VOLUME
83, NUMBER 3 PHYSICAL REVIEW LETTERS 19J
ULY
1999
Synchronization in Nonidentical Extended Systems
S. Boccaletti,
1
J. Bragard,
2,3
F. T. Arecchi,
2,4
and H. Mancini
1
1
Department of Physics and Applied Mathematics, Universidad de Navarra, Irunlarrea s/n, 31080 Pamplona, Spain
2
Istituto Nazionale di Ottica, Largo E. Fermi, 6, I50125 Florence, Italy
3
Department of Physics, University of Liege, Liege, Belgium
4
Department of Physics, University of Florence, Florence, Italy
(
Received 25 March 1999)
We report the synchronization of two nonidentical spatially extended fields, ruled by one-dimensional
complex Ginzburg-Landau equations, both in the phase and in the amplitude turbulence regimes. In
the case of small parameter mismatches, the coupling induces a transition to a completely synchronized
state. For large parameter mismatches, the transition is mediated by phase synchronization. In the
former case, the synchronized state is not qualitatively different from the unsynchronized one, while in
the latter case the synchronized state may substantially differ from the unsynchronized one, and it is
mainly dictated by the synchronization process of the space-time defects.
PACS numbers: 05.45.Xt, 05.45.Pq, 05.45.Jn, 47.54.+r
Coupled chaotic concentrated systems may display four
degrees of synchronization, namely, complete synchro-
nization (CS) [1], phase synchronization (PS) [2], lag
synchronization (LS) [3], and generalized synchronization
(GS) [4]. In CS, a perfect hooking of the chaotic trajec-
tories of two systems is achieved by means of a coupling
signal, in such a way that they remain in step with each
other in the course of time. This mechanism has been
shown to occur when two identical chaotic systems are
coupled, provided that all the sub-Liapunov exponents of
the subsystem to be synchronized are negative [1]. Non-
identical systems can reach a regime (PS), wherein a lock-
ing of the phases is produced, while the amplitudes remain
uncorrelated [2]. The transition to PS for two coupled
oscillators has been characterized with reference to the
Rössler system [3,5]. LS is an intermediate step between
PS and CS. In this case, the two signals lock their phases
and amplitudes, but with a time lag, s
1
共t兲⯝s
2
共t 2t
lag
兲
[3]. Finally, GS consists in the hooking of the output of
one system to a given function of the output of the other
system [4].
The generic scenario for concentrated symmetrically
coupled nonidentical systems consists in successive tran-
sitions between PS, LS, and CS when increasing the cou-
pling strength [3].
When passing to space-extended systems, space-time
chaos synchronization has been studied in discrete sys-
tems, such as populations of coupled dynamical systems
[6], systems formed by globally coupled Hamiltonian
[7] or bistable elements [8], and neural networks [9].
As for continuous systems, the emergence of complete
synchronized states has been studied for one-dimensional
chemical models [10], and for two fields obeying the
identical one-dimensional complex Ginzburg-Landau
(CGL) equation [11].
In this Letter we characterize the emergence of syn-
chronized behaviors in continuous nonidentical space-
extended symmetrically coupled pattern forming systems.
For the sake of exemplification, and without signifi-
cant loss of generality, we will refer to a pair of one-
dimensional fields A
1,2
共x, t兲 evolving in space and time
following the CGL equation. This equation describes the
universal pattern forming features close to the emergence
of a Hopf bifurcation [12]. It has been used to describe
many different situations in laser physics, fluid dynamics,
chemical turbulence, bluff body wakes, etc. The system
under study is
ᠨ
A
1,2
苷 A
1,2
1 共1 1 ia
1,2
兲≠
2
x
A
1,2
2 共1 1 ib
1,2
兲 jA
1,2
j
2
A
1,2
1´共A
2,1
2 A
1,2
兲 ,
(1)
where A
1,2
共x, t兲⬅r
1,2
共x, t兲 exp关ic
1,2
共x, t兲兴 are two com-
plex fields of amplitudes r
1,2
and phases c
1,2
, respec-
tively, ≠
2
x
A
1,2
stays for the second derivative of A
1,2
with
respect to the space variable 0 # x # L, L represents
the system size, dot denotes temporal derivative, the con-
trol parameters a
1,2
, b
1,2
are real numbers, and ´ is the
strength of the symmetric coupling. The boundary condi-
tions are chosen to be periodic.
