olume 5 | Article 8
Edited by:
Ulrich Parlitz,
Max-Planck-Institute for Dynamics
and Self-Organisation, Max Planck
Society (MPG), Germany
Reviewed by:
Isao T . T okuda,
Ritsumeikan University , Japan
Anastasiia Panchuk,
Institute of Mathematics
(NAN Ukraine), Ukraine
*Correspondence:
T anmoy Banerjee
[email protected] .ac.in
Eckehard Schöll
[email protected]
Specialty section:
This article was submitted to
Dynamical Systems,
a section of the journal
Frontiers in Applied Mathematics and
Statistics
Received: 30 November 2018
Accepted: 24 January 2019
Published: 11 February 2019
Citation:
Banerjee T , Bandyopadhyay B,
Zakharova A and Schöll E (2019)
Filtering Suppresses
Amplitude Chimeras.
Front. Appl. Math. Stat. 5:8.
doi: 10.3389/fams.2019.00008
Filtering Suppr esses Amplitude
Chimeras
T anmoy Banerjee 1 * , Biswabibek Bandyopadhyay 1 , Anna Zakharova 2 and
Eckehard Schöll 2 *
1 Chaos and Complex Systems Research Laboratory , Department of Physics, University of Burdwan, Burdwan, India,
2 Institut für Theoretische Physik, T echnische Universität Berlin, Berlin, Germany
Amplitude chimera (AC) is an interesting chimera pattern that has been discovered
r ecently and is distinct fr om other chimera patter ns, like phase chimeras and amplitude
mediated phase chimeras. Unlike other chimeras, in the AC patter n all the oscillators
have the same phase velocity , however , the oscillators in the incoher ent domain show
periodic oscillations with randomly shifted origin. In this paper we investigate the effect
of local filtering in the coupling path on the occurr ence of AC patterns. Our study is
motivated by the fact that in the prac tical coupling channels filtering effects come into play
due to the presence of dispersion and dissipation. W e show that a low-pass or all-pass
filtering is actually detrimental to the occurrence of AC. W e quantitatively establish that
with decreasing cut-of f fr equency of the filter , an AC transforms into a synchr onized
patter n. W e also show that the symmetry-breaking steady state, i.e., the oscillation death
state can be revoked and rhythmogenesis can be induced by local filtering. Our study
will shed light on the understanding of many biological systems where spontaneous
symmetry-breaking and local filtering occur simultaneously .
Keywords: chimera, amplitude chimera, oscillation death, filtering, contr ol, rhythmogenesis, all-pass filter
1. INTRODUCTION
Networks of coupled identical oscillators show various cooperative beha viors. From the symmetry
considerations they can be categorized into two broad types: (i) symmetric (or symmetry
preser ving) states, like synchronization, phase locking, and amplitude de ath (AD) state [ 1 , 2 ], and
(ii) symmetry-breaking states, such as oscillation deat h (OD) [ 2 ] and chimera states [ 3 ]. Among all
these cooperative beha viors, in the center of recent research is the chimera state [ 4 , 5 ] discovered
by Kuramoto and Battoghtok h in 2002. Chimera is a counterintuitive spa tiotemporal pattern in
which coherence and incoherence coexist in a network of identical oscillators [ 3 , 6 ]. In the initial
years studies on chimeras focused on exploring several aspects of chimera theoretically (see two
recent reviews on chimeras in [ 3 , 6 ] for a detailed discussion). Later on experiment al obser vations
of chimeras established their robustness in real systems. After the first experimental evidence of
chimeras in optical systems [ 7 ] and chemical oscillators [ 8 ], they have been observed experimentally
in several other systems also, e.g., in me chanical systems [ 9 , 10 ], electronic [ 11 , 12 ], optoelectronic
delayed-feedback [ 13 – 16 ], electrochemical [ 17 – 19 ] oscillator systems and Boole an networks [ 20 ].
Studies on chimeras are continuing to be a vibrant area of research owing to its connection to
various natural phenomena and systems, including epileptic seizure [ 21 ], unihemispheric sleep
[ 22 , 23 ], ecological synchrony [ 24 , 25 ], social systems [ 26 ], and quantum systems [ 27 ].
