Experimental observations of time delay induced amplitude death in a pair of
coupled nonlinear electronic circuits that are individually capable of
exhibiting limit cycle oscillations are described. In particular, the existence
of multiply connected death islands in the parameter space of the coupling
strength and the time delay parameter for coupled identical oscillators is
established. The existence of such regions was predicted earlier on theoretical
grounds in [Phys. Rev. Lett. 80, 5109 (1998); Physica 129D, 15 (1999)]. The
experiments also reveal the occurrence of multiple frequency states, frequency
suppression of oscillations with increased time delay and the onset of both
in-phase and anti-phase collective oscillations.Comment: 4 aps formatted RevTeX pages; 6 figures; to appear in Phys. Rev. Let

A. A. Brailove

A. Hohl

A. Sen

D. G. Aronson

D. V. Ramana Reddy

E. Niebur

E. Sismondo

G. L. Johnston

J. Benford

J. J. Collins

M. A. Branham

M. F. Crowley

M. G. Rosenblum

M. K. S. Yeung

M. Kawato

M. Komuro

M. Shiino

M. Yoshimoto

P. C. Matthews

R. Herrero

S. H. Strogatz

S. Kim

Y. Kuramoto

English

arXiv

Crossref

Experimental Evidence of Time-Delay-Induced Death in Coupled Limit-Cycle Oscillators

Experimental observations of time-delay-induced amplitude death in two coupled nonlinear electronic circuits that are individually capable of exhibiting limit-cycle oscillations are described. The existence of multiply connected death islands in the parameter space of coupling strength and time delay for coupled identical oscillators is established. The existence of such regions was predicted earlier on theoretical grounds [Phys. Rev. Lett. 80, 5109 (1998); Physica (Amsterdam) 129D, 15 (1999)]. The experiments also reveal the occurrence of multiple frequency states, frequency suppression of oscillations with increased time delay, and the onset of both in-phase and antiphase collective oscillations

Ramana Reddy, D. V.

Sen, A.

Johnston, G. L.

