Antiphase Synchronization in Environmentally coupled Rossler Oscillators

Abstract

We study the manifestation of antiphase synchronization in a system of n Rossler Oscillators coupled through a dynamic environment. When the feedback from system to environment is positive (negative) and that from environment to system is negative (positive), the oscillators enter into a state of anti phase synchronization both in periodic and chaotic regimes. Their phases are found to be uniformly distributed over 2pi, with a phase lag of 2pi /n between neighbors as is evident from the similarity function and the phase plots. The transition to anti phase synchronization is marked by the crossover of (n-1) zero Lyapunov Exponents to negative values. If the systems are individually in chaotic phase, with strong enough coupling they end up in periodic states which are in antiphase synchronization

Full text

arXiv:0812.3753v1 [nlin.CD] 19 Dec 2008
Antiphase Synchronization in Environmentally coupled Rossler Oscillators
G. Ambika∗
Indian Institute of Science Education and Research, Pune-411 021, India
Sheekha Verma†
Indian Institute of Technology, Kharagpur-721302, India
We study the manifestation of antiphase synchronization in a system of n Rossler Oscillators
coupled through a dynamic environment. When the feedback from system to environment is positive
(negative) and that from environment to system is negative (positive), the oscillators enter into a
state of antiphase synchronization both in periodic and chaotic regimes. Their phases are found to
be uniformly distributed over 2π, with a phase lag of 2π/n between neighbors as is evident from the
similarity function and the phase plots. The transition to antiphase synchronization is marked by
the crossover of (n-1) zero Lyapunov Exponents to negative values. If the systems are individually
in chaotic phase, with strong enough coupling they end up in periodic states which are in antiphase
synchronization
PACS numbers: 05.45.+b
Synchronization of mutually coupled chaotic systems
has been an area of intense research activity in recent
times [1, 2]. Depending upon the strength of coupling
and the nature of coupling, such systems are capable of
entering into a state of phase (antiphase) [3], lag [4], an-
ticipatory [5], generalized [6] and complete synchroniza-
tion [7]. Among these, antiphase synchronization with
repulsive or inhibitory coupling is the least studied. In
the case of Rossler oscillators with normal diffusive cou-
pling antiphase synchronization is reported for the funnel
type attractors [8]. So also cases of inhibitory (repulsive)
coupling in coupled map lattices leading to synchronous,
traveling wave and spatiotemporally chaotic states have
been studied from the point of view of persistence [9].
The role of repulsive coupling in enhancing synchroniza-
tion in complex heterogenous networks has been reported
recently[10]. An array of coupled phase oscillators with
repulsive coupling is found to result in a family of syn-
chronized regimes with zero mean field [11].
The work reported in this paper is motivated by the
fact that biological systems utilize different types of con-
nections among them to realize synchronization or trans-
mission of performances. For them the inhibitory cou-
pling is as relevant and useful as excitatory coupling
[12] and biological networks often evolve with positive
and negative connection between their components [13].
An interesting case of agent-environment interaction us-
ing the dynamical systems theory has been explained,
wherein the mutual interaction decides the adaptive fit
of the agent [14]. Our study is focused on the coupling
between the systems and the environment and its effect
in inducing synchronized behavior. Such an indirect cou-
pling through environment has been studied in a system
of van der Pol oscillators in the periodic state where neg-
∗Electronic address: g.ambika@iiserpune.ac.in
†Electronic address: sheekha.iitkgp@gmail.com
ative feedback with environment results in complete syn-
chronization while positive feedback leads to antiphase
synchronization. The stability of the former state is ana-
lyzed in detail and applied to a model of pulsating secre-
tion of GnRH [15]. In the present work, we take Rossler
oscillators in the chaotic state and show that antiphase
synchronization occurs when the feedback from system
to environment is negative (positive) and that from the
environment to system is positive (negative). The rele-
vance of the work lies in the fact that the systems are not
in direct interaction with one another but get feedback
from a dynamical environment which is influencing all
systems. It is interesting to note that even this can in-
duce a collective behavior which is phase correlated and
can lead to zero mean field if the number of systems is
large. Such a behavior can be found in many biologi-
cal systems that communicate through their environment
and often a large number of them synchronise to produce
macroscopic oscillations of any desirable nature.
Our basic model system is n Rossler systems coupled
indirectly through an environment with a dynamics gov-
erned by the following set of equations:
˙xk=−yk−zk+ǫ1w(1)
˙yk=xk+αyk(2)
˙zk=β+zk(xk−µ) (3)
where(k= 1,2, ...n)
˙w=−w+ǫ2
nX(xk) (4)
where (xk, yk, zk) are the states of the oscillator,
(α, β, µ) are the oscillator parameters. Depending on pa-
rameter values the oscillators are in periodic or chaotic
mode of operation. wdenotes the state of the environ-
ment, ǫ1, the coupling coeffient of feedback to the system

