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Effects of periodically modulated coupling on amplitude death in nonidentical
oscillators
Weiqing Liu,1, ∗Xiaoqi Lei,1and Jiangnan Chen2
1School of Science, Jiangxi University of Science and Technology, Ganzhou 341000, China
2School of Information and computer engineering, Pingxiang University, Pingxiang, 337055, China
The effects of periodically modulated coupling on amplitude death in two coupled nonidentical
oscillators are explored. The AD domain could be significantly influenced by tuning the modulation
amplitude and the modulation frequency of the modulated coupling strength. There is an optimal
value of modulation amplitude for the modulated coupling with which the largest AD domain is
observed in the parameter space. The AD domain is enlarged with the decrease of the modulation
frequency for a given small modulation amplitude, while is shrunk with decrease of the modulation
frequency for a given large modulation amplitude. The mechanism of AD in the presence of periodic
modulation in the coupling is investigated via the local condition Lyapunov exponent of the coupled
system. The stability of AD state can be well characterized by conditional Lyapunov exponent. The
coupled system experiencing from the oscillatory state to AD is clearly indicated by the observation
that the conditional Lyapunov exponent transits from positive to negative. Our results are helpful
to many potential applications for the research of neuroscience and dynamical control in engineering.
I. Introduction
By modeling the coupled nonlinear oscillators, a rich
source of ideas and insights into understanding the emer-
gence of self-organized behaviors in physics, biology,
chemistry and neuroscience has been explored [1–3] dur-
ing the last few decades. Quenching of oscillation, as
one of the basic collective dynamic behaviors, has at-
tracted many attentions in various fields of nonlinear sci-
ence since it has potential application on the control of
chaotic oscillations and stabilization of various unstable
dynamics in the aspects of mechanical engineering [4],
synthetic genetic networks [5, 6], and laser systems [7, 8].
Two main categories are described according to their gen-
eration mechanisms and manifestations, amplitude death
(AD) and oscillation death (OD) [9, 10]. In AD, oscil-
lations are suppressed to the same original homogeneous
steady state (HSS) [9]. It is mainly applied as a control
mechanism in physical and engineering systems. In con-
trast, OD occurs due to a stabilization of an newborn
inhomogeneous steady state (IHSS), where the individ-
ual units stay in different branches of the IHSS [10]. The
main implications of OD are in biological systems, since
it has been interpreted as a background mechanism of
cellular differentiation [5, 11, 12] and related neurologi-
cal conditions [13, 14]. AD is manifested to transit to
OD via Turing bifurcation [15], mean-field diffusive cou-
pling, dynamics coupling [16], and time delay coupling in
experimental observations [17, 18].
Generally, there are three types of main factors in-
fluencing the oscillation suppression phenomena as fol-
lows, (1) the parameter mismatches between the nodes
in coupled oscillators [19–21]; (2) The structure of the
interaction networks; (3) The coupling schemes between
interacting nodes. In Ref. [10], the standard oscillatory
solutions are eliminated in a large region of the parame-
∗Electronic address: wqliujx@gmail.com
ter mismatches by establishing the dominance of oscilla-
tion death under strong coupling in a set of qualitatively
different models of coupled oscillators (such as genetic,
membrane, Ca metabolism, and chemical oscillators). In
our previous work [20], AD are general regimes in a ring
of coupled oscillators with parameter mismatches and the
spatial distribution of the parameter mismatches signifi-
cantly influences the critical coupling strength needed for
amplitude death. Rich dynamics of oscillation quenching
are observed in regular networks such as all-to-all coupled
oscillators [22] and diffusively coupled oscillators [23–25],
as well as in the complex networks such as small-world
network [26] and BA network [27] where the topological
property of network distinctly influences the AD dynam-
ics. Furthermore, a rich of complex spatiotemporal pat-
terns of coupled oscillators in networks is explored such
as transition from amplitude chimeras states to chimera
death states [28] where the population of oscillators splits
into distinct coexisting domains of spatially coherent am-
plitude of oscillation (or oscillation death) and spatially
incoherent amplitude of oscillation (or oscillation death).
The transition process to oscillation death can be con-
tinuous one [23] or discontinuous one [29–31] named as
explosive death.
