We investigate the dynamical behaviour of two limit cycle oscillators that
interact with each other via time delayed coupling and find that time delay can
lead to amplitude death of the oscillators even if they have the same
frequency. We demonstrate that this novel regime of amplitude "death" also
exists for large collections of coupled identical oscillators and provide
quantitative measures of this death region in the parameter space of coupling
strength and time delay. Its implication for certain biological and physical
applications is also pointed out.Comment: 4 aps formatted revtex pages; 3 figures; to be published in Phys.
  Rev. Let

Johnston, G. L.

Reddy, D. V. Ramana

Sen, A.

English

arXiv

We investigate the dynamical behavior of two limit cycle oscillators that interact with each other via time delayed coupling and find that time delay can lead to amplitude death of the oscillators even if they have the same frequency. We demonstrate that this novel regime of amplitude "death" also exists for large collections of coupled identical oscillators and provide quantitative measures of this death region in the parameter space of coupling strength and time delay. Its implication for certain biological and physical applications is also pointed out

Ramana Reddy, D. V.

Publications of the IAS Fellows

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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.
(February 5, 2008)
We investigate the dynamical behaviour of two limit cycle oscillators that interact with each other
via time delayed coupling and find that time delay can lead to amplitude death of the oscillators
even if they have the same frequency. We demonstrate that this novel regime of amplitude ”death”
also exists for large collections of coupled identical oscillators and provide quantitative measures of
this death region in the parameter space of coupling strength and time delay. Its implication for
certain biological and physical applications is also pointed out.
PACS numbers : 05.45.+b,87.10.+e
Coupled limit cycle oscillators provide a simple but
powerful mathematical model for simulating the collec-
tive behaviour of a wide variety of systems that are of
interest in physics [1–10], chemistry [11,12] and biologi-
cal sciences [13,14]. These oscillators have also attracted
some large scale numerical [15] and novel experimental ef-
forts [16]. For weakly coupled oscillators the predominant
effect is a synchronization of the frequencies of the indi-
vidual oscillators to a single common frequency once the
coupling strength exceeds a certain threshold, while the
amplitudes remain unaffected. For stronger couplings the
amplitudes also play an important role and give rise to
interesting phenomena like the Bar-Eli effect [11] where
all the oscillators suffer an amplitude quenching or death
[6,9]. In general there can be a wide variety of collective
behaviour including partial synchronisation, phase trap-
ping, large amplitude Hopf oscillations and even chaotic
behaviour [3–5]. In recent times there have been exten-
sive investigations of coupled oscillator systems including
elegant statistical mechanics formulations in the limit of
infinite number of oscillators [7,10].
The salient features of the behaviour of a finitely large
number of oscillators (usually obtained from numerical or
approximate analytic means) can often be understood by
analysing just two coupled oscillators. We have carried
out such an analysis to investigate the effect of time de-
lay on the interaction between two limit cycle oscillators.
Time delay is ubiquitous in most physical and biological
systems [17–19], arising from finite propagation speeds
of signals for example, and have not been widely studied
in the context of coupled limit cycle oscillator systems.
Niebuhr et al [20] and Schuster and Wagner [21] who are
one of the few who have carried out such an investigation,
have restricted themselves to the simpler coupled phase
models where the phenomenon of amplitude death does
not exist. In our model equations we have retained both
the phase and amplitude response of the oscillators and
we find that time delay has a significant effect on the
characteristics of all the major cooperative phenomena
like frequency locking, phase drift and amplitude deaths.
In particular our detailed numerical investigations show
that in the presence of time delay the parameter regime
of amplitude death can extend down to the region of
zero frequency mismatch between the oscillators. This
is in sharp contrast to the situation with no time de-
lay where all previous numerical and analytical studies
[3,5,6,9] show that amplitude death can occur only if
the coupling between oscillators is sufficiently strong and
when the frequencies are sufficiently disparate. In this
Letter we confine ourselves primarily to the effect of time
delayed coupling on the phenomenon of amplitude death
and present a detailed numerical and analytical estimate
of the parameter space in coupling strength and time de-
lay where such a death can occur for identical oscillators.
We also establish that this effect is not an artefact of the
simple two oscillator model, but can occur for a system
of large number of globally (or locally) coupled identical
oscillators (including the continuum limit of N →∞).
We analyse the following model equations:
Z˙1(t) = (1 + iω1− | Z1(t) |2)Z1(t)
+K[Z2(t− τ) − Z1(t)], (1)
Z˙2(t) = (1 + iω2− | Z2(t) |2)Z2(t)
+K[Z1(t− τ) − Z2(t)], (2)
where τ is a measure of the time delay, K is the cou-
pling strength, ω1,2 are the intrinsic frequencies of the
two oscillators and Z1,2 are complex. The model is a
generalization of the diffusively and linearly coupled os-
cillators studied extensively for example in [3,11]. The
time delay parameter is introduced in the argument of the
coupling oscillator (e.g. Z2 in (1)) to physically account
for the fact that its phase and amplitude information is
received by oscillator Z1 only after a finite time τ (due to
finite propagation speed effects). In the absence of cou-
pling (K = 0) each oscillator has a stable limit cycle at
| Zi |= 1 on which it moves at its natural frequency ωi.
The coupled equations represent the interaction between
two weakly nonlinear oscillators (that are near a Hopf bi-
furcation) and whose coupling strength is comparable to
the attraction of the limit cycles. It is important then to
retain both the phase and amplitude response of the os-
cillators [3]. The state Zi = 0 is an equilibrium solution
of the system of equations [(1) and (2)]. For K = 0 this
1
equilibrium state is linearly unstable since the individual
oscillators tend to stable limit cycle states | Zi |= 1. The
stability of amplitude death for K 6= 0 has been studied
in great detail by Aronson et al [3] for system ((1,(2)) in
the absence of any time delay in the coupling (i.e. for
τ = 0). The conditions for stability found by them are,
K > 1 and ∆ =| ω1 − ω2 |> 2
√
2K − 1, (3)
which shows that amplitude death can occur in this case
only for sufficiently large values of ∆ provided K > 1.
In Fig. 1(a), we reproduce the bifurcation diagram
of Aronson et.al [3] where the region marked I repre-
sents the amplitude death region and the dotted curves
mark the boundary as defined by condition (3). The two
bounding curves intersect at the point (K = 1,∆ = 2).
Regions marked (II) and (III) represent phase locked
and phase drift regions respectively which we will not
discuss in detail here. In Fig. 1(b) we present the bi-
furcation diagram of (1-2) for τ = 0.0817. Note that
in contrast to the diagram of Fig. 1(a), the amplitude
death region now extends down to ∆ = 0 and has a fi-
nite extent along the coupling strength (K) axis. We
find that the phenomenon persists for a range of τ af-
ter which the bifurcation curve lifts up from the ∆ = 0
line and identical oscillators can no longer suffer death.
Fig. 2(a) shows this region, for different values of ω,
in τ − K space for which amplitude death of identical
oscillators can occur. The size of this death island is a
function of the frequency of the oscillators (ω), as shown
by the other curves. We shall soon show that the size
is also a function of N , the number of oscillators. The
bifurcation curves (including the island boundaries) have
been obtained from a linear stability analysis of (1),(2)
about the origin (Z1 = Z2 = 0) as well as direct numer-
ical integration of the equations. Assuming the linear
perturbations to vary as eλt the
0 1 2 3
0
 