For ´ 苷 0, Eq. (1) describes two uncoupled fields
A
1,2
each one obeying a separate CGL. This equation
has plane wave solutions A
q
苷
p
1 2 q
2
e
i共qx1vt兲
where
21 # q # 1, with q being the wave number in the
Fourier space, and v 苷 2b 2 共a2b兲q
2
. Such solu-
tions become unstable in the parameter region ab . 21
outside the range 2q
c
# q # q
c
共q
c
苷
q
11ab
2共11b
2
兲111ab
兲
through the so-called Eckhaus instability. Since q
c
vanishes as the product ab approaches 21, all plane
waves become unstable when crossing from below the
ab 苷 21 line in the parameter space, which is called
Benjamin-Feir or Newell line. Above this line, Ref. [13]
identifies three different turbulent states, namely, phase
turbulence (PT), amplitude or defect turbulence (AT), and
bichaos. We will specialize our analysis on PT and AT,
since they have received special attention in the scientific
community [14].
536 0031-9007兾99兾83(3)兾536(4)$15.00 © 1999 The American Physical Society
VOLUME
83, NUMBER 3 PHYSICAL REVIEW LETTERS 19J
ULY
1999
PT occurs just above the ab 苷 21 line, and is
characterized by a chaotic phase c, whereas the amplitude
r remains approximately constant. By further moving
away from the ab 苷 21 line, a transition is encountered
toward AT, wherein the amplitude dynamics becomes
dominant, leading to large amplitude oscillations which
can occasionally drive r to zero. The vanishing of r
causes the occurrence of a space-time defect.
In the following we will discuss the effect of ´fi0 in
Eq. (1). GS was shown to hold for two identical extended
fields in a particular dynamical situation [11], in which
two identical CGL’s (with a, b below the Benjamin-
Feir line) are coupled by an extra cubic term. Here
we deal with nonidentical systems (a
1
fia
2
, b
1
fib
2
),
and consider both the case of small and large parameter
mismatches. In the former case, the systems are prepared
in the same dynamical regime, e.g., both in PT or in AT.
In the latter case, the parameters a
1
, a
2
, b
1
, b
2
are chosen
so that one system is in the PT regime, while the other is
in the AT regime.
We first consider small parameter mismatches, and
select a
1
苷 a
2
苷 2.1, b
1
苷 21.25, and b
2
苷 21.2 in
Eq. (1) (both fields in AT). Figure 1 reports the space-time
plots of r
1
[Figs. 1(a), 1(c), 1(e)] and r
2
[Figs. 1(b), 1(d),
1(f)] for ´ 苷 0.05 [Figs. 1(a), 1(b)], ´ 苷 0.09 [Figs. 1(c),
1(d)], and ´ 苷 0.15 [Fig. 1(e), 1(f)]. The moduli are
coded into a 256 gray levels scale. The dark lines trace
the positions of the space-time defects. The simulations
have been performed with L 苷 64, periodic boundary
conditions, random initial conditions. The numerical code
is based on a semi-implicit scheme in time with finite
differences in space. The precision of the code is first order
in time and second order in space. A space discretization
dx 苷 0.125 (512 mesh points) and an integration time step
dt 苷 0.001 have been used. Figure 1 shows a gradual
passage from a nonsynchronized AT state [Figs. 1(a), 1(b)]
to a completely synchronized AT state [Figs. 1(e), 1(f)],
through an intermediate state [Figs. 1(c), 1(d)] wherein
only partial synchronization is built. Notice that here
the synchronization of the global structure implies the
synchronization of each localized space-time defect [as it
appears evident from Figs. 1(e), 1(f)].
At variance with what happens in concentrated systems,
the transition from no synchronization to CS seems here
not associated with the presence of an intermediate PS
regime. This feature is confirmed by the measurements
of Dr 苷 具j r
1
2r
2
j典 and Dc 苷 具j c
1
2c
2
j典 as func-
tions of ´. Here 具···典 stays for an averaging both in time
and space. As it can be seen in Fig. 2, 具Dr典 and 具Dc典
both show a smooth decreasing behavior as functions of
´, thus meaning that amplitude and phase synchronization
processes occur at the same time. The scenario is there-
fore consistent with what is already observed for small
parameter mismatches in chemical models [10].