Although chimeras were discovered in phase os cillators, later on the notion was extended to
the general class of oscillators ha ving both phase as well as amplitude dynamics. Those oscillators

Banerjee et al. Filtering Suppresses Amplitude Chimeras
may show amplitude mediated phase chimeras (AMC) [ 28 ],
which is the coexistence of synchrony and asynchrony in both
phase and amplitude: here in the incoherent (coherent) domain
oscillators ha ve disparate (same) phase velocities. Recently,
a new type of chimera has been discovered by Z akharova
et al. [ 29 ] called amplitude chimera (A C), in which all the
oscillators of the network are correlated in phase, however , in
the incoherent domain nodes have uncorrelated amplitude. The
distinct signature of an A C state is that in its coherent domain
nodes oscillate around the origin and have equal amplitude,
however , nodes belonging to the incoherent domain show limit
cycles of disparate amplitude and those limit cycles are shifted
from the origin.
In contrast to other chimera patterns, A C has strong
connections to another symmetry-breaking steady state, namely
the oscillation de ath state (OD) [ 2 , 30 – 34 ]. The bridge between
A C and OD is mediated by an interesting emergent spatial
pattern called chimera death [ 29 , 35 ], which carries the attributes
of both A C and OD. Since A C is the coexistence of spatially
homogeneous and inhomogeneous limit cycles, therefore, it is
believed to have relevance in the underlying mechanism for
cellular differentiation [ 36 , 37 ] and ecological oscillations [ 2 4 , 25 ,
38 , 39 ] where coexistence of inhomogeneity and homogeneity
appears naturally.
As amplitude c himeras are a recently discovered variant of
chimera patterns, therefore, it is less explored: the effect of
node dynamics and coupling on the occurrence of AC demands
further investigations. Specifically, in reali stic networks, where
signals often suffer from time delay [ 40 ], noise, dispersion
and dissipation [ 41 ], their effect on th e A C pattern will be
important to explore. Althoug h, the effect of noise and time
delay has recently been explored in detail in Loos et al. [ 42 ]
and Gjur chinovski et al. [ 43 ], however , t he effect of dispersion
and dissipation on the A C state has not been studied yet. In
the presence of dispersion, signals ha ving different frequencies
propagate with different velocities. Whereas, dissipation causes
attenuation and signal loss. A channel ha ving both dispersion
and dissipation is said to beha ve like a low-pass filter. On the
other hand, a channel ha ving only dispersion is said to beha ve
as an all-pass filter [ 44 ]. Several physical and biological systems
contain inherent local low-pass filters (LPFs): For example , the
musculoskeletal system of human body acts as low-pass filter
[ 45 ], the abdominal ganglion of crayfish contains local LPFs [ 46 ],
LPF is one of the building blocks of phase-locked loops [ 47 ]. On
the other hand, in the case of electronic communications a nd
neuronal systems the presence of local amplifiers or ion channels
[ 48 ], respectively, compensate for the dissipation, howe ver , in
those systems signals still suffer dispersions making the coupling
path to beha ve as an all-pass filter (APF). The effect of low-pass
filtering was studied before in the context of synchronization
[ 49 , 50 ] and rhythmogenesis from an amplitude or oscillation
death state [ 41 ] (by rhyth mogenesis we mean the process by
which the rhythmic beha vior of individual nodes in a network
of coupled oscillators is restored from the state of suppressed
oscillations without changing the intrinsic parameters associated
with the individual nodes); Baner jee et al. [ 51 ] reported a novel
transition from homogeneous to inhomogeneous limit cycle as a
consequence of low-pass or all-pass filtering. However , hit herto
the effect of filtering on the chimera stat e in coupled oscillators
has not been explored.
Motivated by the above discussion, in this paper we study
the effect of local filtering on t he occurrence of amplitude
chimera (A C) in a network of nonlocally coupled Stuart-Landau
oscillators. By local filtering we mean t hat the filtering effect is
considered in the self-feedback path only. We consider local low-
pass and all-pass filter s in the network and for the first time we
show that both types of filtering have a detrimental effect on
the occurrence of amplitude chimeras: filtering always suppresses
amplitude chimeras. W ith t he variation of a filtering parameter
(namely, the corner or cut-off frequency) we obser ve transitions
from the oscillation de ath and amplitude chimera state to the
globally synchronized state.
