Anticipating, complete and lag synchronizations in RC phase-shift network based coupled Chua's circuits without delay.
ABSTRACT We construct a new RC phase shift network based Chua's circuit, which exhibits a period-doubling bifurcation route to chaos. Using coupled versions of such a phase-shift network based Chua's oscillators, we describe a new method for achieving complete synchronization (CS), approximate lag synchronization (LS), and approximate anticipating synchronization (AS) without delay or parameter mismatch. Employing the Pecora and Carroll approach, chaos synchronization is achieved in coupled chaotic oscillators, where the drive system variables control the response system. As a result, AS or LS or CS is demonstrated without using a variable delay line both experimentally and numerically.
Anticipating, Complete and Lag Synchronizations in RC Phase-Shift Network Based
Coupled Chua’s Circuits without Delay
K. Srinivasan1, D. V. Senthilkumar2,3, I. Raja Mohamed4, K. Murali5, M. Lakshmanan1, and J. Kurths3,6,7
1Centre for Nonlinear Dynamics,
Department of Physics, Bharathidasan University,
Tiruchirapalli 620024, India
2Centre for Dynamics of Complex Systems,
University of Potsdam, 14469 Potsdam, Germany
3Potsdam Institute for Climate Impact Research,
14473 Potsdam, Germany
4Department of Physics,
B.S.Abdur Rahman University, Chennai-600048, India
5Department of Physics,
Anna University, Chennai, India
6Institute of Physics, Humboldt University,
12489 Berlin, Germany
7Institute for Complex Systems and Mathematical Biology,
University of Aberdeen, UK
(Dated: April 27, 2012)
We construct a new RC phase shift network based Chua’s circuit, which exhibits a period-doubling
bifurcation route to chaos. Using coupled versions of such a phase-shift network based Chua’s
oscillators, we describe a new method for achieving complete synchronization (CS), approximate lag
synchronization (LS) and approximate anticipating synchronization (AS) without delay or parameter
mismatch. Employing the Pecora and Carroll approach, chaos synchronization is achieved in coupled
chaotic oscillators, where the drive system variables control the response system. As a result, AS or
LS or CS is demonstrated without using a variable delay line both experimentally and numerically.
Synchronization of coupled chaotic systems is a
fundamental nonlinear phenomenon observed in
diverse areas of science and technology.
its detection, different kinds of synchronizations
have been demonstrated both theoretically and
experimentally. The existence and/or transition
between different kinds of synchronization in a
single coupled system have also been reported
by tuning certain system parameters. In partic-
ular, transitions between anticipatory, complete
and lag synchronizations have been demonstrated
in dynamical systems described by both ordi-
nary and delay differential equations by tuning
the delay coupling and also in systems with pa-
rameter mismatch without delay coupling.
this investigation, we will demonstrate the exis-
tence of all the above three types of synchroniza-
tions in coupled RC phase-shift network based
Chua’s circuits by using the Pecora and Carroll
method without any parameter mismatch or de-
lay coupling both experimentally (by using elec-
tronic circuits) and theoretically (by simulating
the normalized evolution equations). The novelty
of our approach is that we introduce a RC phase-
shift network circuit to the coupled Chua’s circuit
which results in complete, lag and anticipating
synchronizations depending upon the drive vari-
able. The method is particularly simple and el-
egant to implement and control. Just by simply
switching the connection of response circuit with
the drive system variable, different kinds of syn-
chronization is shown to result in.
Synchronization of chaotic oscillations has been an
area of extensive research since the pioneering works of
Fujisaka and Yamada  and of Pecora and Carroll .
Chaos synchronization properties of uni- or bidirection-
ally coupled chaotic systems have attracted the attention
of many researchers due to their potential applications in
a variety of fields [3, 4]. Apart from identical or complete
synchronization (CS), other important forms of synchro-
nization have also been identified [3, 5, 6]. Among other
forms of interesting types are the lag  and anticipat-
ing synchronizations [8–10], where coupled systems fol-
low identical phase space trajectories but shifted in time
relative to each other. The anticipating and lag synchro-
nization have been observed in lasers [11, 12], neuronal
models [13, 14], and electronic circuits [6, 15–18].