Indian Academy of Sciences

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Experimental Evidence of Time Delay Induced Death in Coupled Limit Cycle
Oscillators
D. V. Ramana Reddy, A. Sen, and G. L. Johnston∗
Institute for Plasma Research, Bhat, Gandhinagar 382428, India
Experimental observations of time delay induced amplitude death in a pair of coupled nonlinear
electronic circuits that are individually capable of exhibiting limit cycle oscillations are described. In
particular, the existence of multiply connected death islands in the parameter space of the coupling
strength and the time delay parameter for coupled identical oscillators is established. The existence
of such regions was predicted earlier on theoretical grounds in [Phys. Rev. Lett. 80, 5109 (1998);
Physica 129D, 15 (1999)]. The experiments also reveal the occurrence of multiple frequency states,
frequency suppression of oscillations with increased time delay and the onset of both in-phase and
anti-phase collective oscillations.
PACS numbers: 05.45.Xt, 87.10.+e
Coupled limit cycle oscillator models have been exten-
sively studied in recent years because of the useful in-
sights they provide into the collective behaviour of many
physical, chemical and biological systems [1–4]. One of
the simplest such models, the so-called Kuramoto model
[1], which retains only the phase information of each os-
cillator and is valid in the limit of weak mutual cou-
plings, displays a spontaneous transition to a synchro-
nized collective state of a single frequency when the cou-
pling strength exceeds a critical value. Similar collective
behaviour is observed in many natural systems, such as
the synchronous flashing of fireflies, the phase locking of
cardiac pacemaker cells, and the collective chirping of
crickets [4]. Phase locking has also been demonstrated
experimentally in arrays of coupled nonlinear electronic
circuits [5]. When the coupling becomes stronger, am-
plitude effects become important and give rise to other
interesting collective states, such as that of amplitude
death in which the various oscillators pull each other off
their periodic states and collapse to a state of zero am-
plitude. The condition for such a state to occur is for
the oscillators to have a broad dispersion in their natu-
ral frequencies and for the coupling strength to exceed
a threshold value. Thus, as has been pointed out in a
number of theoretical studies [6], a collection of identi-
cal limit cycle oscillators cannot display amplitude death.
However, more recent investigations [7] indicate that the
presence of finite propagation time delays in the coupling
removes this restriction and predict the possibility of in-
ducing the death state even in a system of two coupled
identical limit cycle oscillators. Time delay is ubiquitous
in most physical systems due to finite propagation speeds
of signals, finite chemical reaction times, finite response
times of synapses, etc., and its influence on the collective
dynamics of coupled systems can have wide-ranging im-
plications [8]. It is important therefore to establish the
experimental feasibility of such a death phenomenon.
In this Letter we present experimental observations on
time delay induced death in two coupled nonlinear cir-
cuits that are individually capable of exhibiting limit cy-
cle oscillations. A specially designed digital delay line
gives precise control over the delay time in the coupling
and permits us to explore a large area of parameter space.
Observations on the phenomenon of amplitude death have
recently been reported [9] for a pair of optothermal oscil-
lators that are thermally coupled and for which the oc-
currence of death for strong couplings and its relation to
Hopf bifurcations of the uncoupled and coupled systems
have been experimentally verified. However, as the au-
thors themselves point out, their experimental results are
not conclusive about the role of delay in the death phe-
nomenon and in particular they have not investigated the
phenomenon of death islands or, indeed, that of the mul-
tiplicity of death islands predicted by the theory of time
delay induced death [7]. In our experiments we provide
clear evidence of time delay induced death islands and
their multiple connectedness in the parameter space de-
fined by the coupling strength, time delay and frequency,
for the case of two coupled identical oscillators. We also
find the existence of multiple frequency states, frequency
suppression of oscillations with increased time delay and
the onset of both in-phase and anti-phase collective os-
cillations.
Our experimental system, schematically shown in
Fig. 1, consists of two nonlinear LCR circuits coupled
through a digital delay line (DDL). The individual oscil-
lator circuits are a variant of the so-called Chua circuit,
which has been widely employed in a number of nonlinear
dynamical studies [10]. The nonlinear resistive element
RN has the typical V −I characteristic, with negative re-
sistance around the origin and positive slopes away from
the origin as shown in Fig. 2. This enables each indi-
vidual oscillator circuit in the uncoupled state to sustain
limit cycle oscillations with a characteristic frequency of
1
1/
√
LC. The two channel DDL taps the input signal at a
certain rate using an analog to digital converter (in this
case an ADC0809 with a conversion rate of 100µs) and
stores it in a random access memory (RAM) bank. The
stored bits are read using a logical circuit and converted
back to an analog signal to be fed back into the circuit.
The coupling between the oscillators is linear, resistive
and proportional to the difference in the signal strengths
of the two oscillators with a time delay. The coupling
strength is varied by changing the resistances (RK) that
join the two oscillators. By using the current summation
rules at nodes V1 and V2, a theoretical description of the
circuit can be given as,
V̈i + g(Vi) V̇i + ω
2
i Vi = Ki[V̇j(t − τ) − V̇i(t)], (1)
where i, j = 1, 2, i 6= j and g(Vi) = (1/Ci)(−0.95 +
0.66V 2i +322.0V
4
i )×10−5 Amp/F, the nonlinear damping
factor, is obtained from an empirical fit of the V −I char-
