Though the notion of phase synchronization has been well studied in chaotic
dynamical systems without delay, it has not been realized yet in chaotic
time-delay systems exhibiting non-phase coherent hyperchaotic attractors. In
this article we report the first identification of phase synchronization in
coupled time-delay systems exhibiting hyperchaotic attractor. We show that
there is a transition from non-synchronized behavior to phase and then to
generalized synchronization as a function of coupling strength. These
transitions are characterized by recurrence quantification analysis, by phase
differences based on a new transformation of the attractors and also by the
changes in the Lyapunov exponents. We have found these transitions in coupled
piece-wise linear and in Mackey-Glass time-delay systems.Comment: 4 pages, 3 Figures (To appear in Physical Review E Rapid
  Communication

A. S. Pikovsky

D. V. Senthilkumar

H. Fujisaka

J. Kurths

M. Lakshmanan

English

arXiv

Crossref

Phase synchronization in time-delay systems

Though the notion of phase synchronization has been well studied in chaotic dynamical systems without delay, it has not been realized yet in chaotic time-delay systems exhibiting non-phase-coherent hyperchaotic attractors. In this paper we report identification of phase synchronization in coupled time-delay systems exhibiting hyperchaotic attractor. We show that there is a transition from nonsynchronized behavior to phase and then to generalized synchronization as a function of coupling strength. These transitions are characterized by recurrence quantification analysis, by phase differences based on a transformation of the attractors, and also by the changes in the Lyapunov exponents. We have found these transitions in coupled piecewise linear and in Mackey-Glass time-delay systems

Senthilkumar, D. V.

Lakshmanan, M.

Kurths, J.

Publications of the IAS Fellows

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Phase synchronization in time-delay systems
D. V. Senthilkumar1, M. Lakshmanan1,∗ and J. Kurths2†
1Centre for Nonlinear Dynamics,
Department of Physics, Bharathidasan University,
Tiruchirapalli - 620 024, India
and
2 Institute of Physics, University of Potsdam,
Am Neuen Palais 10, 14469 Potsdam, Germany
(Dated: February 8, 2008)
Though the notion of phase synchronization has been well studied in chaotic dynamical systems
without delay, it has not been realized yet in chaotic time-delay systems exhibiting non-phase coher-
ent hyperchaotic attractors. In this article we report the first identification of phase synchronization
in coupled time-delay systems exhibiting hyperchaotic attractor. We show that there is a transition
from non-synchronized behavior to phase and then to generalized synchronization as a function
of coupling strength. These transitions are characterized by recurrence quantification analysis, by
phase differences based on a new transformation of the attractors and also by the changes in the Lya-
punov exponents. We have found these transitions in coupled piece-wise linear and in Mackey-Glass
time-delay systems.
PACS numbers: 05.45.Xt,05.45.Pq
Synchronization is a natural phenomenon that one en-
counters in daily life. Since the identification of chaotic
synchronization [1, 2, 3], several papers have appeared
in identifying and demonstrating basic kinds of synchro-
nization both theoretically and experimentally (cf.[4, 5]).
Among them, chaotic phase synchronization (CPS) refers
to the coincidence of characteristic time scales of the
coupled systems, while their amplitudes of oscillations
remain chaotic and often uncorrelated. Phase synchro-
nization (PS) plays a crucial role in understanding the
behavior of a large class of weakly interacting dynam-
ical systems in diverse natural systems. Examples in-
clude circadian rhythm, cardio-respiratory systems, neu-
ral oscillators, population dynamics, electrical circuits,
etc [4, 5, 6].
The notion of CPS has been investigated so far in os-
cillators driven by external periodic force [7, 8], chaotic
oscillators with different natural frequencies and/or with
parameter mismatches [9, 10, 11, 12], arrays of coupled
chaotic oscillators [13, 14] and also in essentially differ-
ent chaotic systems [15, 16]. On the other hand PS in
nonlinear time-delay systems, which form an important
class of dynamical systems, have not yet been identified
and addressed. A main problem here is to define even
the notion of phase in time-delay systems due to the in-
trinsic multiple characteristic time scales in these sys-
tems. Studying PS in such chaotic time-delay systems
is of considerable importance in many fields, as in un-
derstanding the behavior of nerve cells (neuroscience),
where memory effects play a prominent role, in patho-
logical and physiological studies, in ecology, in lasers etc
∗Electronic address: lakshman@cnld.bdu.ac.in
†Electronic address: jkurths@gmx.de
[4, 5, 6, 17, 18, 19, 20, 21].
In this paper, we report the first identification of phase
synchronization (PS) in nonidentical time-delay systems
in the hyperchaotic regime with non-phase coherent at-
tractors with unidirectional nonlinear coupling. We will
show the entrainment of phases of a coupled piecewise
linear time-delay system for weak coupling from the
non-synchronized state. Phase is calculated using the
Poincare´ method [4, 5] after a new transformation of at-
tractors of the time-delay systems, which looks then like
a smeared limit cycle. The existence of PS and general-
ized synchronization (GS) in coupled time-delay systems
is characterized by recently proposed methods based on
recurrence quantification analysis and also in terms of
Lyapunov exponents of the coupled time-delay systems.
We first consider the following unidirectionally coupled
drive x1(t) and response x2(t) systems, which we have
recently studied in detail in [22, 23, 24],
x˙1(t) = −ax1(t) + b1f(x1(t− τ)), (1a)
x˙2(t) = −ax2(t) + b2f(x2(t− τ)) + b3f(x1(t− τ)),(1b)
where b1, b2 and b3 are constants, a > 0, τ is the delay
time and f(x) is the piece-wise linear equation of the
form
f(x) =