Various kinds of coupling schemes are available for os-
cillation quenching in coupled oscillators, such as dy-
namic coupling [32], conjugate coupling [33], nonlinear
coupling [34], gradient coupling [35], mean-field diffusive
coupling [36], amplitude dependent coupling [37], time-
delay in coupling due to a finite propagation of the signal
[38–40]. In all these existing studies, the interaction or
coupling among the systems works continuously or per-
manently with time. However, the continuous interac-
tion does not always keep in many real systems such
as the biological signal transmission between synapses
and mechanical control of engineering. The strengths of
synapses in neuronal networks are modified according to
external stimuli, the links in metabolic networks are acti-
vated only during specific tasks which lead to non-static
arXiv:1810.12548v2 [nlin.CD] 31 Oct 2018
2
interactions. On-off coupling, one manifestation of dis-
continuous coupling, has been verified to optimize the
synchronization stability and speed [41, 42]. Schroder
et al [43] explored a scheme for synchronizing chaotic
dynamical systems by transiently uncoupling them and
revealed that systems coupled only in a fraction of their
state space may transit to synchronous state from non-
synchronous state of formally full coupling interaction.
The synchronous efficiency may be improved in the as-
pect of control. Periodic coupling, another time-varying
coupling scheme, are also verified to be a available candi-
date to maximize the network synchronizability by prop-
erly selecting coupling frequency and amplitude [44].
Compared to the focus of time-varying coupling on
synchronizability of coupled system, the AD dynamics
under the effects of discontinuous coupling are rarely ex-
plored. Recently, AD was observed theoretically [45] and
experimentally [46] in two coupled oscillators by intro-
ducing the time-varying coupling. Sun et. [1] extended
the study of AD under the influence of on-off coupling
and found that AD domains are enlarged in the parame-
ter space with a proper switching frequency and switch-
ing rate of coupling. However, the discontinuous form of
on-off switches is sharp and difficult to realize physically
owning to a finite response time of the switcher. There-
fore, it is natural to reveal the effects and mechanisms of
a kind of more flexible discontinuous coupling (i.e. pe-
riodic coupling) on the oscillation quenching. The main
goal in this work is to investigate the effects of periodi-
cally modulated coupling on the emergence of AD in the
coupled nonidentical oscillators. In particular, we show
that the occurrence of AD in nonidentical oscillators with
time-varying coupling can be well characterized by the
conditional Lyapunov exponent of the coupled system.
I. II. MODELS
In this section, a periodically modulated coupling
scheme is introduced to the following system of coupled
oscillators of general form:
˙
X1(t) = f1(X1(t)) + (t)Γ(X2(t)−X1(t)),
˙
X2(t) = f2(X2(t)) + (t)Γ(X1(t)−X2(t)),(1)
where Xi∈Rn(i= 1,2), f:Rn→Rnis nonlinear and
capable of exhibiting rich dynamics such as limit cycle
or chaos, and Γ is a constant matrix describing coupling
scheme. and (t) is a periodically modulated coupling
strength as shown in Fig. 1, and can be described as,
(t) = 0[1 + αcos(ω0t)],(2)
where 0,ω0and α∈[0,2] are the average coupling
strength, modulation frequency (ω0=2π
T0), and modu-
lation amplitude of the periodically modulated coupling
strength, respectively. The coupling may vary in the pos-
itive range for all time as α∈(0,1) and keep constant if
α= 0, otherwise the coupling strength may vary between
FIG. 1: The periodically modulated coupling strength versus
time, where α,0, and T0are the modulation amplitude, av-
erage coupling strength, modulation period of the coupling
term.
positive and negative, i.e. the coupling interchanges be-
tween attractive and repulsive as α∈(1,2].
II. III. RESULTS
A. A. Coupled Stuart-Landau oscillators
In order to observe the effects of the periodically
modulated coupling on the AD dynamics, let’s firstly
consider the coupled nonidentical Stuart-Landau oscil-
lators whose dynamics can be described as ˙
Zi(t) =
[1 + jωi− |Zi(t)|2]Zi(t), where Zi(t) = xi(t) + j yi(t),
i= 1,2,j=√−1, ωiis the intrinsic frequency of oscilla-
tor i. Without coupling (0= 0), each oscillator has an
unstable focus at the origin |Zi|= 0 and an attracting
limit cycle with a oscillating frequency ωi. Considering
the coupling scheme Γ = 1 0
0 1, AD can be stabilized
in two coupled oscillators in Eq. 1 with frequency mis-
matches ∆ω=|ω2−ω1|for constant coupling strength
(α= 0) and ω1= 2. The results [19] indicate that the AD
domain is bounded in the interval of coupling strength
for given frequency mismatches while keeps stable when
the frequency mismatch is larger than a critical value for
given coupling strength as shown in Fig. 3(a). Since
the modulated coupling strength has three control pa-
rameters, i.e. average coupling strength 0, modulation
frequency ω0, and modulation amplitude α. Let’s firstly
explore the effects of the modulation amplitude on AD
with the modulation frequency ω0= 4 fixed, and the av-
erage coupling strength 0= 7.0. Figs. 2(a)-(d) present
the bifurcation diagram of x1for α= 0.0,0.8,1.0,1.8,
respectively. For α= 0, the coupling strength is con-
stant, and the coupled system transits to AD from the
oscillating state (i.e. the time series of x1(t) display
3
FIG. 2: (Color online) The bifurcation diagram of x1(gray
dots) versus frequency mismatch ∆ωfor 0= 7 and (a) α=
0.0, (b) α= 0.8, (c)α= 1.0, (d) α= 1.8, respectively. The
insets are the time series of x1(t) for (a) α= 0.0,∆ω= 2,
(b)α= 0.8, ∆ω= 6, (c) α= 1.0, ∆ω= 8 (d)α= 1.8,
∆ω= 5.