By using an explicit time delay or memory both lag
and anticipating synchronizations between unidirection-
ally coupled oscillators can be obtained . While lag
synchronization is acheived by coupling the response sys-
tem to a past state of the drive, anticipating synchro-
nization can be obtained by a feedback control using the
current state of the drive compared to the past state
of the response [8, 9]. In both cases, an explicit time
delay appears in the coupling. In particular, the tran-
sition between anticipatory, complete and lag synchro-
nizations has been demonstrated in dynamical systems
described by delay differential equations by tuning the
delay coupling [6, 20]. Another way to achieve approx-
imate lag synchronization in mutually coupled chaotic
oscillator is by using parameter mismatch . Notably,
intermittent and continuous lag synchronizations have
been observed as intermittent steps in a route from phase
to complete synchronization by increasing the coupling
strength [7, 18]. In general, both lag and anticipating
synchronizations with some finite amplitude error in uni-
directionally coupled chaotic oscillators can be achieved
using specific intentional parameter mismatch between
the drive and the response systems .
method for estimating the correlation and time shift be-
tween drive and response oscillators, using a new cou-
pling scheme and linear filter theory, has been demon-
strated in Ref. . Hence, it is of interest to investi-
gate other potential simple methods which can identify
lag/anticipating synchronizations similar to the above
procedures (that is time delay or parameter mismatch).
In this connection, Chua’s circuit and its variants are
well known chaotic circuits that exhibit a wide variety
of nonlinear dynamics phenomena, such as bifurcations
and chaos [4, 22–28]. Pecora and Carroll have proposed
that a subsystem of a chaotic system can be synchro-
nized with a separate chaotic system under certain con-
ditions [2, 29]. This idea of synchronization has been suc-
cessfully applied to a variety of nonlinear systems includ-
ing phase-locked loops, hysteresis circuits etc. [2, 29–31].
In this paper, we have constructed a well-known simple
RC phase shift network  based Chua’s circuit, which
exhibits a period-doubling bifurcation route to chaotic
attractor. Then, we present a new method for achieving
complete, lag and anticipating synchronizations in uni-
directionally coupled chaotic Chua’s oscillators. In this
method, by switching the parameter in the drive sys-
tem, the response system is shown to exhibit CS or LS
or AS, where neither a time delay nor a parameter mis-
match is necessary. In short, the mechanism proposed
in this paper is a very simple and elegant way of achiev-
ing different types of synchronizations in unidirectionally
coupled chaotic oscillators. This method can also be ex-
tended to various applications including signal process-
ing, temporal pattern recognition, secure communication
and cryptography. We demonstrate complete, lag and
anticipating synchronizations in the designed circuit both
numerically and experimentally.
The organization of this paper is as follows. In Sec.
2, the circuit realization of the RC phase shift network
based Chua’s circuit is presented, while in Sec. 3, the
dynamics of the modified Chua’s circuit is given. In Sec.
4, we discuss the three types of synchronizations (com-
plete, lag and anticipating) exhibited by a single set of
Also a new
values of the other elements are fixed as L = 18.0 mH,
C1 = C2 = C3 = 4.7 nF, CL = 100.0 nF and R1 = R2 =
R3 = 610 Ω.
Circuit realization of phase shift network based
Here, N is Chua’s diode.The parameter
unidirectionally coupled Chua’s circuits of the above type
through appropriate switching.
with a summary in Sec. 5.
The paper concludes
The standard Chua’s circuit [4, 26] contains an LC
oscillator connected to a nonlinear element, namely a
Chua’s diode through an RC circuit. The possibility of
constructing an n-dimensional Chua’s circuit using either
an LC or an RC ladder network is indicated in Ref. .
In this paper we design and implement a phase shift net-
work of modified Chua’s circuit by introducing three RC
circuits in it as in Fig. 1. Each RC circuit introduces a
finite phase shift with an attenuation of the signal. The
main objective to design such a circuit is to study dif-
ferent types of chaotic synchronizations in a simple and
elegant way without the introduction of an explicit time
delay or parameter mismatch. Two or more circuits of
this kind can be coupled to form a network and this cou-
pling is made by connecting any one of the three RC cir-
cuits of the drive system to the response system. We find
that depending upon the choice of the RC circuit used
for coupling, the system exhibits different kinds of syn-
chronization and this is the prime reason for introducing
the RC circuits for phase shift.