2
 
4
 
6
 
K
∆
I
II
III
 (a)
0 10 20 30
0
 
2
 
4
 
6
 
K
I
II II
 (b)
FIG. 1. Bifurcation diagram of Eqns.(1-2). (a) τ = 0. Re-
gion I is the amplitude death region, II is the phase locked
region, and III corresponds to the phase drift or incoherent
region. (b) τ = 0.0817, and ω¯ = 10. The death region ex-
tends down to ∆ = 0 indicating that identical oscillators can
suffer amplitude death. The phase locked region is split into
two disjoint regions.
characteristic eigenvalue equation we get is,
(1 −K + iω1 − λ)(1 −K + iω2 − λ) −
K2e−2λτ = 0, (4)
where λ is the complex eigenvalue and the complete set
of eigenvalues includes those arising from the complex
conjugate equations of (1) and (2) . Setting Real(λ) = 0
in (4) and separating the real and imaginary parts, the
equations for the critical curves (i.e. the marginal stabil-
ity condition) are,
λ2I + 2λI ω¯ + ω¯
2 − ∆
2
4
− (1 −K)2
+K2 cos(2λIτ) = 0, (5)
2(1−K)(ω¯ + λI) +K2 sin(2λIτ) = 0. (6)
where λI = Imag(λ) and ω¯ = (ω1 + ω2)/2 is the mean
frequency. Eliminating λI between (5) and (6) and con-
sidering the full set of eigenvalues , we obtain the follow-
ing transcedental relation between K,∆ and τ which is
now the modified marginal stability condition in place of
(3),
gα = K2 sin(ατ ± 2ω¯τ), (7)
where α =
√
∆2 − 4g2 ∓ 4
√
K4 − g2∆2, and g = 1−K.
Note that for τ = 0, the above relation readily simplifies
to (3) and yields the marginal stability curvesK = 1 and
2K = 1 + ∆
2
4
. Figure 1(b). is a numerical plot of (7) for
τ = 0.0817.
To obtain a condition for the death of identical oscilla-
tors we repeat the analysis with ω1 = ω2 = ω in (4) and
after eliminating λI , obtain the relations
τ =
cos−1(1 − 1/K)
ω −√2K − 1 ; τ =
pi − cos−1(1− 1/K)
ω +
√
2K − 1 . (8)
0 0.2 0.4
0
5
10
15
20
25
30
35
40
τ
K
ω = 9
6
5
  (a)
0 0.2 0.4
0
5
10
15
20
25
30
35
40
τ
N=2
3
4
∞
  (b)
FIG. 2. (a) The region of amplitude death for N = 2 as a
function of the common intrinsic frequency. The size of the
the island decreases with decreasing frequency and vanishes
below a certain threshold. (b) The death islands as a function
of the number of globally coupled oscillators. Each oscillator
is assumed to have an intrinsic frequency of ω = 10. The
death island survives even in the limit of N →∞.
2
The region of intersection between the two curves (8)
corresponds to the death island region of Fig. 2(a). It
demonstrates the only stability switches (of the origin)
that take place as a function of τ . For a given value
of ω and at a fixed K we move (by varying τ) from an
unstable region into a stable region as we cross the left
boundary of the island to emerge again into an unstable
region as we cross the right boundary of the island. No
other amplitude death islands are seen for larger values
of τ (for a fixed ω). We have also confirmed this ana-
lytically by looking at the behaviour of the supremum of
the real parts of the roots of the transcedental equation,
dλR/dτ (obtained from (4) with ∆ = 0) as a function
of τ in the various parameter regimes [18]. The detailed
mathematical proof of this result will be published else-
where.
Can this phenomenon occur for an arbitrary number
of oscillators? To answer this question we have inves-
tigated the following generalized set of globally coupled
equations:
Z˙i(t) = (1 + iωi− | Zi(t) |2)Zi(t) +
2K
N
N∑
j=1
[Zj(t− τ)− Zi(t)]−
2K
N
[Zi(t− τ)− Zi(t)], (9)
where i = 1, ...., N and the last term on the right hand
side has been included to remove the self-coupling term.
For τ = 0, (9) reduces to the set of equations that have
been extensively studied by Ermentrout [9], Mirollo and
Strogatz [6], and others [4]. Mirollo and Strogatz [6]
have provided rigorous analytical and numerical condi-
tions for amplitude death in such a system. Their con-
clusions, in general, are similar to the case of N = 2,
namely, that one needs a sufficiently large variance in fre-
quencies for death to occur and K has to be sufficiently