periodic oscillation for ∆ω= 2.0 in Fig. 2(a)) when
the frequency mismatch increases from zero to the value
larger than ∆ωc= 7.3. As α= 0.8, the critical value
of ∆ωcfor AD becomes 5.6 which is less than that of
the constant coupling strength. The AD state is dis-
played by the time series of x1(t) in the insets of Fig.
2(b) for ∆ω= 6. The coupled system has two inter-
vals of AD domain ∆ω∈[4.6,7.1] and ∆ω∈[9.85,20] as
α= 1.0. Then, three disconnected interval of AD domain
along the direction of ∆ωare observed as ∆ω∈[1.1,1.2],
∆ω∈[15.1,16.0], ∆ω∈[18.3,20] for α= 1.8. The oscil-
lating state is in periodic 2 for ∆ω= 8 and α= 1.0 and
in multi-period state for ∆ω= 5 and α= 1.8 as shown
in the insets of Figs. 2(c)(d),respectively.
To integrally figure out how the periodically modu-
lated coupling influences the AD domain of the coupling
oscillators, it is natural to reveal the mechanical regimes
of the stabilities of AD under the periodically modulated
coupling. Generally the stability of the AD domain in
coupled oscillators is obtained from linear stability anal-
ysis of Eq. 1 around Zi= 0 if the coupling strength is
constant. The characteristic eigenvalue is [19]
λ1,2,3,4= 1 −±r2−(ω2−ω1)2
4±j(ω1+ω2)
2,(3)
then the AD domain is determined by Re(λ)<0, that
is, 1 ≤≤∆ω/2 and ∆ω/2< < 1/2 + ∆ω2/8
which is right the boundary lines of the AD domain as
shown in Fig. 3(a). However, when is varying with
time, the linear stability analysis around the original
fixed points is not available any more. We numerically
record the AD domain in parameter space ∆ωversus 0
(both are in the range of [0,20]) for a given ω0= 4.0,
and α= 0,0.5,1.0,1.1,1.4,1.8 as shown in Figs. 3(a)-
(f), respectively. When the parameters of the coupled
oscillators are in the blue (red) area, the oscillators are
in AD (oscillating) states. With the increment of the
modulation amplitude α, the AD domain firstly expands
by decreasing the critical frequency mismatches, which
is the lower boundary of the AD domain for given aver-
age coupling strength 0. Then the AD domain shrinks
with the increment of αwhen αis larger than a critical
value αc. Moreover, the AD domains split into two parts
when α > 1 (it is repulsively coupled in some interval
of each period) leading to a kind of ragged AD along
the direction of frequency mismatch ∆ω. It should be
emphasized that the AD is firstly observed to be ragged
in the direction of parameter space ∆ω, as a compari-
son, OD domain is ragged in the direction of parameter
space in the coupled system with a certain spatial fre-
quency distribution [47]. There are two segments of AD
domains in parameter space of ∆ωversus 0, as an ex-
ample, AD state occurs in the intervals of ∆ω∈[3.6,5.4]
and ∆ω∈[9.3,20] as α= 1.1 and 0= 5, which can
be also seen from the vertical line in Fig. 3(d). The
two ragged AD domains keep shrinking and leaving away
from each other with the increment of the modulation
amplitude α. Meanwhile, for a given α= 1.5, we explore
the effects of the modulation frequency ω0by numeri-
cally plotting the phase diagram in parameter space ∆ω
versus 0for ω0= 1,5,10,13,16,19 in Figs. 4(a)-(f ), re-
spectively. The results show that the increment of the
modulation frequency ω0firstly splits the AD domain
into two parts with the upper one larger than the lower
one, then the lower larger one expands while the upper
larger one shrinks. Finally, the two parts merged into one
large AD domain again. To present detailed insight of the
effects of αon the AD domain, the normalized ratio fac-
tor R=S(α)/S(α= 0) is defined to qualify the change
of AD domains under designated regions (0∈[0,20] and
δω ∈[0,20]) in Fig. 5(a) where the modulation frequency
ω0= 3,5,10,15,20, respectively. S(α) is the area of the
AD domains for given αwhile S(α= 0) represents the
area of AD domains for α= 0 (i.e., the red domains in
Fig. 3(a)).
