We analyze the size limits of coupled map lattices with diffusive coupling at the crossover of low-dimensional to high-dimensional chaos. We investigate the existence of standing-wave-type periodic patterns, within the low-dimensional limit, in addition to the stable synchronous chaotic states depending upon the initial conditions. Further, we bring out a controlling mechanism to explain the emergence of standing-wave patterns in the coupled map lattices. Finally, we give an analytic expression in terms of the unstable periodic orbits of the isolated map to represent the standing-wave patterns

Lakshmanan, M.

Muruganandam, P.

Palaniyandi, P.

English

Publications of the IAS Fellows

ar
X
iv
:n
lin
/0
50
10
46
v3
  [
nli
n.C
D]
  1
6 S
ep
 20
05
Desynchronized wave patterns in synchronized chaotic regions of coupled map lattices
P. Palaniyandi, P. Muruganandam, and M. Lakshmanan
Centre for Nonlinear Dynamics, Department of Physics,
Bharathidasan University, Tiruchirapalli 620 024, India
(Dated: February 8, 2008)
We analyze the size limits of coupled map lattices with diffusive coupling at the crossover of low-
dimensional to high-dimensional chaos. We investigate the existence of standing-wave-type periodic
patterns, within the low-dimensional limit, in addition to the stable synchronous chaotic states
depending upon the initial conditions. Further, we bring out a controlling mechanism to explain
the emergence of standing-wave patterns in the coupled map lattices. Finally, we give an analytic
expression in terms of the unstable periodic orbits of the isolated map to represent the standing-wave
patterns.
PACS numbers: 05.45.Ra, 05.45.Xt
I. INTRODUCTION
Coupled dynamical systems often arise in nature
whenever a collective or cooperative phenomenon is
favoured [1, 2, 3, 4]. In particular, the coupled map lat-
tice with diffusive coupling (CML) provides a prototype
model to study various features associated with the co-
operative evolution of constituent systems [5, 6, 7, 8, 9].
One of the important properties of such CMLs is that
they exhibit size instability, that is, there is a criti-
cal size on the number of constituents for which stable
synchronous chaotic state exists. Increasing the num-
ber of constituents beyond this limit leads to the oc-
currence of spatially incoherent behaviour (eg., high-
dimensional chaos). For example, Bohr and Chris-
tensen [10] have studied such size instability behaviour in
a two-dimensional coupled logistic lattice. Similar desyn-
chronization has been found in arrays of coupled systems
represented by nonlinear oscillators [4, 11, 12, 13]. The
stability of synchronous chaos in coupled dynamical sys-
tems plays an important role in the study of pattern for-
mation, spatiotemporal chaos, etc. [4, 10, 11, 14, 15].
In general, these studies on size instability are valid in
most situations. However, we have noted that in certain
circumstances there is an ambiguity in dealing with these
systems below the critical system sizes. To be specific,
there exist certain nontrivial ranges of initial conditions
for which the CML admits spatial and temporally peri-
odic solutions in contrast to the usually expected stable
synchronous chaos. In this brief report, we show numer-
ically the coexistence of such periodic states with the
stable synchronous chaotic state well below the critical
system size and explain the underlying mechanism.
II. SIZE INSTABILITY IN COUPLED MAP
LATTICES WITH DIFFUSIVE COUPLING
Consider an one-dimensional coupled map lattice with
nearest-neighbour diffusive coupling [5, 6]
x
j
n+1 = f(x
j
n) + ǫ
[
f(xj−1n ) + f(x
j+1
n )− 2f(x
j
n)
]
, (1)
where j (= 0, 1, 2, · · · , L − 1) represents the lattice sites
and L is the system size, subject to periodic boundary
conditions.
The stability analysis of the synchronized chaotic state
defined by sn = x
0 = x1 = · · · = xL−1 in the above CML
using the procedure derived originally for coupled oscil-
lators by Heagy et al [11], gives the relation connecting
transverse Lyapunov exponents (TLEs), in terms of the
Lyapunov exponent of single (isolated) map, λ0 as
λk = λ0 + ln
[
1− 4ǫsin2
(
πk
L
)]
. (2)
The synchronous state is stable only if the TLEs (λk,
k = 1, 2, · · · L− 1) are all negative.
The above relation (2) can also be obtained by means
of a direct perturbation of the form
xjn = sn + δ exp
(
i
2πk
L
)
exp
(
λkn
)
, δ ≪ 1, (3)
as considered by Bohr and Christensen [10].
The synchronous state loses its stability when the long
waves (lowest mode) are unstable [10]. This means that
for λ1 > 0 the synchronous state is unstable. Thus,
substituting λ1 = 0 in Eq. (2), one obtains the maxi-
mum/critical lattice size (Lc) that supports stable syn-
chronous state as
Lc = int