2. WITHOUT FIL TERING
We consider N = 200 Stuart-Land au oscillators interacting
through nonlocal symmetry-bre aking coupling (i.e., only
through the x -variable). The mathematical model of the coupled
system is given by,
˙ x i = (1 − x 2
i − y 2
i ) x i − y i ω + ε
2 P
i + P
X
j = i − P
( x j − x i ), (1a)
˙ y i = (1 − x 2
i − y 2
i ) y i + x i ω , (1b)
with i = 1 ··· 200. The individual Stuart-L andau oscillators
ha ve unit amplitude and eigenfrequency ω . Here ε denotes
the coupling strength and P is the coupling range of t he
nonlocal coupling.
To explore the dynamics of the coupled network we
numerically solve Equation (1) using the fourth-order Runge-
Kutta method (step size = 0.01). Throug hout this paper we
consider ω = 2 and use the following initial conditions [ 29 ]:
x i = 1 and y i = − 1 for 1 ≤ i ≤ N
2 and x i = − 1 and y i = 1
for N
2 < i ≤ N .
Figure 1A shows the phase diagram in the P − ε space: we
can see that the amplitude chimera (A C) state is interspersed in
between the completely synchronized oscillation zone (Sync) and
the oscillation de ath (OD) zone. This is in accordance with t he
results of Z akharova et al. [ 29 ], Schneider et al. [ 52 ], Z ak harova
et al. [ 53 ], and Tumash et al. [ 54 ] where this system was studied
in detail. Figures 1B–D illustrate the spatiotemporal evolution
of the synchronized state ( ε = 5), AC pattern ( ε = 20) and
multicluster OD state ( ε = 30) at P = 10. Figure 2 depicts the
manifestation of A C and OD in the phase space for an exemplary
coupling range P = 10. Figure 2 (Left panel) shows A C for
ε = 20: here the small amplitude and shifted-origin limit cycles
represent incoherent nodes and those having lar ge amplitude
oscillating around the origin represent the coherent nodes (for
clarity only a few nodes from coherent and incoherent domains
are shown). For higher coupling strengths a symmetry-breaking
steady state (OD st ate) emerges, which is shown in F igure 2
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Banerjee et al. Filtering Suppresses Amplitude Chimeras
FIGURE 1 | W ithout filtering: (A) phase diagram in the P − ε space for
N = 200 nonlocally coupled Stuart-Landau oscillators ( ω = 2). Sync,
synhronized state; AC, amplitude chimera; OD, oscillation death. (B–D) The
spatiotemporal plots at P = 10 for three dif fer ent coupling strengths ε : (B)
synchronized state for ε = 5 [shown by  in (A) ], (C) AC for ε = 20 [shown by
⋆ in (A) ] and (D) multicluster OD for ε = 30 [shown in N in (A) ].
FIGURE 2 | W ithout filtering: Phase-space plot of a few nodes of the network
from the coher ent and incoher ent domains (Left panel) AC ( ε = 20, ⋆ point
in Figure 1A ), and (Right panel) OD ( ε = 30, N point of Figur e 1A ). Other
parameter values are P = 10, ω = 2, N = 200.
(Right panel) for ε = 30. In the next se ction we will explore how
filtering affects this dynamical landsc ape in parameter space.
3. EFFECT OF LOW-P ASS FIL TERING
3.1. Mathematical Model
We consider N = 200 Stuart-Landau oscillators interacting
through nonlocal symmetry-brea king coupling as in
Equation (1), but here we consider local low-pass filter in
the coupling path. The mathematic al model of the coupled
system is given by,
˙ x i = (1 − x 2
i − y 2
i ) x i − y i ω + ε
2 P
i + P
X
j = i − P
( x j − z i ), (2a)
˙ y i = (1 − x 2
i − y 2
i ) y i + x i ω , (2b)
˙ z i = α ( − z i + x i ). (2c)
Equation ((2c)) is the mathematic al equation of a low-pass filter
whose input is x i and output is z i . This z i is fed to the coupling
part of Equation (2a). Here α represents the corner or cut-off
frequency of the LPF: the lower is the value of α , the highe r is
the effect of filtering. For larger α , filtering effects be come lesser:
if we put α → ∞ in Equation (2a), it simply gives z i = x i , i.e., no
filtering effect is present and Equation (2) reduces to the original
Equation (1). Since in the literature of filters we are conversant
with the frequency domain representation, therefore, at first it
is difficult to realize the role of α in Equation (2c). Howe ver , a
close inspection reve als that α controls both phase and amplitude
of the output signal z i by the following way: the phase shift
between input and output is given by φ i = arctan ( ωα − 1 ), the
ratio of output and input (called gain of the filter) is G =
1
√ 1 + ω 2 α − 2 (see [ 51 ] for details). Another equivalent form is
the representation of Equation (2c) as a distributed delayed
coupling term in Equation (2a) with an exponential delay kernel
exp ( − α τ ) [ 55 , 56 ].