acteristics of the nonlinear resistive element as shown in
Fig. 2. The frequencies are ωj = 1/
√
LjCj and the cou-
pling strengths Kj = 1/(CjRKj ). For our experiments
on identical oscillators we have fixed C1 = C2 = 0.1µF
and varied L = L1 = L2 and R = RK1 = RK2 such that
ω = ω1 = ω2 spans the range of 100s
−1 to 1000s−1.
-
+
C1
1
RK K2R
RN1L1 C22NR
V2
τ V1τ
L2
1V V2
++
-
DDL
FIG. 1. The circuit diagram of two delay coupled limit cy-
cle oscillators consisting of a C, L and a nonlinear resistance
RN connected in parallel. The OP-AMPs are buffer ampli-
fiers and ideally draw no input currents. The digital delay
line outputs V τ1,2 = V1,2(t − τ ).
−0.8 −0.4 0 0.4 0.8
−3
−2
−1
0
1
2
3
x 10
−5
 V (Volts)
 I 
(A
m
ps
)
 o :  Experimental
 −−  : polynomial fit
FIG. 2. The V-I characteristics of the nonlinear component
RN . The continuous line is a polynomial fit of the experimen-
tal points.
The coupling strength varies in the range of 10s−1 to
106s−1. Examples of our experimental results for the
time evolution of the oscillator voltages as a function of
the delay parameter are shown in Fig. 3 for K = 1000s−1
and ω = 837s−1. The identical oscillators acquire an in-
phase locked state (for τ = 0.514 ms), death (for τ = 2
ms) and an anti-phase locked state (for τ = 4.428 ms).
A series of such observations are recorded for a fixed fre-
quency and at different fixed values of K by varying the
time delay parameter. The output voltage V of the os-
cillator is monitored on the oscilloscope up to the point
where it shows a sudden drop in value by several orders of
magnitude, i.e., nearly to zero. The value of τ is further
increased up to the point where the signal from the oscil-
lator again suddenly revives to a large value. An entire
parameter space in K − τ for amplitude death is thereby
explored. In Fig. 4 the solid squares are the experimental
data points marking the loci of the death regions. They
clearly reveal closed death island regions in the plane of
the coupling strength, K, and the time delay, τ , for the
common frequency of ω = 837s−1.
FIG. 3. Oscilloscope traces of the temporal behavior of the
voltages of the two delay coupled oscillators for different delay
times with K = 1000s−1 and ω = 837s−1. Time is marked
in units of 5 ms along the horizontal axis. The oscillators
show (top panel) in-phase locking for τ = 0.514 ms, (middle
panel) amplitude death for τ = 2.0 ms, and (bottom panel)
anti-phase locking for τ = 4.428 ms.
2
1.0e1
1.0e2
1.0e3
1.0e4
1.0e5
1.0e6
0 0.001 0.002 0.003 0.004 0.005 0.006 0.007
K
τ (in sec.)
τ1(1,K)
τ2(1,K)
τ2(2,K) τ1(2,K)
ω = 837
FIG. 4. Multiply connected death islands in K − τ space
for ω = 837s−1. Solid squares mark the experimental points
while the solid and dashed lines are theoretical curves ob-
tained from model equation (1).
The solid and dashed lines are analytic curves obtained
from a linear stability analysis of the origin for the model
equation (1). These curves are derived in a standard
manner [7] from the characteristic equation for the eigen-
value λ, namely, λ2(λ− a1
C
+ K)2 + 2ω2(λ− a1
C
+ K)λ +
ω4 − K2λ2e−2λτ = 0, and are given by,
τ1(n, K) =
(n − 1)π + cos−1(1 − a1
CK
)
−A
2
+
√
(A
2
)2 + ω2
, (2)
τ2(n, K) =
nπ − cos−1(1 − a1
CK
)
A
2
+
√
(A
2
)2 + ω2
, (3)
where A =
√
2a1
C
K − (a1
C
)2, a1 = 0.95 × 10−5 and
n = 1, 2, . . .. The region enclosed by the intersection
of τ1(1, K) and τ2(1, K) is the primary death island and
that enclosed between τ1(2, K) and τ2(2, K) is the sec-
ondary (higher order) death island. We observe that the
experimental observations not only display topological
similarity to the analytic predictions but are also in good
quantitative agreement with them. The deviations ob-
served are primarily due to two reasons. First, the ana-
lytic domains are sensitive to the values of the coefficients
used in fitting the V − I characteristic of the nonlinear
resistive element. The relative downward shift and larger
width of the experimental islands compared to the ana-
lytic region are due to these estimation errors. Second,
the deviation at very high values of K (and hence low
RK) is related to limitations on the maximum current
that can flow through RKj for the rated biased voltages
of our experimental circuit. The existence of the higher
order island is particularly significant since it confirms
the theoretical finding of multiple connectivity of the sta-
bility region of the origin [7].
0 200 400 600 800 1000
0
1
2
3
4
5
6
7
8
  ω (in s  )−1
τ  
(in
 m
ill
is
ec
.)
 K = 500
 D
 E
 A
 T
 H
τ1(1,K)
τ2(1,K)
FIG. 5. Amplitude death region in the plane of (ω, τ ) for a
fixed coupling strength of K = 500s−1.
In Fig. 5, the experimental death region is shown at
a constant coupling strength K = 500s−1 for different
values of the frequency. The death region ceases to exist
below a certain threshold frequency, in accordance with
theoretical predictions [7]. On either side of this region,
there are phase locked states exhibiting in-phase locked
oscillations for small time delays and anti-phase locked
oscillations for longer time delays.
Another important characteristic of time delay sys-
tems in general [11,12], and delay coupled limit cycle
oscillators in particular [7], is the existence of multi-
ple phase locked states. When the oscillators are iden-
tical, these phase locked states coincide with in-phase
(phase difference, φ = 0) and anti-phase (φ = π) so-
lutions. In Fig. 6(a), the domains of existence of these
solutions in τ space are indicated. Here In represent
the in-phase solutions and An represent the anti-phase
solutions. These domains have been determined exper-
imentally. For a fixed τ the various existing multiple
states can be accessed with slight perturbations to the
system, i.e., by slightly changing the initial conditions.
We have also observed hysteretic behaviour between the
various branches. In Fig. 6(b), the frequencies of the first
few multiple states are plotted for increasing values of τ .
Note that the collective states can be much higher than
the individual oscillator frequencies and that
0 5 10 15 20 25 30 35
τ (millisec)
A1 A2
A3
A4
I1 I2
I3In-
Phase
Anti-
Phase
(a)
500
600
   