0, x ≤ −4/3
−1.5x− 2, −4/3 < x ≤ −0.8
x, −0.8 < x ≤ 0.8
−1.5x+ 2, 0.8 < x ≤ 4/3
0, x > 4/3.
(2)
We have chosen the values of parameters as a = 1.0, b1 =
1.2, b2 = 1.1 and τ = 15, which are outside the region
of complete, lag and anticipatory synchronizations dis-
cussed in [22, 23]. For this parametric choice, in the ab-
sence of coupling, the drive x1(t) and the response x2(t)
2 0.7
 0.8
 0.9
 1
 0.7  0.8  0.9  1
x 1
(t)
x1(t+15)
(a)
 0.6
 0.8
 1
 1.2
 0.7  0.8  0.9  1
z(t
+
15
)
x1(t+15)
(b)
 0
 50
 100
 150
 200
 4700  4800  4900  5000
φ 1z (
t),
φ 2z (
t),
 ∆
φ
t
(c)φ1z
φ2z
∆φ
FIG. 1: Phase synchronization between the systems (1a) and
(1b). (a) The non-phase coherent hyperchaotic attractor of
the uncoupled drive (1a). (b) Transformed attractor in the
x1(t + τ ) and z(t + τ ) space along with the Poincare´ points
represented as open circles. (c) The phases of the drive system
(φz1), the response system (φ
z
2) calculated from the new state
variable z(t+ τ ) and their difference (∆φ) for b3 = 1.5.
systems evolve independently. Further in this case, the
drive x1(t) exhibits a hyperchaotic attractor (Fig. 1a)
with five positive Lyapunov exponents (See [22] for the
spectrum of Lyapunov exponents) and the response x2(t)
has four positive Lyapunov exponents, i.e. both subsys-
tems are qualitatively different (due to b1 6= b2). The
parameter b3 is the coupling strength of the unidirec-
tional nonlinear coupling (1b), while the parameters b1
and b2 play the role of parameter mismatch resulting in
nonidentical coupled time-delay systems.
Now the important questions we encounter are whether
PS exists in the time-delay system (1) when the coupling
is included (b3 > 0) and, if so, how to characterize the
possible transition to PS in such systems which possess
in general highly non-phase coherent attractors having
a broad-band power spectrum. In low-dimensional sys-
tems, a few methods [4, 5] have been developed to define
phase in phase coherent chaotic attractors, which have
a dominant peak in the power spectrum. Definition of
phase is not so clear in non-coherent chaotic attractors,
in particular in high-dimensional systems having broad-
band power spectra such as time-delay systems. Meth-
ods to calculate phase of non-coherent attractors of time-
delay systems is not readily available. One approach to
calculate the phase of system (1) is based on the concept
of curvature [25], which is often used in low-dimensional
systems. However, we find that this procedure does not
work in the case of time-delay systems in general, and
in particular for Eqs. (1). We present here three other
approaches to study PS in systems as (1):
i) We introduce a new transformation to successfully
capture the phase in the present problem. It transforms
the non-phase coherent attractor (Fig. 1a) into a smeared
limit cycle-like form with well defined rotations around
one center (Fig. 1b). This transformation is performed
by introducing the new state variable
z(t+ τ) = x1(t)x1(t+ τˆ)/x1(t+ τ), (3)
where τˆ is the optimal value of delay time to be chosen
(so as to rescale the original non-phase coherent attractor
into a smeared limit cycle-like form), we plot the above
attractor (Fig. 1a) in the (x1(t+τ), z(t+τ)) phase space.
The functional form of this transformation has been iden-
tified by generalizing the transformation used in the case
of chaotic atractors in the Lorenz system [4]. We find
the optimal value of τˆ to be 1.65. The above transfor-
mation is obtained through a suitable functional form
(along with a delay time τˆ ), so as to unfold the origi-
nal attractor (Fig. 1a) into a phase coherent attractor.
Now the attractor (Fig. 1b) looks indeed like a smeared
limit cycle with nearly well defined rotations around a
fixed center. Hence, we can calculate the phase using
the Poincare´ method [4, 5] from the rescaled attractor.
The Poincare´ points are shown as open circles in Fig. 1b.
The phases of both the drive φz1(t) and the response φ
z
2(t)
systems calculated from the new state variable z(t + τ)
are shown in Fig. 1c along with their phase difference
∆φ = φz1(t)−φ
z
2(t) for the value of the coupling strength
b3 = 1.5, showing a high quality PS. This strong bound-
edness of the phase difference obtains for b3 ≥ 1.382.
(Note that the transformed attractor (Fig. 1b) does not
have any closed loops as in the case of the original at-
tractor (Fig. 1a). If it is so, such closed loops will lead
to phase mismatch, and one cannot obtain exact match-
ing of phases of both the drive and response systems as
shown in Fig. 1c).
ii) Next, we analyze the complex synchronization phe-
nomena in the coupled time-delay systems (1) by means
of very recently proposed methods based on recurrence
plots [26]. These methods help to identify and quan-
tify PS (particularly in non-phase coherent attractors)
and GS. For this purpose, the generalized autocorrela-
tion function P (t) [26] was introduced
P (t) =
1
N − t
N−t∑
i=1
Θ(ǫ− ||Xi −Xi+t||), (4)
where Θ is the Heaviside function, Xi is the ith data
of the system X , ǫ is a predefined threshold. ||.|| is the
Euclidean norm and N is the number of data points.
Looking at the coincidence of the positions of the maxima
of P (t) for both systems, one can qualitatively identify
PS.
A criterion to quantify PS is the cross correlation coef-
ficient between the drive, P1(t), and the response, P2(t),
which can be defined as Correlation of Probability of Re-
currence (CPR)
CPR = 〈P¯1(t)P¯2(t)〉/σ1σ2, (5)
where P¯1,2 means that the mean value has been sub-
tracted and σ1,2 are the standard deviations of P1(t) and
3 0
 0.25
 0.5
 0  100  200  300  400  500
P 1
 