It is obvious that the ratio Rfirstly increases slightly
(expansion of AD domain) and then decreases sharply
to small value (reduction of AD domain) as αincreases
from 0 to 2 for all modulation frequencies ω0. There
is a critical value αcwith which the AD domain gets
to the largest value for each given modulation frequency
ω0. That is to say there is an optimal modulation am-
plitude of the coupling strength with which the coupled
system has the largest AD domain. When α < αc, the
increment of αtends to enlarge the area of AD domain
by shrinking that of the oscillating domain, otherwise,
the AD domain is torn into multi domains by the birth
of oscillating domain and shrinks with the increment of
modulation amplitude α. Interestingly, the optimal value
of αcincreases with the increment of the modulation fre-
quency ω0as shown the inset in Fig. 4(a). The larger
the modulation frequency ωis, the larger the modulation
4
FIG. 3: (color online) The phase diagram of the parameters
αversus ∆ωfor ω0= 4.0 and (a) α= 0.0, (b) α= 0.5, (c)
α= 1.0, (d) α= 1.1, (e) α= 1.4, (f) α= 1.8, respectively.
Where the blue domains are in AD states and the red domains
are the oscillating states.
FIG. 4: (color online) The phase diagram of the parameters
∆ωversus 0for α= 1.4 and (a) ω0= 1.0, (b) ω0= 5, (c)
ω0= 10, (d) ω0= 13, (e) ω0= 16, (f) ω0= 19, respectively.
Where the blue domains are in AD states and the red domains
are the oscillating states.
amplitude is needed to maximize the AD domain.
Now let’s focus on the effects of the modulation fre-
quency ω0on the AD domain. Define the proportion of
the AD domain on the designated regions 0∈[0,20] and
δω ∈[0,20] as P(ω0) = S(ω0)/Stot for given ω0, where
S(ω0) is the area of the AD domain for given ω0, and
Stot is the area of the designated region. Then P(ω0)
versus the modulation frequency ω0can be numerically
presented for an arbitrarily given modulation amplitude
α. Figure. 5(b) presents the results of P(ω0) versus ω0
for given α= 0.6,1.8 respectively. P(ω0) linearly de-
creases with a slow speed for α= 0.6 while obviously
increases for α= 1.8 as ω0increase from 7 to 20. There-
FIG. 5: (color online) (a) The ratio Rdefined in the con-
text versus αfor ω0= 3,5,10,15,20, respectively. The inset
presents the critical value αcversus the modulation frequency
ω0, where αcis the value when Rgets the maximum (the cou-
pled oscillators have largest AD domains in parameter space
(0,∆ω)) (b) The proportion P(ω0) of AD states versus ω0
for a given α= 0.6 (black hollow-dotted line),α= 1.8 (red
solid-dotted line), respectively.
fore, the modulation frequency ω0is beneficial to shrink
AD domain for small modulation amplitude α, otherwise,
the AD domain expands quickly with the increment of
the modulation frequency ω0as αis large (larger than
the maximal αc.
B. B. Coupled Rossler oscillators
AD can also be observed in coupled chaotic oscillators
with time delay coupling [8], and on-off coupling schemes
[1]. It is nature to explore the effects of the periodi-
cally modulated coupling on the AD domain in coupled
chaotic nonidentical oscillators. Let’s consider the cou-
pled chaotic Rossler oscillators with two different time
scales.
˙xi(t) = −ωi(yi(t) + zi(t)),
˙yi(t) = ωi(xi(t) + ayi(t)),
˙zi(t) = ωi(b+zi(t)(xi(t)−c).(4)
where ωirescales the rolling speed of single chaotic
oscillator. The single uncoupled oscillator is in chaotic
regime for given parameters a= 0.15, b= 0.4, c= 8.5,
and has an unstable fixed point (−ay∗,−z∗, z∗) with
z∗= (c−√c2−4ab)/(2a). ω1is arbitrarily set to 2 ,
and the frequency mismatch of two coupled oscillators
is ∆ω=|ω1−ω2|. The periodically modulated cou-
pling term (t) is the same as Eq. 2 with the mod-
ulation frequency ω0= 1. The coupling scheme is
set as Γ =
100
000
000
, (i.e. the interacting variable is
5
FIG. 6: (color online) The phase diagram of the parameters
∆ωversus 0in coupled Rossler oscillators for (a) α= 0.0,
(b) α= 0.5, (c) α= 1.0, (d) α= 1.2, (e) α= 1.4, (f )
α= 1.6, respectively, where the blue domain is AD state, the
red domain is the oscillating state, and the blue domain is the
blowup state.