   
   
    
    
1200
1300
6 10 14 18
C
o
m
m
o
n
 f
re
q
.
τ (millisec)
(b)
A1 A2I1
I2
FIG. 6. (a) Coexistence of in-phase and anti-phase locked
states, and (b) suppression of the phase locked states as τ is
increased for K = 1000s−1 and ω = 837s−1.
3
their magnitude decreases with increasing values of time
delay. Such a phenomenon of time delay induced fre-
quency suppression has been discussed earlier in the
context of phase only oscillators [11].
In conclusion, we have experimentally demonstrated
for the first time the existence of delay induced am-
plitude death in a pair of coupled identical nonlinear
oscillators [7]. We have also demonstrated the coex-
istence of in-phase and anti-phase states with multiple
frequencies and the suppression of the common frequency
with time delay. The observation of multiply connected
death islands (for identical oscillators) in the param-
eter space of the coupling strength and time delay is
our most important result. This result not only bears
topological resemblance to earlier model calculations [7],
but also shows good quantitative agreement with ana-
lytic estimates obtained from a theoretical model of the
experimental circuits. It should be noted that our exper-
imental system of coupled nonlinear electronic circuits
(and its theoretical model) is capable of richer nonlin-
ear behaviour and goes beyond the simple normal form
model used in previous theoretical studies [7]. In this
sense the results on delay induced amplitude death can
be said to have a broader validity and therefore wider
applicability to physically realistic systems.
We now briefly discuss a few interesting directions for
future experiments that are relevant to our present re-
sults as well as to practical applications. A natural next
step is to couple more than two oscillators to explore
some of the other theoretical predictions discussed in [7].
The principal technical challenge and the major expen-
sive elements for such an endeavor are the construction of
additional delay lines with enhanced number of channels.
Such an effort is presently underway. Another interesting
possibility is to move away from the limit cycle regime
and investigate the effect of time delay on the synchro-
nization of coupled chaotic oscillators - an area of much
current scientific interest [13]. In principle this should be
possible with our present experimental system since the
basic Chua circuits are capable of displaying chaotic oscil-
lations. It would be necessary however to employ a faster
delay line. Time delay could facilitate phase synchroniza-
tion in view of its significant influence on phenomena like
phase slips [14]. A final area of practical interest is the
stability of these limit cycle collective states to external
noise. Past numerical studies indicate that time delay
induced higher frequency states can exhibit metastable
behaviour and decay to the lowest periodic state in the
presence of sufficient external noise [11]. We do not ob-
serve this behaviour since by design our system has very
low intrinsic (thermal) noise in order to facilitate access
to the higher frequency states. It would be interesting
to introduce a suitably designed noise source in the in-
dividual oscillator circuits to experimentally test these
theoretical predictions including the influence of noise on
the threshold for onset of synchronization [15].
We acknowledge the important contribution of H. S.
Mazumdar in the design, development and construction
of the digital delay line.
∗ Present Address: EduTron Corp., 5 Cox Rd., Winch-
ester, MA 01890.
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4


Experimental evidence of time-delay-induced death in coupled limit-cycle oscillators

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Experimental Evidence of Time Delay Induced Death in Coupled Limit Cycle
  Oscillators

Experimental Evidence of Time Delay Induced Death in Coupled Limit Cycle Oscillators

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