(t)
,P
2 
(t)
t
(a)
P1 (t)
P2 (t)
 0
 0.25
 0.5
 0  100  200  300  400  500
P 1
 
(t)
,P
2 
(t)
t
(b)
 0
 0.25
 0.5
 0  100  200  300  400  500
P 1
 
(t)
,P
2 
(t)
t
(c)
FIG. 2: Generalized autocorrelation functions of both the
drive P1(t) and the response P2(t) systems. (a) Non-phase
synchronization for b3 = 0.6, (b) Phase synchronization for
b3 = 1.5 and (c) Generalized synchronization for b3 = 2.3.
P2(t) respectively. CPR ≈ 1 indicates that the systems
are in complete PS, whereas for non-PS one obtains low
values of CPR.
To characterize GS, the authors of [26] proposed the
first index as the Joint Probability of Recurrences (JPR),
JPR =
1
N2
∑N
i,j Θ(ǫx − ||Xi −Xj||)Θ(ǫy − ||Yi − Yj ||)−RR
1−RR
(6)
where RR is rate of recurrence, ǫx and ǫy are thresholds
corresponding to the drive and response systems respec-
tively. RR measures the density of recurrence points and
it is fixed as 0.02 [26]. JPR is close to 1 for systems in
GS and is small when they are not in GS. The second
index depends on the coincidence of probability of re-
currence, which is defined as Similarity of Probability of
Recurrence (SPR),
SPR = 1− 〈(P¯1(t)− P¯2(t))
2〉/σ1σ2. (7)
SPR is of order 1 if both the systems are in GS and
approximately zero if they evolve independently.
Now, we will apply this concept to the original (non-
transformed) attractor (Fig. 1a). We estimate these re-
currence based measures from 5000 data points after suf-
ficient transients with the integration step h = 0.01 and
sampling rate ∆t = 100. The generalized autocorre-
lation functions P1(t) and P2(t) (Fig. 2a) for the cou-
pling b3 = 0.6 show that the maxima of both systems
do not occur simultaneously and there exists a drift be-
tween them, so there is no synchronization at all. This
is also reflected in the rather low value of CPR = 0.381.
For b3 ∈ (0.91, 1.381), we observe the first substantial
increase of recurrence reaching CPR ≈ 0.5 − 0.6. Look-
ing to the details of the generalized correlation functions
P (t), we find that now the main oscillatory dynamics
becomes locked, i.e. the main maxima of P1 and P2 co-
incide. For b3 ∈ (1.382, 2.2) CPR reaches almost 1, i.e.
now all maxima of P1 and P2 are in agreement and this is
in accordance with strongly bounded phase differences.
This is a strong indication for PS. Note, however that the
heights of the peaks are clearly different (Fig. 2b). The
differences in the peak heights exhibit that there is no
strong interrelation in the amplitudes. Further increase
of the coupling (here b3 = 2.3) leads to the coincidence of
both the positions and the heights of the peaks (Fig. 2c)
referring to GS in systems (1). This is also confirmed
from the maximal values of the indices JPR = 1 and
SPR = 1, which is due to the strong correlation in the
amplitudes of both systems. The transition from non-
synchronized to PS and then to GS is characterized by
the maximal values of CPR, SPR and JPR (Fig. 3b). As
expected from the construction of these functions, CPR
refers mainly to the onset of PS, whereas JPR quanti-
fies clearly the onset of GS. The existence of GS is also
confirmed using the auxiliary system approach[27].
iii) The transition from non-synchronization to PS is