large. We have been able to carry out a similar linear
stability analysis of (9) around the origin for the case
of finite τ and for a large number of identical oscillators
(ωj = ω, j = 1, ..., N). The resulting stability condition
yields the following bounding curves for the death island
region:
τ =
cos−1 p
ω −√4Kb− 1 ; τ =
2pi − cos−1 p
ω +
√
4Kb− 1 ,
τ =
cos−1( b
b−1
p)
ω +
√
q
; τ =
2pi − cos−1( b
b−1
p)
ω −√q ,
(10)
where p = 1− 1/(2Kb), q = 4K2− 1+ 4Kb(1− 2K) and
the factor b = (1 − 1/N) introduces the N dependence
of the island size explicitly. In Fig. 2(b) we have plotted
these islands for N = 2, 3, 4 and N = ∞. To confirm
these results we have also numerically scanned the re-
gion with a direct numerical integration of (9) for a large
number of oscillators, upto N = 10000, and found excel-
lent agreement. We have also carried out a similar study
for a large number of locally coupled identical oscillators
(nearest neighbour coupling [2]) with periodic boundaries
and find that time delay introduces death islands in such
systems as well. Thus it appears that in the presence of
finite time delay in the mutual coupling, amplitude death
of identical oscillators is a fairly universal phenomenon
and occurs for any arbitrary number N of oscillators ex-
tending upto N = ∞ over a range of τ and K values.
To the best of our knowledge such a result has not been
realised in the past and may have important applications
in biological or physical systems. There are many phys-
ical examples of amplitude death in real systems. One
of the earliest that was investigated both theoretically
and experimentally is that of coupled chemical oscilla-
tor systems e.g. coupled Belousov-Zhabotinskii reactions
carried out in coupled stirred tank reactors [11,12]. They
can also occur in ecological contexts where one can imag-
ine two sites each having the same predator-prey mech-
anism which causes the number density of the species to
oscillate. If the species are capable of moving from site
to site at a proper rate (appropriate coupling strength)
the two sites may become stable (stop oscillating) and
acquire constant populations. Another important appli-
cation of this concept is in pathologies of biological os-
cillator networks e.g. an assembly of cardiac pacemaker
cells [13]. Amplitude death signifies cessation of rhyth-
micity in such a system which is otherwise normally spon-
taneously rhythmic for other choices of parameters. For
the onset of such an arrhythmia, current models based
on coupled oscillator networks need to assume a signifi-
cant spread in the natural frequencies of the constituent
cells (oscillators) [6]. Our work demonstrates that this
assumption may not be necessary if one takes into ac-
count time delay effects arising naturally from the finite
propagation times of the signals exchanged between the
cells. Another possible application is in the area of high
power microwave sources where it is proposed to enhance
the microwave power production by phase locking a large
number of sources such as relativistic magnetrons [22].
Time delay effects, arising from the finite propagation
time of information signals traveling through the con-
necting waveguide bridges, could impose important limi-
tations on the connector lengths and geometries in these
schemes. Our findings could provide a guideline in this
direction. It should be noted that a form of oscillator
death described in [23] for identical oscillators is not a
genuine amplitude death since it occurs in the context of
a phase only model. Time delay in our study provides a
new mechanism for genuine amplitude death to occur in
coupled identical oscillators.
Finally, it is worthwhile to mention that time delay can
introduce other interesting phenomena as well, some of
which have been studied in the context of the phase only
model and need to be investigated for the more general
phase and amplitude model. Our numerical results, for
3
0    1    2    3    
0
 
