NATIONAL CONFERENCE ON NONLINEAR SYSTEMS & DYNAMICS, NCNSD-2003
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Experiment on Transition to Phase Synchronization
in Coupled Chua's Oscillators
Syamal Kumar Dana, Prodyot Kumar Roy, Satyabrata Chakraborty

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S.K.Dana (e-mail:sdana_ecsu@yahoo.com) and S.Chakraborty
(e-mail:satya_iicb@yahoo.com) are with the Instrument Division, Indian
Institute of Chemical Biology, Kolkata 700032. P.K.Roy is with the
Department of Physics, Presidency College, Kolkata 700073 (email:pkpresi@yahoo.co.in)
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YNCHRONIZATION of coupled oscillating systems is
an important topic [1-3] of research in nonlinear
dynamics. One can observe different regimes of
synchronization in coupled chaotic systems, namely, complete
synchronization (CS), generalized synchronization (GS), lag
synchronization (LS), intermittent lag synchronization (ILS)
and phase synchronization (PS) with decreasing coupling
strength. CS is observed for strong coupling when the
interacting states of coupled systems coincide both in
amplitude and phase, the state vectors x1(t)=x2(t), t, while
in GS, a functional relationship x1(t)=F(x2(t)), t is found
true between interacting nonidentical oscillators. PS is
observed for weaker coupling when the phases 1,2 of the
interacting oscillators satisfy a n:m locking relation, |n1m2|<constant. On the other hand, LS is characterized by the
coincidence of the state of drive system with the state of
response system shifted in time i.e., x1(t-)=x2(t), where  is
the lag time. ILS has been characterized in terms of the
existence of a set of lag times n (n=0, 1, 2, 3…) such that the
system always verifies x1(t-n)=x2(t) for a given n. In this
regime, the coupled systems most of the time remain in the
principal LS configuration (=0) but occasionally visits other
close LS configurations (n, n=1, 2, 3…) too. Two
transition routes from CS to PS with decreasing coupling
strength have been observed one through LS and ILS [4-7]
and another through intermittency [8-9]. In this paper,
experimental observations on transition route to PS are
presented using two nonidentical Chua's oscillators
where x1=[Xd Yd Zd]T and x2=[Xr Yr Zr]T are the state vectors of
the driver and response oscillators respectively. T denotes
transpose of the matrices. S() gives a measure of phase lag
between the driver and response signals. It shows a global
minimum 0=S2min(0) with a lag minima min=0 that
indicates the existence of a principal lag time between the
interacting oscillators. There also exist other local minima
n=S2min(n) for n0 (n=1, 2, 3….). In case of CS, Xd(t0)=Xr(t) when 0=0 for 00, while in case of LS, Xd(t0)=Xr(t) is satisfied with global minimum  0 at nonzero lag
minimum 00. ILS regime is characterized by nonzero 0
with comparatively larger values of n and nreflecting the
coexistence of many lag configurations. However, the
nonzero 0 is small in the ILS regime in comparison to its
larger value in the PS regime. But PS cannot be identified
from the similarity function. To identify PS, the instantaneous
phase of each oscillator (drive and response) is estimated
from measured scalar signals by using Hilbert transform [13]. Phase difference between the oscillators remains bounded
in the PS regime above a critical coupling but it shows regular
2 phase slips for coupling weaker than the critical value.
V-
S
(1)
( X d 2 (t ) X r 2 (t ) )1/2
4
INTRODUCTION
[ X d (t   )  X r (t )]2
7
I.
S 2 (t ) 
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Keywords- Chaotic oscillators, Complete synchronization, Lag
synchronization, Phase synchronization.
coupled in unidirectional mode. In order to verify the extent
of synchronization between the coupled oscillators, a
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Abstract- Experimental observations on the effect of coupling
on synchronization of two coupled nonidentical Chua’s
oscillators are presented in this paper. Two oscillators are
coupled in unidirectional drive-response mode. The driver is
always kept in single scroll chaotic state while the response
oscillator is kept in various dynamical states as point attractor
and chaotic. With decreasing coupling strength, two routes of
transitions to phase synchronization, one through lag and
intermittent lag synchronization and another route through
intermittency have been observed in single scroll cases.