3.2. Results
We investigate the effect of local low-pass filtering on the
occurrence of amplitude chimera. Since α is the only control
parameter , we will explore the effect of α on the dynamics of
the network. We keep all the parameters and initial conditions
the same as in the unfiltered c ase; the initial conditions for
the filter variable z i are chosen the same as those of x i for the
unfiltered case.
Figures 3A,B demonstrate the phase diagram of the network
in the P − ε space for three different (decre asing) values
of α . It can be obser ved from Figure 3 that t he smaller the
value of α is, the more the network dynamics de viates from
the original scenario shown in Figure 1A . It is apparent from
Figure 3 that with decreasing α (i.e., increasing filtering effect)
the synchronized portion dominates and therefore suppresses the
A C and OD regions: a lower α shifts the A C and OD zone to a
higher P region and also quenches the area of the AC and OD
zone. E ventually, below a critical value of α (say α c ) t he A C and
OD state disappe ar and only the synchronized state pre vails in
the whole P − ε space. This suppression of the A C and OD zone
is shown in Figure 3C for α = 10.
The scenario can be understood more clearly in t he ε − α
space for a fixed P . Figure 4A shows this for P = 10: we c an
obser ve that for comparatively high values of α the dynamics
of the system remains unchanged. However , as the value of α is
decreased the system goes to a synchronized state irrespectively
of ε . It also shows that there exists a critical value α c of α ,
below which the synchronized state is the only possible state.
Figures 4B–D illustrate how decreasing α leads to t he transition
from OD to synchrony via A C ( ε = 25 and P = 10): for α = 45
the network shows a multi-clustered OD state ( Figure 4B ), and
the A C state is shown for α = 35 ( Figure 4C ), and finally global
synchrony (a coherent tra veling wa ve or a splay state) appears for
further lowering of α ( Figure 4D for α = 25). It is noteworthy
that in a range of lower ε , no OD state occurs and in this zone a
decreasing α leads to a dire ct transition from A C to synchrony.
Figure 4E shows the scenario in t he P − α space ( ε = 35):
here also we can see that at lower α values the completely
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Banerjee et al. Filtering Suppresses Amplitude Chimeras
FIGURE 3 | W ith local low-pass filtering: Phase diagram in the P vs ε space ( N = 200, ω = 2) for (A) α = 50 and (B) α = 20. (C) shows complete suppression of AC
for α = 10.
synchronized state emerges out of either A C or OD. From
Figures 4A,E we see that OD state is predominant for higher
coupling strength ( ε ) and near-global coupling range (i.e., P →
N / 2): it is interesting to note that a suitably chosen filtering
parameter α can suppress the steady state and t herefore results
in rhythmogenesis in the network. In Zou et al. [ 41 ] and Banerjee
et al. [ 51 ] filtering-induced rhythmogenesis in coupled oscillators
was reported, however , in contrast to Zou et al. [ 41 ] and Banerjee
et al. [ 51 ] here we show the existence of a broad parameter
zone where OD does not transform into oscillation (SYNC)
directly, but another symmetry-bre aking emergent state, i.e., an
amplitude chimera, mediates the transition. Therefore, filtering
plays an important role in networks of physical, biologic al,
and physiological systems where the occurrence of os cillation
suppression often leads to a fatal system degradation and an
irrecoverable malfunctioning [ 57 – 59 ]. A similar enhancement of
the stability domain of the synchronized solutions for small α
was found for distributed delayed coupling with an exponential
kernel [ 56 ].