 π
sin−1
(√
1−e−λ0
4ǫ
)

 . (4)
Now, let us consider a coupled logistic lattice with diffu-
sive coupling (CLL) where each lattice site in Eq. (1) is
occupied by the logistic map
f(x) = µx(1− x), x ∈ (0, 1), µ ∈ (0, 4). (5)
In particular for the choice µ = 3.5732, the Lyapunov
exponent of single (isolated) map is positive (i.e., λ0 ∼
0.057 > 0) and chaotic. In this case, for coupling strength
2ǫ = 0.2, the critical lattice size (Lc) is found to be 11 from
Eq. (4). That is, upto the lattice size 11 the CLL exhibits
synchronous chaos and for lattice size ≥ 12 the synchro-
nization is found to be lost, thereby confirming the size
instability in the diffusively coupled logistic lattices.
III. EXISTENCE OF MULTIPLE STABLE
STATES IN COUPLED MAP LATTICES
When random initial conditions are assumed, in most
cases, the above CLL with diffusive coupling exhibits
stable synchronous chaos for 2 ≤ L ≤ 11 as predicted
by Eq. (4). However, there are certain ranges of initial
conditions for which CLL shows some interesting asyn-
chronous spatiotemporal patterns even for L < Lc. For
example, for L = 8 and with the choice of initial con-
 0
 0.2
 0.4
 0.6
 0.8
 1
 1  2  3  4  5  6  7  8
x
j
 0
 0.2
 0.4
 0.6
 0.8
 1
 1  2  3  4  5  6  7  8
x
j
 0
 0.2
 0.4
 0.6
 0.8
 1
 1  2  3  4  5  6  7  8
x
j
 0
 0.2
 0.4
 0.6
 0.8
 1
 1  2  3  4  5  6  7  8
x
j
(a) (b)
(c) (d)
FIG. 1: The possible spatiotemporal patterns in coupled lo-
gistic lattice (1) and (5) with L = 8 for different initial con-
ditions. (a) single standing wave, (b) double standing waves
(c) single standing wave with a temporal period-2 orbit in one
half and period-4 orbit in the other half of the lattice and (d)
synchronized chaos.
ditions {xj0}
L−1
j=0 = {0.1, 0.01, 0.7, 0.2, 0.65, 0.1, 0.15,
0.001} or several nearby initial conditions, the CLL ex-
hibits a standing wave type pattern as shown in Fig. 1(a).
But for the same lattice size, if we choose a different set
of initial conditions, {xj0}
L−1
j=0 = {0.0004, 0.0001, 0.0003,
0.0005, 0.00045, 0.00018, 0.00016, 0.00001} or the nearby
points, two standing waves with different amplitudes are
produced within the CLL as in Fig. 1(b). Similarly a
disturbed standing wave pattern as shown in Fig. 1(c)
is possible to be exhibited by the CLL for many choices
of initial conditions. A synchronous chaos, as one would
expect for L = 8 from the theory, is also exhibited by
the CLL as depicted in Fig. 1(d), for most of the ran-
dom choices of initial conditions. This kind of multiple
stable solutions is also observed in the CLL for other
lattice sizes, namely L = 6, 7, 9 and 10 which are also
less than Lc. The occurrence of various spatiotemporal
patterns for different lattice sizes of the CLL and their
percentage of occurrence are shown in Table I. In order
to quantify the percentage of occurrence of different spa-
tiotemporal patterns, we have used 106 sets of random
initial conditions (i.c.’s) in the interval 0 to 1 and identi-
fied the number of i.c.’s which lead to a specific pattern
as indicated in Table I. We have further confirmed our
assertion by analyzing the same systems in different com-
puting environments such as Intel Pentium 4, Sun Sparc
server/workstation and Compaq Alpha workstation.
In addition, by considering the whole CML of size L
as a single L-dimensional map we have verified that the
above spatiotemporal periodic structures are essentially
the stable fixed points of this map. For example, the pe-
riodic structure in Fig. 1(a) represents a fixed of point of
period-2 of the eight-dimensional map and the eigenval-
ues of the corresponding Jacobian matrix all are having
magnitude less than unity. In a similar fashion one can