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Fig.1. Two Coupled Chua’s Circuit
(d)
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INDIAN INSTITUTE OF TECHNOLOGY, KHARAGPUR 721302, DECEMBER 28-30, 2003
duration =40s.
Fig.2. Oscilloscope display (X-Y plot): Drive in CH-1 and response in CH-2, (a) CS for RC=51, (b) LS for RC=718, (c) ILS for RC=2256,
(d) PS for RC=4857. Drive is single scroll chaotic and response is in stable equilibrium
(excitable state),
d=1449, Rr=2099.
couplingstate
resistance)
with Rsuccessive
transitions of PS,
II. EXPERIMENTAL RESULTS
Experimental circuit of two coupled Chua’s oscillators in
drive-response configuration is shown in Fig.1. Each Chua's
oscillator consists of inductor L1,2 with a leakage resistance
R01,02, capacitors C1,3 and C2,4 and a nonlinear resistance,
which is approximated by a piecewise linear function and
designed by using two Op-amps (uA741). The coupling is
made unidirectional by using another Op-amp (uA741) with a
series resistance Rc. When Rc is increased the coupling
strength decreases and vice versa.
The driver is always kept in single scroll chaotic state. The
response is only changed to either of the states, namely, stable
excitable state (point attractor), single scroll chaotic. The
coupling strength has been decreased from very strong to
weaker limit by increasing Rc when transitions from CS to PS
and then to nonsynchronous state have been observed. One
transition route to PS is through LS and ILS as shown in Fig.2
for single scroll driver and stable excitable (point attractor)
response. Oscilloscope pictures of Xd vs Xr plots are shown in
Figs.2(a)-(d) for CS, LS, ILS and PS regimes respectively for
increasing (decreasing) coupling resistance (strength). The
corresponding similarity function is shown in Fig.3. Figure
3(a) shows a global minimum 0=S2min(0) with a lag
minimum min=0 for each regime, which is the principal lag
time of the interacting oscillators. There also exist other local
minima S2min(n) forn 
the enlarged version of principal lag time (first both 0 and
minima) in Fig.3(a). It may be noted that the values
of 0 decrease with increasing coupling strength (decreasing
Fig.3. Similarity function with lag time shows periodic variations
in (a) for CS (bold line, 0=.004,0=0), LS (dotted line, 0=0.01,
0=4s), ILS (dashed line, 0=.028, 0=12s ), PS (dashed-dotted
line, 0=0.062, 0=24s). They are enlarged in (b) for lag time
1
0
Xd (t-0 )
1
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0.5
1
Xr(t)
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Xd (t-0 )-Xr(t)
0
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10
20
t(ms)
ILS,
LS and CS. In case of CS (solid line), Xd(t-0)=Xr(t) where
4
(a)
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S2 ()
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400
600
800
0.25
(b)
S2 ()
0.13
0
0
10
20
30
40
s)
00 for 00, whereas in case of LS (dotted line), Xd(t0)=Xr(t) is satisfied for global minimum 00.01 for 00
(4s). The delayed difference Xd(t-0)-Xr(t) is bounded to
zero line for global lag minimum 0=4s as shown in Fig.4(b)
in the LS regime. While Xd(t-0) vs. Xr(t)
Fig.4. Lag synchronization: single scroll chaotic driver and stable excitable
response, Rd=1449, Rr=2099, Rc=718 (a) phase portrait of Xd(t-0)
vs.(t), (b) delayed difference between drive and response. 0=4s.
NATIONAL CONFERENCE ON NONLINEAR SYSTEMS & DYNAMICS, NCNSD-2003
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In the PS regime, amplitudes are very weakly correlated
although the instantaneous phases of the oscillators have a
locking relation. In simplest case of 1:1 locking the phase
difference is given by =|d-r|<constant as mentioned
1
Xd (t-0 )
(a)
0
1
2
0
1
2
0
3
Xr(t)
2

1
4
Xd (t-0 )-Xr(t)
6
0
(b)
1
10
8
20
t ( ms )
plot shows a 45 line (cf. Fig.2(b), Xd(t) vs. Xr(t) plot shows
a loop due to LS) defined as the synchronization subspace.