In the above results we use suitable measures, such as the
measure of spatial correlation ( g 0 ) and the center of mass ( y cm i ) to
ensure the occurrence of the synchronized stat e and A C state and
also to distinguish them (distinction of the OD st ate is relatively
simple as we have to c heck whether a steady st ate is reached or
not). A ccording to Kemeth et al. [ 60 ], the measure of spa t ial
correla tion is defined in terms of the normalized probability
density function g as
g 0 ( t ) ≡
δ th
X
| ˆ
L ψ i ( t ) |= 0
g ( | ˆ
L ψ i ( t ) | ). (3)
Here ˆ
L ψ i ( t ) represents the local cur vature at e ach node i at time
t given by
ˆ
L ψ i ( t ) = ψ ( i − 1) ( t ) − 2 ψ i ( t ) + ψ ( i + 1) ( t ), (4)
where ˆ
L is the discrete L aplacian operator on each snapshot
{ ψ i } . In our present case the state variable ψ i ( t ) = y i (one can
use x i as well). In Equation (3) we consider a threshold value
δ th = 0.01 L max , where L max is the maximum cur vature in the
network [ 60 ]. The measure of spatial correlation g 0 ( t ) = 1 for
a fully synchronized network and g 0 ( t ) = 0 for a completely
unsynchronized network. Therefore, 0 < g 0 ( t ) < 1 represents
partial synchronization ensuring the occurrence of chimera state.
Although g 0 ( t ) can ensure t he occurrence of a chimera state, it
cannot distinguish between phase and amplitude chimeras. To
ensure that A C indeed emerge s in the network, we compute the
center of mass of each oscillator defined by [ 29 ]
y cm i = 1
T Z T
0
y i dt , (5)
where y i represents the state of the i -t h oscillator and T is a
sufficiently large time. The quantity y cm i gives a measure of the
shift of a limit cycle from the origin. Therefore, it can distinguish
the homogeneous limit cycles from inhomogeneous ones.
Figures 5A,C , respectively, show g 0 ( t ) and y cm i of each
oscillator corresponding to the synchronized state of Figure 4D
( α = 25): we obser ve that all the os cillators in the network
ha ve g 0 ( t ) = 1 and y cm i = 0 indicating that the whole network
is synchronized. On the other hand, Figures 5B ,D , respectively,
show g 0 ( t ) and y cm i corresponding to the A C state of Figure 4C
( α = 35): we can see that 0 < g 0 ( t ) < 1 indic ating the occurrence
of chimeras and at the same time y cm i in t he incoherent region
exhibits a random sequence of shifts to positive and negative
values, however , in the coherent region y cm i = 0 indicating that
the resulting chimera is indeed an A C pattern.
A ccording to Tumash et al. [ 54 ], a strong measure that
distinguishes an A C state from the synchronized state is the
Floquet exponent. We study the stability of the periodic solution
of nonlocally coupled Stuart-L andau oscillators given by (2)
using Floquet theory [ 54 ]. We rewrite (2) as
˙ x = f ( x ( t )), (6)
with x ( t ) ∈ R n and also consider that a periodic solution ψ ( t ) =
ψ ( t + T ) exists. In our case, we ha ve three equations, therefore,
n = 3 N . The linearized equation is written as,
δ ˙ x ( t ) = J ( ψ ( t )) δ x ( t ), (7)
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Banerjee et al. Filtering Suppresses Amplitude Chimeras
FIGURE 4 | W ith local low-pass filtering: (A) phase diagram in the ε − α space for P = 10 ( ω = 2). Three points at three α values at a particular ε = 25 ar e marked by
• ( α = 45), H ( α = 35) and  ( α = 25). (B–D) spatiotemporal plots corresponding to those thr ee points (decrea sing α ): (B) multicluster OD, (C) AC, (D) synchronized
state (coherent traveling wave or splay state). (E) phase diagram in the P − α space for ε = 35.
FIGURE 5 | W ith local low-pass filtering: (A,B) The time evolution of g 0
corresponding to the synchr onized (A) and AC (B) state as marked in
Figure 4A by (  ) ( α = 25) and ( H ) ( α = 35), r espectively . (C,D) The
corresponding center of mass ( y cm i ) for the above two points, showing
synchronized (C) and AC (D) states, r espectively . Other parameters are
P = 10, ε = 25 and ω = 2.
where J ( ψ ( t )) is the J acobian matrix evaluated at ψ ( t ) and has the
following solution:
δ x ( t ) = M ( t ) δ x (0). (8)
Here δ x (0) is the initial condition. The fundamental matrix M ( t )
obeys the equation,
˙
M ( t ) = J ( ψ ( t )) M ( t ), (9)
where M (0) = 1, and M ( t + T ) = M ( t ) M ( T ). M ( T ) is
the monodromy matrix whose eigenvalues are called Floquet
multipliers ( µ k ). Each Floquet multiplier can be expressed as
µ k = exp (( 3 k + i  k ) T ), where ( 3 k + i  k ) is the Floquet
exponent. The stability of the periodic orbit can be analyzed
by determining the sign of the real part of these exponents.