The RC phase shift network based Chua’s circuit is
shown in Fig. 1. It contains four capacitors C1,C2,C3
and CL, an inductor L, three linear resistors R1,R2and
R3and only one nonlinear element, namely Chua’s diode
By applying Kirchhoff’s laws to this circuit , the gov-
erning equations for the voltage v1across the capacitor
C1, voltage v2across the capacitor C2, voltage v3across
the capacitor C3, voltage vLC across the capacitor CL
and the current iLthrough the inductor L are given by
the following set of five coupled first-order autonomous
nonlinear (piecewise) differential equations
R1(vLC− v1) +
R2(v1− v2) +
R3(v2− v3) − iN,
R1(v1− vLC) + iL,
The term iN= f(v3) represents the v−i characteristics
of Chua’s diode and is given by
f(v3) = Gbv3+ 0.5(Ga− Gb)[|v3+ Bp| − |v3− Bp|], (2)
where Gaand Gbare the inner and outer slopes of the
characteristic curve respectively. Here ±Bpdenotes the
break point of the characteristic curve.
mental parameters of the circuit elements are fixed at
L = 18.0 mH, C1= C2= C3= 4.7 nF, CL= 100.0 nF,
R1 = R2 = R3 = 610 Ω, Ga = −0.76 mS,Gb =
−0.41 mS and ±Bp= ±1.0 V .
Eq. (1) can be converted into a normalized form, con-
venient for numerical analysis by using the following
rescaled variables and parameters, v1= xBp, v2= yBp,
v3 = zBp, vLC = ωBp, iL = (BpG)h, t = CLt?/G,
G = 1/R3. Note that here t?is in dimensionless unit.
The set of normalized equations so obtained are
˙ x = β1(ω − x) + β2(y − x),
˙ y = β3(x − y) + β4(z − y),
˙ z = α[(y − z) − g(z)],
˙ ω = γ(x − ω) + h,
˙h = −β0ω,
where α = (CL/C3R3G), γ
CL/(LG2), β1 = CL/(C1GR1), β2 = CL/(C1GR2),
β3 = CL/(C2GR2) and β4 = CL/(C2GR3). The term
g(z) is obviously represented in the rescaled form as
= 1/(GR1), β0
g(z) = bz + 0.5(a − b)[|z + 1| − |z − 1|].
Here, a = Ga/G and b = Gb/G. The dynamics of Eq. (3)
now depends on the rescaled parameters α, γ, β0, β1,
β2, β3, β4, a and b. The parameter values are fixed as
α = β1= β2= β3= β4= 21.2765, γ = 1, a = −0.462
and b = −0.25, while varying β0.
FIG. 2: Phase portraits of Eq. (3). (a) β0 = 3.383 (Period-T),
(b) β0 = 3.2356 (Period-2T), (c) β0 = 3.194 (Period-4T), (d)
β0 = 3.05 (One-band chaos) and (e) β0 = 2.953 (Double-band
NETWORK BASED CHUA’S CIRCUIT
DYNAMICS OF RC PHASE SHIFT
In this section, we first present the results of our nu-
merical study of system (3) so as to make the under-
lying dynamics clear and then present the correspond-
ing experimental results of the associated circuit (Fig. 1)
described by (1).We fix all the parameter values as
mentioned in the previous section. From the nature of
the numerical results obtained by solving Eq. (3), us-
ing the standard fourth order Runge-Kutta algorithm,
we infer the following picture. We use the system pa-
rameter β0as the control parameter. When β0is varied
from 3.381 downwards, the system exhibits the familiar
period-doubling bifurcation route to chaos, followed by
periodic windows, etc. In addition, a few other interest-
ing dynamical phenomena are also identified by a careful
study through β0scanning. This is illustrated in Figs. 2
in the (z−ω) phase plane. Experimentally, the phase tra-
jectory is obtained by measuring the voltage levels v3and
vLCin the circuit of Fig. 1 and connected to the X and Y
channels of an oscilloscope. The phase trajectory so ob-
tained is shown in Figs. 3. Similar to numerical studies,
experiments reveal a transition from periodic attractor
to chaotic attractor through universal period doubling
The above details can be easily inferred from the one
parameter bifurcation diagram in the (β0−x) plane and
the corresponding three maximal Lyapunov exponents
in the (β0− λm), m = 1,2,3, plane associated with
FIG. 3: The corresponding experimental results of Figs. 2.
Vertical scale 2v/div., horizontal scale 0.5v/div.