The same qualitative features occur when the parame-
ters are selected so as both fields are in PT. Here again,
FIG. 1. AT-AT case: Space- (horizontal) time (vertical) plots
of the moduli r
1
[(a), (c), (e)] and r
2
[(b), (d), (f)].
a
1
苷 a
2
苷 2.1, b
1
苷 21.25, and b
2
苷 21.2. Time increases
downwards from 300 to 600 (u.t.). The first 300 time units
(not plotted) correspond to the transient before the system
reaches two independent chaotic (AT) states starting from two
independent random initial conditions (the same remarks hold
for Fig. 3). (a) and (b) correspond to ´ 苷 0.05, (c) and (d) to
´ 苷 0.09, and (e) and (f) to ´ 苷 0.15.
the system passes from an unsynchronized PT state at
small coupling to a completely synchronized PT state. In
this case, since defects are not present, the synchroniza-
tion takes place at smaller coupling strengths.
A more interesting scenario emerges in the case of
large parameter mismatches. Let us select in Eq. (1)
a
1
苷 a
2
苷 2.1, b
1
苷 21.2, and b
2
苷 20.83, so as the
field A
1
is evolving in AT, while the field A
2
is evolv-
ing in PT. In Fig. 3 we show the space-time representa-
tion of the patterns of r
1
[Figs. 3(a), 3(c), 3(e)] and r
2
[Figs. 3(b), 3(d), 3(f)] for ´ 苷 0.09 [Figs. 3(a), 3(b)],
´ 苷 0.14 [Figs. 3(c), 3(d)], and ´ 苷 0.19 [Figs. 3(e),
3(f)]. For small coupling strengths, the two systems do
not synchronize, and they hold in their respective regimes
[Figs. 3(a), 3(b)]. At large coupling strengths, the two
systems reach a CS regime, which is realized in PT
[Figs. 3(e), 3(f)]. This means that the final synchronized
state is space-time chaotic, but the complete synchroniza-
tion process is here associated with the suppression of
all defects, which were initially present in the field A
1
.
In other words, since CS implies amplitude synchroniza-
tion, the small amplitude oscillations of A
2
attract the
synchronized set, thus suppressing the defects originally
existing in the dynamics of A
1
. However, the most inter-
esting regime is the intermediate regime [Figs. 3(c), 3(f)],
wherein the two systems partially synchronize, and they
both recover an AT regime. Figure 4 reports the plots of
537
VOLUME
83, NUMBER 3 PHYSICAL REVIEW LETTERS 19J
ULY
1999
FIG. 2. AT-AT case: 具Dr典, 具Dc典 vs ´ (see definitions in the
text). Same parameters as in the caption of Fig. 1. The left
(right) vertical axis reports the 具Dr典 (具Dc典) scale. Notice
that 具Dr典 never vanishes completely, due to the parameter
difference in the evolution equations for the two fields.
具Dr典 and 具Dc典 as functions of ´. At variance with Fig. 2,
here a quite wide range of ´ exists (0.1 #´#0.16),
wherein amplitude synchronization is not yet reached,
while the global phase distance converges to a constant
value, thus meaning the emergence of an intermediate PS
state. In this state, the amplitudes of the two fields are
still uncorrelated, while the phases are strongly coupled.
Now, since the natural evolution of A
1
is in AT,
it shows the presence of many space-time defects. As
discussed above, defects are localized points resulting by
the vanishing of r. Therefore, in each one of them, the
phase c
1
shows a singularity. One can imagine that AT
FIG. 3. AT-PT case: Space- (horizontal) time (vertical) plots
of the moduli r
1
[(a), (c), (e)] and r
2
[(b), (d), (f)].
a
1
苷 a
2
苷 2.1, b
1
苷 21.2, and b
2
苷 20.83. (a) and (b)
correspond to ´ 苷 0.09, (c) and (d) to ´ 苷 0.14, and (e) and
(f) to ´ 苷 0.19.
allows flexibility in the dynamics of the amplitude, but
the variations of the phase are not flexible, and they are
substantially determined by the local amplitude variations.
On the contrary, A
2
would naturally evolve in PT, that is
with a dominant phase dynamics. The phase c
2
is not
naturally bounded, and its oscillations are allowed by the
evolution of the system in the uncoupled case. In
the range 0.1 #´#0.16, a PS state is built. There,
the phases c
1
and c
2
must converge (apart from a
constant). This is possible only when the phase c
2
locally
adjusts on c
1
. The relevant consequence of this process
is the introduction of many phase defects in the field A
2
[Fig. 3d], which would be instead free of them in the
uncoupled state.