FIGURE 6 | W ith local low-pass filtering: Phase diagram of the p eriodic
solutions (Sync and AC) in ε − α space based on the Floquet exponent. For the
synchronized r egion (black), at each point, the largest r eal part of the Floquet
exponents ( 3 max ) is negative (for the Goldstone mode it is app roximat ely
equal to zero). For the AC r egion (orange) at each point it is gr eater than zero
(i.e., 3 max > 0). Other parameters are P = 10, ω = 2, N = 200.
When the real parts of all the Floquet exponents are less than
zero (i.e., 3 k < 0) except the Goldstone mode (which is
equal to zero) then the periodic solution is st able indicating a
synchronized solution [ 54 ]. But according to Tumash et al. [ 54 ]
when at least one or two of them are gre ater than zero ( 3 k >
0), then the solution becomes unstable indicating a saddle
cycle in phase space which corresponds to an A C st ate. In
our computation we average the exponents over 200 T (where
T = π ). Figure 6 shows the zone in black where all the
exponents are negative (except the Goldstone mode), whic h
indicates the synchronized state; Again, at every point in t he
orange region, a few 3 k s ha ve small ( < 0.5) positive values,
which means that the system is in the AC state. Note the
agreement between Figure 4A and Figure 6 , which confirms
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Banerjee et al. Filtering Suppresses Amplitude Chimeras
FIGURE 7 | W ith all-pass filtering: Phase diagram in the P − ε space for (A)
α = 50 and (B) α = 20. (C) spatiotemporal plot of AC corresponding to
α = 50 (shown by H in A ). (D) spatiotemporal plot of the synchronized state
(coherent traveling wave) corr esponding to α = 20 (shown by  in B ). In both
(C,D) ε = 12, P = 15, ω = 2.
that a transition from A C to synchrony indeed occurs with
decreasing α .
4. EFFECT OF ALL-P ASS FIL TERING
Next we consider the effect of all-pass filtering (APF) in the
network of Stuart-Landa u oscillators described in Equation (1).
The mathematical model of the coupled system is given by
˙ x i = (1 − x 2
i − y 2
i ) x i − y i ω + ε
2 P
i + P
X
j = i − P
( x j − U i ) (10a)
˙ y i = (1 − x 2
i − y 2
i ) y i + x i ω (10b)
˙ z i = α ( − z i + x i ) (10c)
U i = 2 z i − x i (10d)
Equations (10c, 10d) jointly represent the differential algebraic
equation of an all-pass filter , whose input is x i and output is U i
[ 51 ]. In this case α has the s ame meaning as in Equation (2c),
but the effect of α is different on U i : Here α does not affect the
amplitude of U i , it only affects the phase part by introducing
a phase shift between the input and output signals, given by
θ = 2 arctan ( ωα − 1 ). Note that for the same α t he phase shift
introduced by a LPF (i.e., φ ) is half of that of an APF (i.e., θ ).
In Figure 7 the effect of a n all-pass filter is shown in the
P − ε space for two α values: Figure 7A is for α = 50 and
Figure 7B is for α = 20. Figures 7C,D show the spatiotemporal
representation of A C (for α = 50) and synchronized state
(coherent traveling wa ve for α = 2 0), respectively: it shows that
α acts as an efficient control parameter for the suppression of A C.