Eq. (3) which are given in Figs. 4. In particular, the
standard period-doubling bifurcation sequence to chaos
and windows have been observed for a range of parame-
ter values, 3.0 < β0< 3.5. For example, it is clear that
for β0 > 3.3 there is a limit cycle attractor of period-
T. At β0 = 3.26, a period doubling bifurcation occurs
and a period-2T limit cycle develops and is stable in
the range 3.26 > β0 > 3.2. When β0 is decreased fur-
ther the period-2T limit cycle bifurcates to a period-4T
(3.19 > β0 > 3.18) attractor. Further period doubling
occurs when β0< 3.18 giving rise to 8T and 16T period
limit cycle, respectively. The chaotic attractor (single
band) is first observed at β0= 3.175. Further decrease in
the parameter (β0) of the system causes it to admit dou-
ble band chaotic nature. For 3.0 > βo> 2.5, the dynam-
ics is even more complicated and intricate. This interval
of β0is not fully occupied by chaotic orbits alone. Many
fascinating changes in the dynamics like reverse period-
doubling bifurcations, periodic orbits (windows), period-
doubling of windows, intermittency and antimonotonic-
ity take place at different critical values of β0 in this
At β = 2.067, the system (3) exhibits double band
chaotic attractor which is shown in Fig. 5(I) for different
projections of phase space and the corresponding exper-
imental analysis is shown in Fig. 5(II). For the same β
value, the time series plot is presented in Fig. 6. The
phase shift in x, y and z can be clearly seen in Fig. 6 due
to the three RC phase shift circuits shown in Fig. 1.
FIG. 4: One parameter bifurcation diagram (β0− x) for the
parameter values α = β1 = β2 = β3 = β4 = 21.2765, γ = 1,
a = −0.462 and b = −0.25.
FIG. 5: (I) Double band chaotic attractors of the system (3):
(a) (x − ω) plane, (b) (y − ω) plane and (c) (z − ω) plane.
(II) Phase portraits (experimental) of chaotic attractor from
the circuit (Fig. 1), (a) (v1−vLC); vertical scale 1v/div., hor-
izontal scale 0.5v/div., (b) (v2− vLC); vertical scale 1v/div.,
horizontal scale 0.5v/div. and (c) (v3− vLC); vertical scale
2v/div., horizontal scale 0.5v/div.
IV. COUPLED RC PHASE SHIFT NETWORK
BASED CHUA’S CIRCUITS: LAG AND
Next, we study the dynamics of coupled phase-shift
network based Chua’s circuits which is shown in Fig. 7.
FIG. 6: Time-series plot for double band chaotic attractor of
system (3). Note the phase shift between the three dynamical
variables x, y and z.
FIG. 7: Circuit realization of coupled RC phase shift network
based Chua’s circuit.
The network is made by connecting the drive circuit to
the response circuit through a buffer and a gain amplifier.
Here, the single phase-shift network based Chua’s circuit
acts as the drive circuit and the governing circuit equa-
tion for the drive part is nothing but Eq. (1). Depending
upon the value of the feedback resistance of the standard
inverting amplifier, the gain can be fixed. Using the Pec-
ora and Carroll approach of building an identical copy
of the response subsystem, we demonstrate three types
of chaos synchronization, namely complete, lag and an-
ticipating synchronizations in the proposed Chua’s cir-
cuit both experimentally and numerically without the
introduction of any time-delay or parameter mismatch.
When the voltage across C2is used to drive the subsys-
tem, complete synchronization is observed. On the other
hand when voltage across C1and C3are used, lag and
anticipating synchronizations are observed. By simply
connecting (switching) the response system to the drive
system through either of the three terminals 1, 2 and 3
(shown in Fig. 7) we observed three types of synchroniza-
tions. The reason for introducing the different coupling
of state variables v1, v2and v3is to observe all the three
types of synchronization, namely lag, identical and an-
ticipating synchronizations, in a simple manner. This is
achieved by exploiting the finite phase-shift that is being
introduced by the individual RC network element.
The governing circuit equations are given below.
(a) Drive system :
Same as Eq. (1), including the circuit parameter values.
(b) Response system :
(? × v2− v?
LC) + i?
The term iN= f(v3) represents as before the v−i char-
acteristics of the Chua’s diode. The parameters of the
circuit elements are fixed at L?= 18.0 mH, C?
the gain term. Eqs. (5) are now rescaled as follows: with
Then the rescaled version of the equation is given as
(a) Drive system :
Same as Eq. (3), including the parameter values.