x(t)). Then the AD domain with different periodic cou-
pling strength can be conveniently observed by present-
ing the phase diagram of parameters ∆ωversus 0for
α= 0,0.5,1.0,1.2,1.4,1.6 as shown in Figs. 6(a)-(f ), re-
spectively. With constant coupling strength (α= 0), the
coupled Rossler oscillator has three states, AD state (v-
shaped blue domain), oscillating state (red domain), and
blowup to infinite (green domain). As α= 0.5 the AD
domain is enlarged while the domain of blowup state is
shrunk. Then the AD domain is ragged into two parts
as αis larger than 1.0. Finally, the AD domain shrinks
while the domain of the blowup state occurs again and
keeps enlarging as αincreases from 1.2 to 2.0.
Figure. 7(a) exhibits the change of areas of AD domain
by the ratio factor Rversus the modulation amplitude α
for given modulation frequency ω0= 1,3,5,10,15,20, re-
spectively. Where the ratio factor Ris defined the same
as the above one with the designated region of (, ∆ω)
space in the range of ∈[0,8] and ∆ω∈[0,8]. The
effects of the modulation frequency and modulation am-
plitude of the coupling strength on AD domains in cou-
pled Rossler oscillators are similar to that in the coupled
Stuart-Landau oscillators. The increment of the modu-
lation amplitude tends to enlarge the AD domain first
then shrinks the AD domain for a given modulation fre-
quency ω0. There is also a critical value αcwith which the
coupled Rossler oscillators have the largest AD domain.
Similarly, we may defined the proportion of blowup do-
main as P(α) = S(α)/Stot for given α, where S(α) is the
area of parameter space of blowup domain and Stot is the
area of domain in the designated area of ∈[0,8] and
∆ω∈[0,8]. Then the effects of αon blowup can be in-
dicated by P(α) versus αas shown in Fig. 7(b) for given
ω0= 1,3,5,8,10, respectively. It is obvious that P(α) is
FIG. 7: (color online) (a) The ratio Rdefined in the context
versus αfor ω0= 1,3,5,10,15,20, respectively. (b) The pro-
portion P(α) of blowup domains versus αfor given ω0= 1
(red line), ω0= 3 (cyan line), ω0= 5 (green line), ω0= 8
(magenta line), ω0= 10 (blue line), respectively. The red
solid dotted line in inset is the proportion of P(α= 2) versus
ω0, and the black hollow dotted line in inset is the critical
value of α0
c, where α0
cis the critical value when p(α)>0.
small and approaches to zero for small αand then grow
up again when αis larger than a critical value α0
c. The
critical value α0
cincreases linearly with the increment of
the modulation frequency ω0as shown the inset in Fig.
7(b). Meanwhile, the increment of ω0tends to decrease
linearly the area of blowup domain according to the rela-
tionship between p(2) versus ω0for given α= 2 as shown
the red solid-dotted line in the inset of Fig.7(b).
III. IV. MECHANISM ANALYSIS
Since the modulated coupling strength varies with
time, the linear stability analysis near the fixed points
is not available to predict the dynamics regimes. Based
on the fact that the conditional Lyapunov exponent [49]
is a valid tool to determine the generalized synchroniza-
tion, it is expected to determine the stability of the
AD state in the coupled nonidentical oscillators. Let
δzi=zi−z∗
i,(i= 1,2) be an infinitesimal perturbation
added to oscillator i, then whether the perturbed trajec-
tories of Eq. 1 could be converged to the fixed point z∗
is mainly determined by the set of variational equations
˙
δz1(t)
˙
δz2(t)=(DF1(z∗
1)) 0
0DF2(z∗
2)δz1(t)
δz2(t)+(t)ΓAδz1(t)
δz2(t)(5)
Here z∗
1= (0,0), z∗
2= (0,0) are the original fixed points
of single oscillators. DF1() and DF2() are the devia-
tions of the two coupled oscillators. Γ is the the cou-
pling scheme (Γ = 1 0
0 1for coupled LS oscillators) and
6
FIG. 8: (color online) The conditional Lyapunov exponent λc
versus ∆ωfor = 7, ω0= 4 and (a) α= 0.0, (b) α= 0.8,
(c)α= 1.0, (d) α= 1.8, respectively. The grey dots are
bifurcation diagram.
A=−1 1
1−1is the link matrix whose eigenvalue is
λ1= 0 and λ2=−2. Solving Eq. 5 numerically for
λ2=−2, we are able to obtain the conditional Lyapunov
exponent λcwith respect to the parameters of the cou-
pling strength and frequency mismatches, based on which
the stable domain of AD, i.e. the domain with λc<0,
can be identified. In Figs. 8(a)-(d), we plot the con-
ditional Lyapunov exponent λcas a function of ∆ωfor
0= 7, and α= 0.0,0.8,1.0,1.8, respectively. The condi-
tional Lyapunov exponent transit to negative from posi-
tive when coupled system transit from oscillating state to
AD state which matches well with the bifurcation results
in grey dots.