Here it is evident t hat local all-pass filtering can also suppress A C
(and OD) and gives rise to the synchronized state. Comparing
Figures 7A,B of t he low-pass filtering case with Figures 7A,B ,
respectively, it is interesting to note that for the s ame α an APF is
more effective than a LPF as far as the suppression of amplitude
chimeras (and OD) is concerned: see for example at α = 20, low-
pass filtering only quenches the A C and OD z one in P − ε space
( Figure 3B ), however , all-pass filtering completely suppresses the
A C and OD zone ( Figure 7B ). We ensure that α c , the critical
value below which A C and OD are completely suppressed , is
much higher for an APF compared to that of a LPF (not shown
here): therefore even a relatively we ak all-pass local filtering
is equivalent to a stronger local low-pass filtering, as far as
suppressing A C and OD is concerned. This is t he consequence of
the fact that at a particular value of α , the phase shift introduced
by an APF is twice of that of a LPF [ 51 ]. Therefore, all the results
suggest that α which is the control parameter of local filters, also
controls the dynamics of the whole network.
5. CONCLUSION
In this paper , we ha ve revealed that the presence of local
filtering (either low-pass or all-pass) suppresses the amplitude
chimera state and therefore gives rise to global synchrony
(coherent traveling waves). Further , it has been shown that local
filtering causes rhythmogenesis by suppressing the steady state
beha vior (i.e., OD state), which has immense importance in
many biological and engineering systems [ 58 , 61 ]. Collectively,
our study has a broad significance: it establishes that local
filtering is detrimental for the symmetry-brea king states
(A C and OD) and fa vors restoration of the symmetry in
the network.
Our study reveals t hat the cut-off frequency α of the local
filter acts as an efficient control parameter of the network that
can be tuned to achieve a desired symmetry-breaking st ate
or synchronized state without changing coupling strength or
range. Several control methods to st abilize phase or amplitude
mediated phase chimeras ha ve recently been proposed [ 62 – 64 ].
In Gjur chinovski et al. [ 43 ] it has been s hown that a constant
time delay in the coupling path can s tabilize amplitude chimeras.
In contrast, here we established that the local filtering has a
destabilizing effect on the occurrence of amplitude chimeras. In
the case of rhythmogenesis, the value of α that suppresses the
steady state depends upon the system and coupling parameters
in a nontrivial manner (see [ 51 ] for two mean-field coupled
oscillators). It has been obser ved that if one wants to ensure
rhythmogenesis (irrespectively of other parameters) the typical
value of α is of the order of the intrinsic frequency of an
individual oscillator (here ω ). Howe ver , depending upon s ystem
and coupling parameters, α (filtering) need not be so small
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Banerjee et al. Filtering Suppresses Amplitude Chimeras
(strong): rhythmogenesis appears much before that, i.e., for
α > ω .
From the perspective of dynamical systems the role of α
can be understood in the following way: α actually controls the
dissipative property of the whole network by controlling the
dissipation and dispersion in the coupling path; a smaller α
imposes a larger filtering effect and therefore smaller dissipation,
which fa vors synchrony and r hythmogenesis. In this context we
obser ve that filtering does not affect the pattern of phase chimera
appreciably. This may be due to the fact that additional phase
shift and/or attenuation caused by filtering has lesser effect on the
mean frequency than on the amplitude dynamics (note that in t he
phase chimera the mean frequency is the determining factor that
distinguishes the coherent and incoherent domains, whereas in
the amplitude chimera, the amplitude of the nodes matters).
In this paper we have considered a network of Stuart-
Landau osci llators. However , we verified that th e filtering affects
the amplitude chimera in a similar way in other systems
also, for example, in a network of Rayleig h oscillators [ 65 ]
(results not shown). Since the Stuart-Land au oscillator is
a generic model for systems near a Hopf bifurcation and
since filtering naturally arises in many biological and physical
systems, we believe that our results c an also be extended to
those systems.
AUTHOR CONTRIBUTIONS
TB , ES, and AZ formulated the problem. BB and TB carried
out the analysis. BB performed the computa tions. All authors
discussed the results and contributed to writing the manuscript,
read and approved the final manuscript.
ACKNOWLEDGMENTS
BB acknowledges University of Burdwan for providing financial
support through the st ate funded research fellowship. AZ and ES
acknowledge the financial support by DFG in the framework of
SFB 910.
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Conflict of Interest Statement: The authors declare that the research was
conducted in the absence of any commercial or financia l relationships that could
be construed as a potential conflict of interest.
Copyright © 2019 Banerjee, Bandyopadhyay, Zakharova and Sc höll. Th is is an open-
access article distributed under the terms of t he Crea tive Commons A ttr ibution
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