also characterized by changes in the Lyapunov exponents
of the coupled time-delay systems (1). The spectrum of
the nine largest Lyapunov exponents of the coupled sys-
tems is shown in Fig. 3a, from which one can find that
the Lyapunov exponents corresponding to the response
system become negative from the value of the coupling
strength b3 > 0.9, where the transition to PS occurs,
except for the largest Lyapunov exponent λ
(2)
max which
continues to remain positive. This is a strong indication
that in this rather complex attractor the amplitudes be-
come somewhat interrelated already at the transition to
PS (as in the funnel attractor [25]). It is interesting to
note that the Lyapunov exponents of the response system
λi (other than λ
(2)
max) are changing already at the early
stage of PS (b3 ≈ 0.91), where the complete PS is not yet
reached. This has been also observed for the onset of PS
in phase-coherent oscillators [9].
We have obtained the same results for different sam-
pling intervals ∆t and for various values of delay time
τ . We have also identified this transition to PS and to
GS in the coupled Mackey-Glass systems [17, 28], and
also in (1) for linear coupling (unidirectional) using these
above three approaches (these results will be published
in a more comprehensive paper).
In conclusion, we have identified for the first time the
4 0
 0.2
 0.4
 0.6
 0.8
 1
 0  0.5  1  1.5  2  2.5  3
CP
R,
JP
R
b3
(b)
CPR
JPR
-0.03
-0.02
-0.01
 0
 0.01
 0.02
 0  0.5  1  1.5  2  2.5  3
λ i
b3
(a)
FIG. 3: (a) Spectrum of first nine largest Lyapunov expo-
nents of the coupled systems (1) as a function of coupling
strength b3 ∈ (0, 3) and (b) Spectrum of CPR and JPR as a
function of coupling strength b3 ∈ (0, 3).
existence of PS in coupled time-delay systems in the hy-
perchaotic regime with highly non-phase coherent attrac-
tors. We have shown that there is a typical transition
from non-synchronized state to PS for weak coupling and
in the range of strong coupling there is a transition to GS
from PS. We have also identified a suitable transforma-
tion, which works equally well for Mackey-Glass system
(having a more complex hyperchaotic attractor), to cap-
ture the phase of the underling attractor. We have also
characterized the existence of PS and GS in terms of
recurrence based indices like generalized autocorrelation
function P (t), CPR, JPR and SPR and quantified the
different synchronization regimes in terms of them. The
above transition is also confirmed by the changes in the
Lyapunov exponents. We have pointed out the existence
of PS in coupled Mackey-Glass systems as well. The re-
currence based technique as well as the new transforma-
tion are also appropriate for the analysis of experimental
data, i.e. we expect experimental verification of these
findings.
Acknowledgments
The work of D. V. S and M. L has been supported by
a Department of Science and Technology, Government
of India sponsored research project. J. K has been sup-
ported by his Humboldt-CSIR research award and NoE
BIOSIM (EU).
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Phase synchronization in time-delay systems

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