2
-30
-20
-10
0
10
20
30
950 955 960 965 970
x(t)
time
FIG. 1: Time series plot of x1(t) and x2(t)
0
0.5
1
1.5
2
0 5 10 15 20
S(τ)
τ
FIG. 2: Similarity function plot
and ǫ2that of feedback to the environment. For simplic-
ity we consider the feedback to the environment and vice
versa through one of the variables xkonly. The intrinsic
dynamics of the environment is assumed to decay expo-
nentially and without feedback from the oscillators it is
incapable of sustaining itself for extended periods of time.
We consider cases when ǫ1>0 and ǫ2<0 or ǫ1<0 and
ǫ2>0, since when both ǫ1, ǫ2<0 or ǫ1, ǫ2>0 yields
uninteresting or unbounded behavior.
For the numerical analysis we have taken α=β= 0.1
and µ= 18 where the individual systems are chaotic.
The system is evolved using fourth order Range Kutta
Method for a time of 20,000 secs. As displayed in Fig 1(a)
the states x1(t) and x2(t) are found to be 180 degrees out
of phase. The similarity function [16] S2(τ), defined as
S2(τ) = <[x2(t+τ)−x1(t)]2>
[< x2
1(t)>< x2
2(t)>]1/2(5)
is computed for τ= (0,21). The x1(t), x2(t) and S2(τ)
plot for two Rossler oscillators are shown in Fig 1
It is clear that the first minima of S2(τ) corresponds to
aτvalue which is the shift between x1(t) and x2(t). This
minimum repeats with an average periodicity of nearly
6.0 seconds that corresponds to the approximate period-
icity of the two individual oscillators.
-0.2
-0.15
-0.1
-0.05
0
0.05
0.1
0.15
0.2
0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000
λ
time
FIG. 3: The first four Lyapunov exponents of two independent
Rossler systems for ǫ1= 0 and ǫ2= 0 in equation(1)
-0.2
-0.15
-0.1
-0.05
0
0.05
0.1
0.15
0.2
0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000
λ
time
FIG. 4: The first four Lyapunov exponents of two environ-
mentally coupled Rossler systems for ǫ1= 0.5 and ǫ2=−0.5
The above behavior is observed for the range ǫ1=
(0.01,0.95) and ǫ2= (−0.5,−1) . Moreover when µ= 4
when the individual systems are in periodic state, qual-
itatively similar behavior with periodically synchronized
states are found to occur. In this case the minimum of
S2(τ) is exactly zero and has a periodicity of 6.0 equal
to the periodicity of the two oscillators.
The analysis is continued by calculating the Lyapunov
spectrum for the 7-dimensional system given by equa-
tion (1) for n=2. When ǫ1=ǫ2= 0, the asymptotic
values of the Lyapunov Exponents are λ1= 0.084, λ2=
0.084, λ3= 0.001, λ4= 0.001, λ5=−1.000, λ6=
−17.880 and λ7= 17.889. The λ1, λ2are positive with
values corresponding to the λ1of an independent Rossler
system [17]. λ3and λ4≈0 and λ5, λ6and λ7are
negative. When ǫ1= 0.5andǫ2=−0.5, correspond-
ing to the antiphase synchronised case, we find that
λ1= 0.064, λ2= 0.015, λ3= 0.000λ4=−0.075, λ5=
−0.089, λ6=−10.927 and λ7=−24.768. Here one of
the zero Lyapunov exponents becomes negative due to
coupling, indicating phase correlation between the sub-
systems [1]. λ1and λ2though different still remain pos-
itive indicating that the amplitudes of the systems are
uncorrelated and chaotic. The behavior of the first four
Lyapunov exponents in both cases are given in Fig 2 and