(b) Response system :
1= 4.7 nF,
L= 100.0 nF, R?
1= 610 Ω and R?
2= 610 Ω. Here ? is
1= x?Bp, v?
LC= ω?Bp, i?
L= (BpG)h?, t = CLt?/G.
˙ω?= γ?(x?− ω?) + h?,
values become (due to the above choice of the circuit pa-
1(ω?− x?) + β?
2(??× y − x?),
2), γ?= 1/(GR?
1) and ??= ?. The parameter
2= 21.2765, γ?= 1.0 and β?
The unidirectionally coupled RC phase shift network
based Chua’s circuit (Fig. 7) shows that the response sub-
system contains an identical oscillator as that of the drive
system. Here, the drive system controls the response,
through the drive component in Eq. (5a) and Eq. (6a).
Connecting the response system to the terminal 2 (see
Fig. 7) of the drive system, we observed complete syn-
chronization. With ? = 1.0 and the y-drive component
coupled through one way coupling with the response sub-
system, the coupled oscillators exhibit identical synchro-
nization. Time series of the drive and response variables
(ω & ω?) are shown in Fig. 8(a) and the corresponding
experimental plot is shown in Fig. 9(a). In Fig. 9 the
horizontal axis is calibrated as 200 µs/div. and the ver-
tical axis is 0.5 v/div. From these figures, it is clearly
seen that both the phase and amplitude of the drive and
response systems are coinciding, proving that the cou-
pled system of Fig. 7 exhibits complete synchronization.
FIG. 8: Numerically observed drive (red, ω) and response
(black, ω?) wave forms showing: (a) identical synchroniza-
tion, (b) approximate anticipating synchronization, (c) ap-
proximate lag synchronization in coupled chaotic oscillators
and (c’) enlarged version of Fig. 8(c).
The same is observed in the phase space plots, shown in
Now, connecting the response system to the terminal 1
(see Fig. 7) of the drive system variable v1is coupled with
the response subsystem and correspondingly Eq. (5a) and
Eq. (6a) get modified respectively as
(? × v1− v?
1(ω?− x?) + β?
2(??× x − x?).
FIG. 9: Experimentally observed drive (yellow, vLC) and re-
sponse (green, v?
chronization, (b) approximate anticipating synchronization,
(c) approximate lag synchronization in coupled chaotic oscil-
lators and (c’) enlarged version of Fig. 9(c). Vertical scale
0.5v/div.: horizontal scale 200µs/div.
LC) wave forms showing: (a) identical syn-
FIG. 10: (i) Numerically observed drive (ω) and response (ω?)
phase plane in (ω−ω?): (a) identical synchronization, (b) ap-
proximate anticipating synchronization and (c) approximate
lag synchronization in coupled chaotic oscillators. (ii) Exper-
imentally observed (vLC) and response (v?
LC) phase plane in
LC). Vertical scale 1v/div., horizontal scale 1v/div.
For ? = 1.66 and x-drive component coupled through one
way coupling with the response subsystem, the coupled
oscillators exhibit anticipatory synchronization.
ures 8(b) and 9(b) depict the time series plot of ω and ω?
numerically and experimentally. From these figures we
can observe that ω?anticipates ω. In other words, the
response system anticipates the drive system, thereby we
can infer the anticipatory synchronization of the system
shown in Fig. 7. Anticipating synchronization can also
be inferred from the phase space plot shown in Fig. 10(b).
The degree of synchronization with the corresponding
time shift τ can be quantified using the similarity func-
tion  defined as
S2(τ) =?[ω?(t?− τ) − ω(t?)]2?
where ?ω? means the time average over the variable ω. If
the signals ω(t?) and ω
ence between them is of the same order as the signals
themselves.If ω(t?) = ω
plete synchronization, the similarity function reaches a
minimum S2(τ) = 0 for τ = 0. But for the case of a
nonzero value of time shift τ, if S2(τ) = 0, then there
exists a time shift τ between the two signals ω(t?) and
larity function S2(τ) as a function of the coupling de-
lay τ for the three different values of ?. One may note
that the minimum of S2(τ) ≈ 0.024 occurs at τ ≈ 0.25
for ? = 1.66. This indicates that there exists a time
shift, corresponding to an anticipating time τ ≈ 0.25
time units, between the two signals in Fig. 8(b) such
that ω?(t?−τ) ≈ ω(t?) demonstrating approximate antic-
ipactory synchronization. Translated into experimental
units (see Sec. III) this time shift works out to be ap-
proximately 15.25µs. The nearness of S2(τ) to the value
zero quantifies the degree of synchronization and hence
S2(τ) ≈ 0.024 attributes to the approximate anticipatory
synchronization as observed in Figs. 8(b) and 9(b). It is
also to be noted that for slightly higher and lower values
of ?, the minimum of S2(τ) occurs at the same τ but with
further reduced degree of synchronization, indicated by
their respective minima of S2(τ), than that for ? = 1.66.