To observe clearly how the varying coupling strength
influences the dynamical of the coupled oscillators, the
phase diagrams of x1(t) versus y1(t) for ∆ω= 4.5 (AD
state in numerical results), and ∆ω= 6.5 (oscillat-
ing state in numerical results) are presented in Figs.
9(a)(b), respectively. The value of the normalized cou-
pling strength ((t)/0) is indicated by the color of the
phase diagram. Fig. 9(a) indicates that the oscillator
may leave away from (stage AB) or approach to (stage
BC) the original fixed point (0,0) in each period of the
modulated coupling strength. The speed of approaching
to the original fixed point is larger than that of leaving
away which results to that the coupled system approaches
to the fixed points gradually as time goes to infinite. The
enlarged diagram indicates that the coupled system may
leave away from the original point in some interval of
each period of the modulated coupling after approaching
to the fixed point. Therefore, the oscillator will never
stay on the original fixed point no matter how near it
approach to the original fixed point (in this sense, the
original fixed point is not stable). However, owing to the
finite preciseness of the computer, AD can be observed
in the numerical results when the distance between the
oscillator and the original fixed point is smaller than the
computer’s preciseness. In Fig. 9(b), the oscillator may
also diverge from (stage DE) and approach to (stage EF)
the original fixed point in each period of the modulated
coupling. Noted that the speed of approaching to fixed
point is larger than that of diverging from the fixed point,
however, the time of the former one is shorter than the
later one. As a result, the coupled oscillator forms an
oscillating state. The speed of approaching to or leaving
away from the fixed point is related to the value of the lo-
cal conditional Lyapunov exponent [50] as shown in Figs.
9(c)(d), respectively. Positive local conditional Lyapunov
exponent makes the oscillator leave away from the fixed
point while negative one drive the coupled system to con-
verge to the fixed point. The speed of approaching to or
diverging from the fixed point is related to the absolute
value of the local conditional Lyapunov exponent. Com-
pared the results between Fig. 9(c) and Fig. 9(d), the
final fate of the coupled oscillator is completely deter-
mined by the average value of the local conditional Lya-
punov exponent in one modulation period. The coupled
system is in AD state if the average value of the local con-
ditional Lyapunov exponent is negative, otherwise it is in
oscillating state. The conditional Lyapunov exponent is
also available to predict the AD dynamics in the coupled
Rossler oscillators. Figs. 10(a)-(d) present the condi-
tional Lyapunov exponent together with the bifurcation
diagram on variable x1versus ∆ω. The conditional Lya-
punov exponent gets to negative value when the coupled
Rossler oscillators experience AD which also agrees well
with the bifurcation diagram.
To verify the stability of the fixed points further, we
add the noise on the coupled system,
˙
X1(t) = f1(X1(t)) + (t)Γ(X2(t)−X1(t)) + ξ1(t),
˙
X2(t) = f2(X2(t)) + (t)Γ(X1(t)−X2(t)) + ξ2(t),(6)
where the independent stochastic variables ξi(t) are the
zero-mean white Gauss noises with strength σ, namely,
< ξi(t)>= 0,< ξi(t)ξj(t0)>= 2σδ(t−t0)δij ,γis the same
as the above. If the stability of AD is strong, then the
periodically modulated coupled system with noise may
meander in the vicinity of fixed point. Define the variable
γas the ability of resisting noise as following,
γ=
50, η >= 20,
40,15 ≤η < 20
30,10 ≤η < 15
20,2≤η < 10
10, η < 2,
(7)
where η=max(x1−x∗
1)/σ is the ratio of the maximum
x1(t)−x∗
1to the noise strength σ. Then AD is more stable
if ηis smaller. The phase diagram of γin the parameter
∆ωversus 0of coupled LS oscillators are presented in
Figs. 10 (a)-(b) for α= 1.1, and ω0= 4, as well as that
of coupled Rossler oscillators presented in Figs. 10 (c)-
(d) for α= 1.0, and ω0= 1. According to Figs. 10(a)(b),
7
FIG. 9: (color online) (a) The phase diagram of y1(t) versus
x1(t) for α= 1.1, = 5, ∆ω= 4.5. The inset is the enlarged
part of the red squared area. A,B,C note the position when
the local conditional Lyapunov exponent λc(t) is crossing x
axis as shown in Fig. 9(c). The colors of lines in the Fig. 9
denote the values of the normalized coupling strength (t)/0.