3
-30
-25
-20
-15
-10
-5
0
5
10
15
20
25
-30 -20 -10 0 10 20 30
y
x
FIG. 5: The snapshot distribution of 3 Rossler Systems
marked as black dots plotted on the x-y projection of the
trajectory of a particular system(shown in dotted lines)
-30
-25
-20
-15
-10
-5
0
5
10
15
20
25
-30 -20 -10 0 10 20 30
y
x
FIG. 6: The snapshot distribution of 4 Rossler Systems
marked as black dots plotted on the x-y projection of the
trajectory of a particular system(shown in dotted lines)
Fig 3.
Extending the analysis to higher number of systems
coupled in the same way, we find that their phases are
distributed over 2πwith a lag of 2π/n between neigh-
bors. This would mean that for large n, the mean field
is zero. Similar results has been reported in [11] in the
context of directly coupled oscillators with repulsive cou-
pling.The phase plots in x-y plane of the oscillator with
a snapshot distribution of the systems marked as black
dots is given in Fig 4 and Fig 5 for n=3 and 4. The simi-
larity function and the spectrum of Lyapunov Exponents
for n=3 and 4 are computed and are found to give qual-
itatively similar behavior as seen for n=2. We conclude
that the onset of antiphase synchronization occurs when
(n-1) zero Lyapunov exponents cross over to the negative
region.
In summary, we have studied the antiphase synchro-
nization in a system of n Rossler oscillators that are
coupled indirectly through a dynamic environment. The
phase relation between the systems is analyzed via the
time series and correlating it with the similarity func-
tion. The onset of phase correlated synchronized state
is evident from the study of Lyapunov exponents also.
When n is large, each of the systems are shifted in phase
from its nearest neighbor by 2π/n resulting in zero in-
stantenous mean field. By varying the coupling strength
the same antiphase synchronization is observed with con-
trol of chaos. Further work aimed at synchronization of
systems like neurons with environmental coupling is in
progress and will be reported elsewhere.
Acknowledgement
One of the authors, SV, thanks IISER, Pune for the facil-
ities and warm hospitability provided during her summer
project.
[1] J.F.Heagy, T.L.Caroll and L.M.Pecorra, Phys. Rev. A
50, 1874 (1994).
[2] I. Schreiber and M. Marek, Physica (Amsterdam) 5D,
258 (1982); S.K. Han, C. Kurrer, and Y. Kuramato,
Phys. Rev. Lett. 75, 3190 (1995).
[3] A. Pikovsky, M.Rosenblum and J.Kurths, Europhys.
Lett. 34, 165 (1996).
[4] Dae-Sic Lee, Guang-Hoon Kim, Uk Kang, Optics Express
10488 Vol 14, No.22 (2006).
[5] D.V.Senthilkumar, M.Lakshmanan, arXiv:nlin/
0510027V1 (2005).
[6] Henry D. I. Abarbanel, Nikolai F. Rulkov, Mikhail M.
Sushchik, Phy. Rev. E Vol. 53 No. 5 (1996).
[7] S. Boccaletti, J. Kurths, G. Osipov, D.L. Valladares, C.S.
Zhou, The synchronization of chaotic systems, Physics
Reports 366 (2002) 1-101.
[8] H.L.Yang, Phy. Review E, Volume 64, 026206.
[9] Prashant M. Gade, D. V. Senthilkumar, Sukratu Barve,
Sudeshna Sinha, Phy. Rev. E 75, 066208 (2007).
[10] I. Leyva, I. Sendina-Nadal, J.A. Almendral and M.A.F.
Sanjuan, Phy. Rev. E 74, 056112 (2006).
[11] L.S. Tsimring, N.F. Rulkov, M.L. Larsen, M. gabbay,
Phy. Review Lett. 95, 014101 (2005).
[12] A.V. Rangan and D. Cai, Phys. Rev. Lett. 96, 178101
(2006).
[13] X. Chen and J.E. Cohen, J. Theo. Biol. 212, 223 (2001);
S. Sinha and S. Sinha, Phys. Rev. E 71, 020902( R)
(2005).
[14] Randall D. Beer, Artificial Intelligence 72 (1995) 173-215.
[15] Guy Katriel, arXiv:0804.3734v1 (2008).
[16] M.G. Rosenblum, A.S. Pikovsky, and J. Kurths, Phys.
Rev. Lett. 78, 4193 (1997).
[17] A. Wolf, J. B. Swift, H. L. Swinney, and J. A. Vastano,
Physica D 16 (1985) 285.