Next, connecting the response subsystem to the ter-
minal 3 (see Fig. 7) of the drive system variable v3 is
coupled and correspondingly Eq. (5a) and Eq. (6a) get
?(t?) are independent, the differ-
?(t?), as in the case of com-
?(t?) such that ω
?(t?− τ) = ω(t?), demonstrating an-
Figure 11 shows the simi-
(? × v3− v?
1(ω?− x?) + β?
2(??× z − x?).
For ? = 0.7 and the z-drive component coupled through
one way coupling to the response subsystem, the coupled
oscillators exhibit lag synchronization. Numerical and
experimental plots of time series of ω and ω?or vLCand
FIG. 11: Similarity function S2(τ) corresponding to Fig. 8(b)
confirming anticipatory synchronization for different values of
? (red, ? = 1.66; black, ? = 1.58; cyan, ? = 1.72).
FIG. 12: Similarity function S2(τ) corresponding to Fig. 8(c)
confirming lag synchronization for different values of ? (red,
? = 0.7; black, ? = 0.72; cyan, ? = 0.68).
response system variable ω?lags the drive system vari-
able ω. In other words the circuit (Fig. 7) exhibits lag
synchronization. The phase space plots for lag synchro-
nizations is shown in Fig. 10(c). Further, in the present
case also we can use the same similarity function S2(τ)
to characterize lag synchronization with a positive time
shift τ instead of the negative time shift τ in Eq. (9). Fig-
ure 12 shows the similarity function S2(τ) vs τ for three
different values of ?. The minimum of similarity function
becomes S2(τ) ≈ 0.004 at τ ≈ 0.078 indicating that there
is a time shift (Fig. 12) between the drive and response
signals ω(t?) and ω
confirming the occurrence of lag synchronization. The
minimum of S2(τ) ≈ 0.004 corresponds to the approx-
imate lag synchronization with lag time τ ≈ 0.078 be-
tween ω(t?) and ω
an experimental lag time 4.76µs, approximately.
LCare shown in Figs. 8(c) and 9(c). In this case, the
?(t?), such that ω
?(t?+ τ) ≈ ω(t?),
?(t?) in Fig. 8(c), which corresponds to
V.SUMMARY AND CONCLUSION
In this paper, we have constructed a three RC phase
shift network based Chua’s circuit, which exhibits a
period-doubling bifurcation route to chaos. Further, we
have confirmed the generation of chaos by calculating
the Lyapunov exponents and have investigated the re-
lated bifurcation phenomena. Without introducing any
time delay or parameter mismatch, different chaotic syn-
chronizations are achieved by switching the drive system
variables, using a one way coupling approach to the re-
sponse subsystem. We have explored the effectiveness of
the approach using numerical simulations and the corre-
sponding experimental results. As a result, the complete,
lag and anticipating synchronizations are controlled by
the drive component on the response system. It is worth
mentioning that the three types of synchronization can
also be realized for a similar identical response circuit
with error feedback coupling, which will be discussed sep-
arately. By extending the RC phase-shift networks cas-
cadingly, one can realize a simple delayed chaotic circuit
with the delay element simply made-up of RC networks.
Further, by exploiting phase synchronization in such net-
works, one can realize phase-synchronization based logic
elements . Such possibilities will be explored in fu-
The authors are very grateful to an anonymous ref-
eree for some very valuable and critical comments which
helped to improve the presentation of the results in Sec.
IV.B. The work of K.S. and M.L. has been supported
by the Department of Science and Technology (DST),
Government of India sponsored IRHPA research project,
and DST Ramanna program and DAE Raja Ramanna
program of M.L. D. V. S. and J. K. acknowledges the
support from EU under Project No. 240763 PHOCUS
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