The above qualitative picture is quantitatively con-
firmed by the measurement of the total number of phase
defects N
d
as a function of ´ for a
1
苷 a
2
苷 2.1, b
1
苷
21.2, and b
2
苷 20.83. Figure 5 reports N
d
vs ´ for A
1
(circles) and A
2
(squares). For small coupling strength
(0 ,´,0.1) the two fields evolve in an unsynchro-
nized manner [see Figs. 3(a), 3(b)]. At intermediate ´
values (0.1 ,´,0.16) there is a process of defect in-
jection into the field A
2
up to the point (´ ⯝ 0.16)
where both fields show the same defect number. No-
tice that this ´ range coincides exactly with the phase
plateau in Fig. 4. The spectral analysis of the PS state
reveals that, while the mean wave number is vanishing
for both fields independently on ´, the mean frequen-
cies are always bounded away from zero and show an
interesting dynamics as functions of ´. Precisely, while
A
1
appears to be robust in its frequency variations, A
2
shows large frequency variations as a function of ´,
thus confirming our heuristic argument about the flexi-
bility of A
2
during the synchronization process. A de-
tailed analysis of this synchronization state will appear
elsewhere.
When all defects have been synchronized, then the
system begins to reach a CS state, which is realized in
FIG. 4. AT-PT case: Indicators of modulus (circles) and
phase (squares) synchronization (same stipulations as in Fig. 2).
Same parameters as in the caption of Fig. 3. Note the phase
plateau for 0.1 #´#0.16.
538
VOLUME
83, NUMBER 3 PHYSICAL REVIEW LETTERS 19J
ULY
1999
FIG. 5. AT-PT case: Total number of phase defects N
d
as a
function of the coupling strength ´ for the fields A
1
(circles)
and A
2
(squares). Same parameters as in the caption of Fig. 3.
PT [see Figs. 3(e), 3(f)], implying the absence of phase
defects in both fields.
All the above scenarios are generally observed in Eq. (1)
regardless of the particular choices of a
1
, a
2
, b
1
, b
2
for
suitable values of the coupling parameter ´.
In conclusion we have shown how CS and PS are built
in nonidentical space-extended systems. In the case of
small parameter mismatches, one observes a passage from
nonsynchronized to a completely synchronized state. For
large parameter mismatches, this transition is mediated
by phase synchronization. While in the former case, the
resulting space-time synchronized state is not qualitatively
different from the unsynchronized one, in the latter case,
the state of the system resulting from the synchronization
process may substantially differ from that present with no
coupling, and it is mainly dictated by the synchronization
process of the space-time defects.
We acknowledge J. Kurths and A. Pikovski for
useful discussions. This work was partly supported
by Integrated Action Italy-Spain HI97-30. S.B.
acknowledges financial support from EU Contract
No. ERBFMBICT983466. J. B. benefits from a EU
Network grant under Contract No. FMRXCT960010
“Nonlinear dynamics and statistical physics of spatially
extended systems.”
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539
![FIG. 3. AT-PT case: Space- (horizontal) time (vertical) plots of the moduli r1 [(a), (c), (e)] and r2 [(b), (d), (f )]. a1 a2 2.1, b1 21.2, and b2 20.83. (a) and (b) correspond to ´ 0.09, (c) and (d) to ´ 0.14, and (e) and (f ) to ´ 0.19.](/figures/fig-3-at-pt-case-space-horizontal-time-vertical-plots-of-the-2898band.webp)
![FIG. 1. AT-AT case: Space- (horizontal) time (vertical) plots of the moduli r1 [(a), (c), (e)] and r2 [(b), (d), (f )]. a1 a2 2.1, b1 21.25, and b2 21.2. Time increases downwards from 300 to 600 (u.t.). The first 300 time units (not plotted) correspond to the transient before the system reaches two independent chaotic (AT) states starting from two independent random initial conditions (the same remarks hold for Fig. 3). (a) and (b) correspond to ´ 0.05, (c) and (d) to ´ 0.09, and (e) and (f) to ´ 0.15.](/figures/fig-1-at-at-case-space-horizontal-time-vertical-plots-of-the-3gdhc59h.webp)