(b) The phase diagram of y1(t) versus x1(t) for α= 1.1, = 5,
∆ω= 6.5. D,E,F note the position when the local conditional
Lyapunov exponent λc(t) is crossing xaxis as shown in Fig.
9(d). The local conditional Lyapunov exponent and the nor-
malized coupling strength (1% of (t)/0) versus time for (c)
∆ω= 4.5 and (d) ∆ω= 6.5
FIG. 10: (color online) The conditional Lyapunov exponent
λc(red lines) versus ∆ωfor = 2 and (a) α= 0.5, (b) α= 1.0,
(c)α= 1.2, (d) α= 1.4, respectively in the coupled Rossler
oscillators (the conditional Lyapunov exponent is magnified
50 times for the convenience of observation in the figure). The
grey dots are bifurcation diagram.
it is easy to conclude that the ragged AD domain in the
upper part (except the edge parts) has stronger stabil-
ity than that in the lower part. Noticed that the upper
part of the ragged AD domain is stable for constant cou-
pling strength (without modulation) while the lower AD
domain is newly born by the modulated coupling which
has less stability than the original AD domain. More-
over, the conclusion is slightly influenced by the noise
FIG. 11: The phase diagram of the parameters ∆ωversus
0with noise strength being (a) σ= 0.001 in coupled SL
oscillators, (b) σ= 0.1 in coupled SL oscillator,(c) σ= 0.001
in coupled Rossler oscillators, (d) σ= 0.1 in coupled Rossler
oscillators. The colorbar determines the values of γin Eq. 7.
intensity.
V. Conclusions
Totally, both the modulation frequency and the modu-
lation amplitude of the periodically modulated coupling
strength significantly influenced the dynamics of the cou-
pled limit cycles or chaotic oscillators. The increment of
the modulation amplitude firstly increases the AD do-
main and then decreases the AD domain as it is larger
than a critical value which is related to the modulation
frequency. That is to say, small modulation amplitude of
the coupling strength is helpful to enlarge the AD domain
of the coupled nonidentical oscillators. However, when
the coupling term is varying between repulsive (negative)
and attractive (positive), the AD domains may be shrunk
and ragged to several parts by the occurrence of oscillat-
ing domain. Meanwhile, the increment of modulation fre-
quency of the periodic coupling tends to slightly decrease
the AD domains for small modulation amplitude while
dramatically increase the AD domains for large modula-
tion amplitude. According to the local conditional Lya-
punov exponent of the periodically coupled oscillators,
one may find that the stability of the AD states is vary-
ing with the coupling strength. Whether the coupled
oscillators can converge to AD state or not is completely
determined by the sign of the averaged conditional Lya-
punov exponent. The periodically modulated coupling
is beneficial to realize AD and is more easy to physical
realization than on-off coupling which needs high speed
stitchers and is difficult to apply, therefore, it has poten-
tial application in the dynamical control in engineering.
Acknowledgement Weiqing Liu is supported by the
National Natural Science Foundation of China (Grant
No. 11765008) and the Qingjiang Program of Jiangxi
University of Science and Technology. Jiangnan Chen is
supported by the Project of Jiangxi province.
8
Reference
[1] Z. Sun,N. Zhao , X. Yang ,W. Xu, Nonlinear Dyn, 92,(3)
1185-1195 (2018).
[2] Y. Kuramoto, Chemical Oscillations, Waves, and Turbu-
lence (Springer, Berlin, 1984).
[3] A. Pikovsky, M. Rosenblum, and J. Kurths, Synchroniza-
tion: A Universal Concept in Nonlinear Sciences (Cam-
bridge University Press, Cambridge, 2001).
[4] G. Song , N. V. Buck and B. N. Agrawal, J. Guid Control
Dyn., 22 (1999) 4330.
[5] A. Koseska, E. Ullner, E. Volkov, J. Kurths, and J.
Garcia-Ojalvo, J. Theor. Biol. 263, 189 (2010).
[6] E. Ullner , A. Zaikin, E. I. Volkov and J. Garcia- Ojalvo,
Phys. Rev. Lett., 99 148103 (2007).
[7] M. Y. Kim , R. Roy, J. L. Aron,T. W. Carr and I. B.
Schwartz, Phys. Rev. Lett., 94 088101 (2005).
[8] A. Prasad, Y. Lai, A. Gavrielides and V. Kovanis , Phys.
Lett. A, 318 (2003) 71.
[9] G. Saxena, A. Prasad, and R. Ramaswamy, Phys. Rep.
521, 205 (2012).
[10] A. Koseska ,E. Volkov and J. Kurths , Europ Lett., 85
28002(2009).