4
 
8
  
12
  
K
∆
τ = 0.4084
III
II
II
III
FIG. 3. The bifurcation diagram of Eqs. (1) and (2) for
τ = 0.4084 and ω¯ = 10. The amplitude death region (I) is
surrounded by the phase locked regions (II). The dotted curve
which separates the incoherent (III) and the phase locked re-
gions is obtained from numerical integration of the original
equations.
example, show that the bifurcation diagram of the sys-
tem in the presence of time delay has a significantly richer
structure. Fig. 3 is an example for the N = 2 system for
τ = 0.4084, which can be contrasted with the Aronson
et al [3] diagram of Fig. 1(a). Note that one no longer
has the clean separation of the Bar-Eli region, the phase
locked region and the phase drift region into three dis-
joint regions that converge at a single degenerate point.
Instead the phase locked region now always surrounds
the Bar-Eli region and the single degenerate point is re-
placed by a series of X points resulting from the braided
structure of the phase locked region in the vertical direc-
tion. At large values ofK other bifurcation curves appear
in the phase locked region indicating the appearance of
higher frequency states [21]. A detailed investigation of
various properties of this rich phase diagram, including
stability studies of the various states, is now in progress
and will be reported elsewhere.
∗ Email address: tapovan@plasma.ernet.in
† Permanent address: 12 Billings St., Acton, MA 01720.
[1] A.K. Jain, K.K. Likharev, J.E. Lukens and J.E.
Sauvageau, Phys. Rep. 109, 309 (1984);D.S. Fisher,
Phys. Rev. B 31, 1396 (1985); P. Hadley, M. R. Beasley
and K. Wiesenfeld, Phys. Rev. B 38, 8712 (1988); Appl.
Phys. Lett. 52, 1619 (1988).
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21, L699 (1988); S. H. Strogatz, R. E. Mirollo, Physica
(Amsterdam) 31D, 143 (1988).
[3] D.G. Aronson, G.B. Ermentrout and N. Koppel, Physica
(Amsterdam) 41D, 403 (1990).
[4] Y. Aizawa, Prog. Theor. Phys. 56, 703 (1976); M. Shiino
and M. Frankowicz, Phys. Lett. A 136, 103 (1989).
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65, 1701 (1990); P. C. Matthews, R. E. Mirollo, and
S. H. Strogatz, Physica (Amsterdam) 52D, 293 (1991);
N. Nakagawa and Y. Kuramoto, Physica (Amsterdam)
D75, 74 (1994).
[6] R. E. Mirollo and S. H. Strogatz, J. Stat. Phys. 60, (1990)
245-262.
[7] Y. Kuramoto, Prog. Theor. Phys. Suppl. No.79, 223-240
(1984); Y. Kuramoto and I. Nishikawa, J. Stat. Phys. 49,
569 (1987).
[8] J. W. Swift, S. H. Strogatz, and K. Wiesenfeld, Physica
(Amsterdam) 55D, 239 (1992).
[9] G. B. Ermentrout, Physica (Amsterdam) 41D, 219
(1990).
[10] H. Daido, J. Stat. Phys. 60, 753 (1990); H. Daido, Phys-
ica (Amsterdam) 91D, 24 (1996) and references there in;
H. Daido, Phys. Rev. Lett. 78, 1683 (1997).
[11] K. Bar-Eli, Physica (Amsterdam) 14D, 242 (1985).
[12] Michael F. Crowley and Irving R. Epstein, J. Phys.
Chem. 93, 2496 (1989).
[13] A. T. Winfree, The Geometry of Biological Time
(Springer-Verlag, New York, 1980); The Three-
Dimensional Dynamics of Electrochemical Waves and
Cardiac Arrythmias (Princeton University Press, Prince-
ton, NJ, 1987).
[14] Mitsuo Kawato and Ryoji Suzuki, J. theor. Biol. 86, 547
(1980).
[15] Kazuhiro Satoh, J. Phys. Soc. Japan 58, 2010 (1989).
[16] A. A. Brailove and P. S. Linsay, Int. J. Bifurcation Chaos
6, 1211 (1996).
[17] L. Glass and M. C. Mackey, Science 197, 287 (1977).
[18] K. L. Cooke and Z. Grossman, J. Math. Anal. App. 86,
592 (1982); F. G. Boese, J. Math. Anal. App. 140, 510
(1989).
[19] J. K. Hale and N. Sternberg, J. Comp. Phys. 77, 221
(1988); W. Wischert, et. al., Phys. Rev. E 49, 203 (1994).
[20] E. Niebur, H.G. Schuster and D. Kammen, Phys. Rev.
Lett. 67, 2753 (1991).
[21] H.G. Schuster and P. Wagner, Prog. Theor. Phys. 81,
939 (1989).
[22] J. Benford, H. Sze, W. Woo, R.R. Smith and B. Harte-
neck, Phys. Rev. Letts. 62, 969 (1989).
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125 (1990).
4