verify that all the periodic structures are the stable fixed
points of corresponding periods. Thus, in addition to the
stable synchronized manifold, there exists other invariant
sets corresponding to stable periodic structures.
TABLE I: The possible spatiotemporal patterns for different
lattice sizes and their percentage of occurrence for the cou-
pled logistic lattice sampled over a set of 106 random initial
conditions (IC’s). The parameters are fixed as µ = 3.5732 in
Eq. (5) and ǫ = 0.2 in Eq. (1). Here, SW: standing wave,
sync.: synchronized.
size spatiotemporal % of size spatiotemporal % of
(L) patterns IC’s (L) patterns IC’s
6 single SW 9 10 double SWs 43
sync. chaos 91 sync. chaos 45
7 single SW 22 others 12
sync. chaos 78 11 double SWs 48
8 single SW 21 four SWs 2
double SWs 10 sync. chaos 34
sync. chaos 66 others 16
others 3 12 double SWs 24
9 double SWs 38 four SWs 28
sync. chaos 54 sync. chaos 0.2
others 8 others 47.8
IV. EMERGENCE OF STANDING WAVE
PATTERNS BY CONTROLLING
The extraordinary behaviour of the CML with diffu-
sive coupling showing standing wave patterns well below
the critical lattice size (Lc) can be explained as follows.
The second term in the right hand side of Eq. (1), can be
considered as a kind of force or perturbation applied to
every lattice point in the CML and we call it as the cou-
pling force. In fact this force on a particular lattice point
is developed either due to a mismatch in the parameters
3of the neighbouring lattice points or due to differences in
their initial conditions or both. If the neighbouring lat-
tice points are identical then this force is formed due to
variation in the initial conditions and our system indeed
falls under this category. In general, the strength of the
coupling force at all the lattice points approaches zero
when they oscillate towards synchronization with their
neighbours, and usually this will happen for L ≤ Lc.
But in the case of L > Lc, this force at every lattice
point oscillates periodically or in a chaotic manner, giving
rise to various spatiotemporal patterns, including stand-
ing waves. However, as we have pointed out above that
under certain circumstances, (i.e., for certain ranges of
initial conditions), even for L < Lc, the coupling force of
each and every lattice point oscillates periodically with
different amplitudes and thereby makes the CML to ex-
hibit spatiotemporal periodic (standing wave) solutions.
In general, the periodic oscillations in the coupling force
may be of any period and this fixes the number of stand-
ing waves produced within the lattice.
In order to understand the mechanism behind the
emergence of spatiotemporal periodic structure, let us
now consider a specific case of the periodically oscillating
coupling force of period two, that is, the force oscillating
between two fixed amplitudes, say, k1 and k2 so that a
single standing wave is formed in the CML. Then the am-
plitudes of the coupling force at the jth lattice site will
alternate between the numerical values kj1 and k
j
2, where
j = 1, 2, . . . , L. Thus, the evolution of jth map in the
CML can be effectively described by the equation
x
j
n+1 = f(x
j
n) + k
j
1, x
j
n+2 = f(x
j
n+1) + k
j
2. (6)
For a given lattice point j, this is nothing but a single
logistic map with a periodic kick of period-2 (modified
map). It is now obvious to note from Eqs. (6) and (1)
that the original coupled map lattice exhibiting a stand-
ing wave pattern can be decomposed into L number of
modified maps such that the dynamics of (1) is essentially
mimicked by the set (6). Thus studying the evolution of
L decoupled modified maps (6) with allowed sets of val-