Global lag minimum of 0=4s in the LS regime is calculated
from the similarity function. In the ILS regime, bubbling in
Xd(t) vs. Xr(t) plot in Fig.5(a) and
corresponding
largeintermittent spike bursts in the delayed difference plot in
Fig.5(b) are observed for principal lag minimum 0=12s. In
this regime, the coupled oscillators also visit other lag
configuration n as shown in Fig.6. A part of the time series of
delayed difference for t=6ms with fewer spiking bursts is
shown in Fig.5(b) for comparison. In the upper trace two
spike bursts are seen for principal lag time 0=12s otherwise
the delayed difference is bounded. In the lower trace, for
other lag configuration n=48s, the delayed difference Xd(tn)-Xr(t) is not bounded except at the spiking instances in the
upper trace. A close inspection can reveal that the delayed
difference is appreciably small in the lower trace at instants of
large spikes in upper trace. This confirms that the coupled
system visits other lag configuration too in the ILS regime.
1
(b)
Xd (t-0 )-Xr(t)
Fig.7.Phase synchronization: [Rd=1449Rr=2099 Rc=4875upper
trace). 2 phase jump in middle trace for weaker coupling (R c=7900 A
few 2 phase jumps are seen in the lower trace for coupling
Rc=8200Middle and lower traces are scaled down for visual clarity.
above. The phases d(t) and r(t) are estimated by using
hilbert transform on measured voltage signals at capacitors C2
and C4 respectively.  is bounded to zero for Rc=4875 as
shown in the upper trace in Fig.7. With decreasing coupling
 remains bounded until at critical coupling
Rc=7900when a 2 phase jump appears. More and more
2 phase slips appear with increasing coupling resistance
(decreasing coupling strength) and eventually the oscillators
move to nonsynchronous state.
Fig.8. Oscilloscope display of Xd(t) vs. Xr(t) plot: Rd=1439,Rr=1441
Both drive and response are single scroll chaotic, (a) CS,
Rc=669bintermittency, Rc=4.36kcPS, Rc=33.79kwhich is
imperfect here.
The intermittency route is shown in Fig.8 when both driver
and response are single scroll chaotic. CS and PS are shown
in the oscilloscope pictures in Fig.8(a) and 8(c) respectively.
Intermittency is observed for intermediate coupling strength
as shown in Xd(t) vs. Xr(t) plot in Fig.8(b). Plots of Xd(t)-Xr(t)
and corresponding phase difference  are given in
Fig.9 for the case shown in Fig.8(b). The trajectories of driver
and response mostly follow each other i.e., remain bounded to
0
Xd (t-n)-Xr(t)
(b)
1
2
20
t(ms)
Fig.5. Intermittent lag synchronization: Rd=1449Rr=20990=12s,
(a) phase plane plot of delayed driver and response signals, (b) delayed
difference shows large intermittent spikes. Coupling resistance R c=2256
(a)
10
4
6
t ( ms )
Fig.6. Coexisting lag configurations: principal lag minimum, 0=12s
(upper trace). Other lag configuration n=48s (lower trace) is scaled
down for visual clarity.
(c)
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(a)
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INDIAN INSTITUTE OF TECHNOLOGY,
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2
Xd (t)-Xr(t)
0
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synchronization subspace tangential to 45 line but
occasionally moves transversely away from the subspace.
Only imperfect PS is observed in this case, when shows
occasional jumps with long epoch between jumps. Details
are not shown due to space limitation.
III. CONCLUSION
Transition from CS to PS with decreasing coupling strength,
mainly, follows two routes. One route follows intermittency,
while other route follows successive regimes of LS and ILS.
LS and ILS are found to be present when parameter mismatch
between drive and response is sufficiently large as observed
in first experiment. The mismatch is reduced as in single
scroll drive and response, the LS and ILS are absent.
Response of excitable driven oscillator forced by the chaotic
drive cannot be called as synchronization, it is rather a
resonance phenomena in true sense of the term. Whether we
call it synchronization or resonance, transition routes remain
same.
Fig.9. Intermittency:  in upper trace remains bounded except few 2
jumps at the instant of intermittent large spikes shown in time domain plot
of Xd(t)-Xr(t) in the lower trace. Lower trace is scaled down for visual
clarity. Rd=1439, Rr= 1441
ACKNOWLEDGEMENTS
This work is partly supported by BRNS/DAE under Grant
no.2000/13/34/BRNS.
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