Time delay induced death in coupled limit cycle oscillators

D. V. Ramana Reddy

A. Sen

G. L. Johnston

Crossref

Physical Review Letters

Time Delay Induced Death in Coupled Limit Cycle Oscillators

Indian Academy of Sciences

ar
X
iv
:c
ha
o-
dy
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98
05
02
9v
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 2
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M
ay
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99
8
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.
(February 5, 2008)
We investigate the dynamical behaviour of two limit cycle oscillators that interact with each other
via time delayed coupling and find that time delay can lead to amplitude death of the oscillators
even if they have the same frequency. We demonstrate that this novel regime of amplitude ”death”
also exists for large collections of coupled identical oscillators and provide quantitative measures of
this death region in the parameter space of coupling strength and time delay. Its implication for
certain biological and physical applications is also pointed out.
PACS numbers : 05.45.+b,87.10.+e
Coupled limit cycle oscillators provide a simple but
powerful mathematical model for simulating the collec-
tive behaviour of a wide variety of systems that are of
interest in physics [1–10], chemistry [11,12] and biologi-
cal sciences [13,14]. These oscillators have also attracted
some large scale numerical [15] and novel experimental ef-
forts [16]. For weakly coupled oscillators the predominant
effect is a synchronization of the frequencies of the indi-
vidual oscillators to a single common frequency once the
coupling strength exceeds a certain threshold, while the
amplitudes remain unaffected. For stronger couplings the
amplitudes also play an important role and give rise to
interesting phenomena like the Bar-Eli effect [11] where
all the oscillators suffer an amplitude quenching or death
[6,9]. In general there can be a wide variety of collective
behaviour including partial synchronisation, phase trap-
ping, large amplitude Hopf oscillations and even chaotic
behaviour [3–5]. In recent times there have been exten-
sive investigations of coupled oscillator systems including
elegant statistical mechanics formulations in the limit of
infinite number of oscillators [7,10].
The salient features of the behaviour of a finitely large
number of oscillators (usually obtained from numerical or
approximate analytic means) can often be understood by
analysing just two coupled oscillators. We have carried
out such an analysis to investigate the effect of time de-
lay on the interaction between two limit cycle oscillators.
Time delay is ubiquitous in most physical and biological
systems [17–19], arising from finite propagation speeds
of signals for example, and have not been widely studied
in the context of coupled limit cycle oscillator systems.
Niebuhr et al [20] and Schuster and Wagner [21] who are
one of the few who have carried out such an investigation,
have restricted themselves to the simpler coupled phase
models where the phenomenon of amplitude death does
not exist. In our model equations we have retained both
the phase and amplitude response of the oscillators and
we find that time delay has a significant effect on the
characteristics of all the major cooperative phenomena
like frequency locking, phase drift and amplitude deaths.
In particular our detailed numerical investigations show
that in the presence of time delay the parameter regime
of amplitude death can extend down to the region of
zero frequency mismatch between the oscillators. This
is in sharp contrast to the situation with no time de-
lay where all previous numerical and analytical studies
[3,5,6,9] show that amplitude death can occur only if
the coupling between oscillators is sufficiently strong and
when the frequencies are sufficiently disparate. In this
Letter we confine ourselves primarily to the effect of time
delayed coupling on the phenomenon of amplitude death
and present a detailed numerical and analytical estimate
of the parameter space in coupling strength and time de-
lay where such a death can occur for identical oscillators.
We also establish that this effect is not an artefact of the
simple two oscillator model, but can occur for a system
of large number of globally (or locally) coupled identical
oscillators (including the continuum limit of N → ∞).
We analyse the following model equations:
Ż1(t) = (1 + iω1− | Z1(t) |2)Z1(t)
+K[Z2(t − τ) − Z1(t)], (1)
Ż2(t) = (1 + iω2− | Z2(t) |2)Z2(t)
+K[Z1(t − τ) − Z2(t)], (2)
where τ is a measure of the time delay, K is the cou-
pling strength, ω1,2 are the intrinsic frequencies of the
two oscillators and Z1,2 are complex. The model is a
generalization of the diffusively and linearly coupled os-
cillators studied extensively for example in [3,11]. The
time delay parameter is introduced in the argument of the
coupling oscillator (e.g. Z2 in (1)) to physically account
for the fact that its phase and amplitude information is
received by oscillator Z1 only after a finite time τ (due to
finite propagation speed effects). In the absence of cou-
pling (K = 0) each oscillator has a stable limit cycle at
| Zi |= 1 on which it moves at its natural frequency ωi.
The coupled equations represent the interaction between
two weakly nonlinear oscillators (that are near a Hopf bi-
furcation) and whose coupling strength is comparable to
the attraction of the limit cycles. It is important then to
retain both the phase and amplitude response of the os-
cillators [3]. The state Zi = 0 is an equilibrium solution
of the system of equations [(1) and (2)]. For K = 0 this
1
equilibrium state is linearly unstable since the individual
oscillators tend to stable limit cycle states | Zi |= 1. The
stability of amplitude death for K 6= 0 has been studied
in great detail by Aronson et al [3] for system ((1,(2)) in
the absence of any time delay in the coupling (i.e. for
τ = 0). The conditions for stability found by them are,
K > 1 and ∆ =| ω1 − ω2 |> 2
√
2K − 1, (3)
which shows that amplitude death can occur in this case
only for sufficiently large values of ∆ provided K > 1.
In Fig. 1(a), we reproduce the bifurcation diagram
of Aronson et.al [3] where the region marked I repre-
sents the amplitude death region and the dotted curves
mark the boundary as defined by condition (3). The two
bounding curves intersect at the point (K = 1, ∆ = 2).
Regions marked (II) and (III) represent phase locked
and phase drift regions respectively which we will not
discuss in detail here. In Fig. 1(b) we present the bi-
furcation diagram of (1-2) for τ = 0.0817. Note that
in contrast to the diagram of Fig. 1(a), the amplitude
death region now extends down to ∆ = 0 and has a fi-
nite extent along the coupling strength (K) axis. We
find that the phenomenon persists for a range of τ af-
ter which the bifurcation curve lifts up from the ∆ = 0
line and identical oscillators can no longer suffer death.
Fig. 2(a) shows this region, for different values of ω,
in τ − K space for which amplitude death of identical
oscillators can occur. The size of this death island is a
function of the frequency of the oscillators (ω), as shown
by the other curves. We shall soon show that the size
is also a function of N , the number of oscillators. The
bifurcation curves (including the island boundaries) have
been obtained from a linear stability analysis of (1),(2)
about the origin (Z1 = Z2 = 0) as well as direct numer-
ical integration of the equations. Assuming the linear
perturbations to vary as eλt the
0 1 2 3
0
 