ues for kj1 and k
j
2 is equivalent to that of the original
CML, given by Eq. (1).
In general, a chaotically evolving system can be con-
trolled to a stable periodic orbit by the addition of an
appropriate constant or periodic external bias [3, 16, 17,
18, 19]. As a consequence, one can expect a stable fixed
point solution for the modified map (6) for appropri-
ate forcing amplitudes (k1 and k2), with same set of
parameters for which the original map, xjn+1 = f(x
j
n),
exhibits chaotic solution. Thus there exists a possibility
for obtaining periodic solutions of period two for the con-
stituent map within the CML even though its parameter
is in the chaotic region of the individual map.
In the case of coupled logistic lattice with parameters
mentioned in the previous section, the allowed region
of forcing amplitudes which lead the decoupled modi-
fied map (6) to exhibit period two solution is shown in
Fig. 2. For L = 8, we have calculated the amplitudes
of the coupling force [that is, the second term in the
right hand side of Eq. (1)] of period-2 for the initial con-
ditions, {xj0}
L−1
j=0 = {0.1, 0.01, 0.7, 0.2, 0.65, 0.1, 0.15,
0.001}, and these are shown in Table II. Now one can
easily check that these amplitudes of the coupling force
fall in the region specified by the phase diagram shown
in Fig. 2. Also, the shift invariant property of the CLL
-0.1 0 0.1
-0.1
0
0.1
k1
k 2
FIG. 2: The phase diagram in the k1 − k2 plane for the mod-
ified map (6) with f(x) = µx(1−x), µ = 3.5732, showing the
regions which correspond to the periodic solution of period 2
(dark region).
ensures that there is no temporal variation if we shift
the initial conditions of each lattice point in the CML
to its neighbour spatially. In this case, the wave pat-
tern will also make only a corresponding shift. We have
made similar investigations for higher periodic standing
waves which lead to same type of conclusions based on
the appropriate periodic nature of the coupling force.
TABLE II: Amplitudes of the coupling force k1 and k2 in
the coupled map lattices for L = 8 with period-2 for initial
conditions, {xj
0
}L−1j=0 = {0.1, 0.01, 0.7, 0.2, 0.65, 0.1, 0.15,
0.001}.
lattice kj
1
k
j
2
lattice kj
1
k
j
2
site(j) (×10−2) (×10−2) site(j) (×10−2) (×10−2)
1 −0.17626 3.24717 5 3.24717 −0.17626
2 −5.98282 2.91190 6 2.91190 −5.98282
3 2.91190 −5.98282 7 −5.98282 2.91190
4 3.24717 −0.17626 8 −0.17626 3.24717
So, if it is possible to control the coupling force to fall in
the region which corresponds to a periodic solution, then
one can obtain standing wave patterns irrespective of the
size of the lattices. This is in fact possible by choosing ap-
propriate initial conditions to each lattice point and this
explains the occurrence of standing waves (asynchronous)
as shown in Figs. 1(a) and 1(b) in the CLL well below
the critical lattice size (Lc). Similar explanation holds
good for Fig. 1(c). The same principle is involved in the
occurrence of standing waves even for L > Lc [6].
4V. ANALYTICAL EXPRESSION FOR
STANDING WAVE PATTERNS
From a careful numerical analysis, we have observed
that the nodes and antinodes of standing waves are
formed at or very close to the UPOs of the isolated logis-
 0
 0.2
 0.4
 0.6
 0.8
 1
 1  2  3  4  5  6  7  8
x
j  1  2  3  4  5  6  7  8j
(a) (b)
FIG. 3: Standing wave patterns obtained from Eq. (7) for
L = 8: (a) standing wave with period-1 UPO at nodes and
period-2 UPO at antinodes and (b) period-2 UPO at nodes
and period-4 UPO at antinodes.
tic map. Keeping this in mind, we propose an expression
for the standing wave pattern [20] of the form
xj = uk + Ak sin
(
mπ(j − δ)
L
)
cos (πi) , (7)
where,
Ak =