[11] Y. Goto and K. Kaneko, Phys. Rev. E 88, 032718 (2013).
[12] N. Suzuki, C. Furusawa, and K.Kaneko, PLoS ONE 6,
e27232 (2011).
[13] R. Curtu, Physica D 239, 504 (2010).
[14] K. P. Zeyer, M. Mangold, and E. D. Gilles, J. Phys.
Chem. 105, 7216 (2001).
[15] A. Koseska, E. Volkov, and J. Kurths, Phys. Rev. Lett.,
111 024103(2013).
[16] W. Zou, D. V. Senthilkumar, A. Koseska, and J. Kurths,
Phys. Rev. E 88, 050901(R) (2013).
[17] A. Koseska, E. Volkov, and J. Kurths, Phys. Rep. 531,
173 (2013).
[18] R. Herrero, M. Figueras, J. Rius, F. Pi, and G. Orriols,
Phys. Rev. Lett., 84 5312 (2010).
[19] D.G. Aronson, G.B. Erementrout, N. Kopell, Physica D
41, 403-449 (1990).
[20] Y. Wu,W. Liu, J. Xiao, W. Zou, and J. Kurths, Phys.
Rev. E 85, 056211 (2012).
[21] A. Koseska, E. Volkov, and J. Kurths, Chaos 20, 023132
(2010).
[22] G.B. Ermentrout, Physica D 41 219 (1990).
[23] J. Yang., Phys. Rev. E 76 (2007) 016204.
[24] L. Rubchinsky, M. Sushchik, Phys. Rev. E 62 6440
(2000).
[25] F. M. Atay, Physica D 183 1 (2003).
[26] Z. Hou and H. Xin, Phys. Rev. E 68 055103R (2003).
[27] W. Liu, X. Wang, S. Guan, C.-H. Lai, New J. Phys. 11
093016 (2009).
[28] A. Zakharova, M. Kapeller, and E. Scholl,Phys. Rev.
Lett. 112, 154101 (2014.
[29] H. Bi, X. Hu, X. Zhang, Y. Zou, Z. Liu, and S. Guan,
Europhys. Lett. 108, 50003 (2014).
[30] N. Zhao, Z. Sun, X. Yang, and W. Xu, Phys. Rev. E 97,
062203 (2018).
[31] U. K. Verma, A. Sharma, N. K. Kamal, J. Kurths, M. D.
Shrimali, Scientific Reports 7 7936 (2017).
[32] K. Konishi, Phys. Rev. E 68, 067202 (2003).
[33] R. Karnatak, R. Ramaswamy, and A. Prasad, Phys. Rev.
E 76, 035201(R) (2007).
[34] A. Prasad, M. Dhamala, B. M. Adhikari, and R. Ra-
maswamy, Phys. Rev. E 81, 027201 (2010).
[35] W., J. Xiao, L. Li, Y. Wu, M. Lu, Nonlinear Dyn 69:1041-
1050(2012)
[36] T. Banerjee and D. Ghosh, Phys. Rev. E 89,
052912(2014).
[37] W. Liu, G. Xiao, Y. Zhu,etc., Phys. Rev. E 91, 052902
(2015).
[38] Zou, W. Zhan, M., Phys. Rev. E 80, 065204 (2009).
[39] W. Zou, D. V. Senthilkumar,J. Duan, J. Kurths, Phys.
Rev. E 90, 032906 (2014).
[40] A. Prasad,. Phys. Rev. E, 72, 056204 (2005).
[41] L. Chen, C. Qiu, H. Huang, Phys. Rev. E 79, 045101
(2009).
[42] A. Buscarino, M. Frasca, M. Branciforte, L. Fortuna, J.
Sprott, Nonlinear Dyn. 88, 673-683 (2017).
[43] M. Schroder, M. Mannattil, D. Dutta, S. Chakraborty,
M. Timme, Phys. Rev. Lett. 115, 054101 (2015).
[44] S. Li, N. Sun, L. Chen, and X. Wang, Phys. Rev. E 98,
012304 (2018).
[45] A. Prasad,PRAMANA journal of physics 81(03) 0407-
0415 (2013).
[46] K. Suresh, M.D. Shrimali , Awadhesh Prasad, K. Thamil-
maran, Physics Letters A 378 2845-2850 (2014)
[47] H. Ma, W. Liu, Y. Wu, M. Zhan, J. Xiao, Communi-
cations in Nonlinear Science and Numerical Simulation
19(8) 2874-2882 (2014).
[48] X. Sun, M. Perc, and J. Kurths, Chaos 27, 053113 (2017)
[49] K. Pyragas, Phys. Rev. E 56(5) 5183 (1997)
[50] A. Pikovsky, Chaos 3, 225 (1993).
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