2
 
4
 
6
 
K
∆
I
II
III
 (a)
0 10 20 30
0
 
2
 
4
 
6
 
K
I
II II
 (b)
FIG. 1. Bifurcation diagram of Eqns.(1-2). (a) τ = 0. Re-
gion I is the amplitude death region, II is the phase locked
region, and III corresponds to the phase drift or incoherent
region. (b) τ = 0.0817, and ω̄ = 10. The death region ex-
tends down to ∆ = 0 indicating that identical oscillators can
suffer amplitude death. The phase locked region is split into
two disjoint regions.
characteristic eigenvalue equation we get is,
(1 − K + iω1 − λ)(1 − K + iω2 − λ) −
K2e−2λτ = 0, (4)
where λ is the complex eigenvalue and the complete set
of eigenvalues includes those arising from the complex
conjugate equations of (1) and (2) . Setting Real(λ) = 0
in (4) and separating the real and imaginary parts, the
equations for the critical curves (i.e. the marginal stabil-
ity condition) are,
λ2I + 2λI ω̄ + ω̄
2 − ∆
2
4
− (1 − K)2
+K2 cos(2λIτ) = 0, (5)
2(1 − K)(ω̄ + λI) + K2 sin(2λIτ) = 0. (6)
where λI = Imag(λ) and ω̄ = (ω1 + ω2)/2 is the mean
frequency. Eliminating λI between (5) and (6) and con-
sidering the full set of eigenvalues , we obtain the follow-
ing transcedental relation between K, ∆ and τ which is
now the modified marginal stability condition in place of
(3),
gα = K2 sin(ατ ± 2ω̄τ), (7)
where α =
√
∆2 − 4g2 ∓ 4
√
K4 − g2∆2, and g = 1−K.
Note that for τ = 0, the above relation readily simplifies
to (3) and yields the marginal stability curves K = 1 and
2K = 1 + ∆
2
4
. Figure 1(b). is a numerical plot of (7) for
τ = 0.0817.
To obtain a condition for the death of identical oscilla-
tors we repeat the analysis with ω1 = ω2 = ω in (4) and
after eliminating λI , obtain the relations
τ =
cos−1(1 − 1/K)
ω −
√
2K − 1
; τ =
π − cos−1(1 − 1/K)
ω +
√
2K − 1
. (8)
0 0.2 0.4
0
5
10
15
20
25
30
35
40
τ
K
ω = 9
6
5
  (a)
0 0.2 0.4
0
5
10
15
20
25
30
35
40
τ
N=2
3
4
∞
  (b)
FIG. 2. (a) The region of amplitude death for N = 2 as a
function of the common intrinsic frequency. The size of the
the island decreases with decreasing frequency and vanishes
below a certain threshold. (b) The death islands as a function
of the number of globally coupled oscillators. Each oscillator
is assumed to have an intrinsic frequency of ω = 10. The
death island survives even in the limit of N → ∞.
2
The region of intersection between the two curves (8)
corresponds to the death island region of Fig. 2(a). It
demonstrates the only stability switches (of the origin)
that take place as a function of τ . For a given value
of ω and at a fixed K we move (by varying τ) from an
unstable region into a stable region as we cross the left
boundary of the island to emerge again into an unstable
region as we cross the right boundary of the island. No
other amplitude death islands are seen for larger values
of τ (for a fixed ω). We have also confirmed this ana-
lytically by looking at the behaviour of the supremum of
the real parts of the roots of the transcedental equation,
dλR/dτ (obtained from (4) with ∆ = 0) as a function
of τ in the various parameter regimes [18]. The detailed
mathematical proof of this result will be published else-
where.
Can this phenomenon occur for an arbitrary number
of oscillators? To answer this question we have inves-
tigated the following generalized set of globally coupled
equations:
Żi(t) = (1 + iωi− | Zi(t) |2)Zi(t) +
2K
N
N
∑
j=1
[Zj(t − τ) − Zi(t)] −
2K
N
[Zi(t − τ) − Zi(t)], (9)
where i = 1, ...., N and the last term on the right hand
side has been included to remove the self-coupling term.
For τ = 0, (9) reduces to the set of equations that have
been extensively studied by Ermentrout [9], Mirollo and
Strogatz [6], and others [4]. Mirollo and Strogatz [6]
have provided rigorous analytical and numerical condi-
tions for amplitude death in such a system. Their con-
clusions, in general, are similar to the case of N = 2,
namely, that one needs a sufficiently large variance in fre-
quencies for death to occur and K has to be sufficiently