Amaxk , if sin
(
mπ(j−δ)
L
)
cos (πi) > 0
Amink , if sin
(
mπ(j−δ)
L
)
cos (πi) < 0
where the discrete index j = 0, 1, 2, · · · , L−1 corresponds
to the lattice site, m denotes the mode of the waves, and
k and i represent the nodes and antinodes of different
standing waves present in the pattern, which can take
values from 1 to p and 1 to 2p, respectively and p is
the period of UPO. Also in the above equation (7), uk’s
represent the values of the UPOs of the isolated logistic
map at the node of the k-th standing wave, and Amaxk and
Amink are the absolute values of the differences between
UPOs at the node and UPOs at high and low amplitudes
at the antinodes of the kth standing waves, respectively.
The wave patterns with one and two number of standing
waves obtained using Eq. (7) for L = 8 are shown in
Fig. 3, whereas for the same lattice size, the numerically
obtained pattern which have been discussed in Sec. III is
shown in Figs. 1(a) and 1(b). One observes that these
two figures coincide very closely.
VI. SUMMARY AND CONCLUSION
In this report, we have pointed out that coupled map
lattices with diffusice coupling exhibit multiple stable
states for the same set of parameters with respect to
the initial conditions. It has also been shown that by
choosing appropriate initial conditions, one can obtain
different standing wave type patterns for such coupled
map lattices even for lattice size much less than the crit-
ical system size Lc, where one normally would expect
synchronized chaos. In addition, we have proposed the
mechanism behind the occurrence of such standing wave
patterns.
Acknowledgments
This work has been supported by the National Board
for Higher Mathematics, Department of Atomic Energy,
Government of India and the Department of Science
and Technology, Government of India through research
projects.
[1] M. Lakshmanan and K. Murali, Chaos in Nonlinear Os-
cillators: Controlling and Synchronization (World Scien-
tific, Singapore, 1996).
[2] A. Pikovsky, M. Rosenblum, and J. Kurths, Synchroniza-
tion: A Universal Concept in Nonlinear Sciences (Cam-
bridge University Press, Cambridge, 2001).
[3] M. Lakshmanan and S. Rajasekar, Nonlinear Dynamics:
Integrability, Chaos and Patterns (Springer-Verlag, New
York, 2003).
[4] P. Muruganandam, K. Murali, and M. Lakshmanan, Int.
J. Bifur. Chaos: Appl. Sci. Eng. 9, 805 (1999).
[5] K. Kaneko, Collapse of Tori and Genesis of Chaos in
Dissipative Systems (World Scientific, Singapore, 1986).
[6] K. Kaneko, Phys. Rev. Lett. 69, 905 (1992).
[7] P. G. Lind, J. Corte-Real, and J. A. C. Gallas, Phys.
Rev. E 66, 016219 (2002).
[8] G. Francisco and P. Muruganandam, Phys. Rev. E 67,
066204 (2003).
[9] K. Kaneko and I. Tsuda, Complex Systems: Chaos and
Beyond (Springer, Berlin, 2000).
[10] T. Bohr and O. B. Christensen, Phys. Rev. Lett. 63, 2161
(1989).
[11] J. F. Heagy, T. L. Carroll, and L. M. Pecora, Phys. Rev.
E 50, 1874 (1994).
[12] L. M. Pecora and T. L. Carroll, Phys. Rev. Lett. 80, 2109
(1998).
[13] J. G. Restrepo, E. Ott, and B. R. Hunt Phys. Rev. Lett.
93, 114101 (2004).
[14] G. Rangarajan and M. Ding, Phys. Lett. A 296, 204
(2002).
[15] Y. Chen, G. Rangarajan, and M. Ding, Phys. Rev. E 67,
026209 (2003).
[16] K. Pyragas, Phys. Lett. A 170, 421 (1992).
[17] A. Y. Loskutov and A. I. Shismarev, Chaos 4, 391 (1994).
[18] P. Palaniyandi and M. Lakshmanan, Phys. Lett. A 342,
134 (2005).
[19] A. Venkatesan, S. Parthasarathy, and M. Lakshmanan,
Chaos, Solitons and Fractals, 18, 891 (2003).
5[20] I. G. Main, Vibrations and Waves in Physics (Cambridge
University Press, Cambridge, 1993), 3rd ed.


Desynchronized wave patterns in synchronized chaotic regions of coupled map lattices

http://repository.ias.ac.in/19590/1/420.pdf

Desynchronized wave patterns in synchronized chaotic regions of coupled map lattices

Abstract

Similar works

Full text

Available Versions

Publications of the IAS Fellows