large. We have been able to carry out a similar linear
stability analysis of (9) around the origin for the case
of finite τ and for a large number of identical oscillators
(ωj = ω, j = 1, ..., N). The resulting stability condition
yields the following bounding curves for the death island
region:
τ =
cos−1 p
ω −
√
4Kb − 1
; τ =
2π − cos−1 p
ω +
√
4Kb − 1
,
τ =
cos−1( b
b−1
p)
ω +
√
q
; τ =
2π − cos−1( b
b−1
p)
ω −√q ,
(10)
where p = 1− 1/(2Kb), q = 4K2 − 1 + 4Kb(1− 2K) and
the factor b = (1 − 1/N) introduces the N dependence
of the island size explicitly. In Fig. 2(b) we have plotted
these islands for N = 2, 3, 4 and N = ∞. To confirm
these results we have also numerically scanned the re-
gion with a direct numerical integration of (9) for a large
number of oscillators, upto N = 10000, and found excel-
lent agreement. We have also carried out a similar study
for a large number of locally coupled identical oscillators
(nearest neighbour coupling [2]) with periodic boundaries
and find that time delay introduces death islands in such
systems as well. Thus it appears that in the presence of
finite time delay in the mutual coupling, amplitude death
of identical oscillators is a fairly universal phenomenon
and occurs for any arbitrary number N of oscillators ex-
tending upto N = ∞ over a range of τ and K values.
To the best of our knowledge such a result has not been
realised in the past and may have important applications
in biological or physical systems. There are many phys-
ical examples of amplitude death in real systems. One
of the earliest that was investigated both theoretically
and experimentally is that of coupled chemical oscilla-
tor systems e.g. coupled Belousov-Zhabotinskii reactions
carried out in coupled stirred tank reactors [11,12]. They
can also occur in ecological contexts where one can imag-
ine two sites each having the same predator-prey mech-
anism which causes the number density of the species to
oscillate. If the species are capable of moving from site
to site at a proper rate (appropriate coupling strength)
the two sites may become stable (stop oscillating) and
acquire constant populations. Another important appli-
cation of this concept is in pathologies of biological os-
cillator networks e.g. an assembly of cardiac pacemaker
cells [13]. Amplitude death signifies cessation of rhyth-
micity in such a system which is otherwise normally spon-
taneously rhythmic for other choices of parameters. For
the onset of such an arrhythmia, current models based
on coupled oscillator networks need to assume a signifi-
cant spread in the natural frequencies of the constituent
cells (oscillators) [6]. Our work demonstrates that this
assumption may not be necessary if one takes into ac-
count time delay effects arising naturally from the finite
propagation times of the signals exchanged between the
cells. Another possible application is in the area of high
power microwave sources where it is proposed to enhance
the microwave power production by phase locking a large
number of sources such as relativistic magnetrons [22].
Time delay effects, arising from the finite propagation
time of information signals traveling through the con-
necting waveguide bridges, could impose important limi-
tations on the connector lengths and geometries in these
schemes. Our findings could provide a guideline in this
direction. It should be noted that a form of oscillator
death described in [23] for identical oscillators is not a
genuine amplitude death since it occurs in the context of
a phase only model. Time delay in our study provides a
new mechanism for genuine amplitude death to occur in
coupled identical oscillators.
Finally, it is worthwhile to mention that time delay can
introduce other interesting phenomena as well, some of
which have been studied in the context of the phase only
model and need to be investigated for the more general
phase and amplitude model. Our numerical results, for
3
0    1    2    3    
0
 
4
 
8
  
12
  
K
∆
τ = 0.4084
III
II
II
III
FIG. 3. The bifurcation diagram of Eqs. (1) and (2) for
τ = 0.4084 and ω̄ = 10. The amplitude death region (I) is
surrounded by the phase locked regions (II). The dotted curve
which separates the incoherent (III) and the phase locked re-
gions is obtained from numerical integration of the original
equations.
example, show that the bifurcation diagram of the sys-
tem in the presence of time delay has a significantly richer
structure. Fig. 3 is an example for the N = 2 system for
τ = 0.4084, which can be contrasted with the Aronson
et al [3] diagram of Fig. 1(a). Note that one no longer
has the clean separation of the Bar-Eli region, the phase
locked region and the phase drift region into three dis-
joint regions that converge at a single degenerate point.
Instead the phase locked region now always surrounds
the Bar-Eli region and the single degenerate point is re-
placed by a series of X points resulting from the braided
structure of the phase locked region in the vertical direc-
tion. At large values of K other bifurcation curves appear
in the phase locked region indicating the appearance of
higher frequency states [21]. A detailed investigation of
various properties of this rich phase diagram, including
stability studies of the various states, is now in progress
and will be reported elsewhere.
∗ Email address: tapovan@plasma.ernet.in
† Permanent address: 12 Billings St., Acton, MA 01720.
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4


Time Delay Induced Death in Coupled Limit Cycle Oscillators

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