PHYSICAL REVIEW E 78, 017203 共2008兲
Experimental observation of ragged synchronizability
P. Perlikowski, B. Jagiello, A. Stefanski, and T. Kapitaniak
Division of Dynamics, Technical University of Lodz, Stefanowskiego 1/15, 90-924 Lodz, Poland
共Received 29 January 2008; revised manuscript received 12 May 2008; published 31 July 2008兲
Synchronization thresholds of an array of nondiagonally coupled oscillators are investigated. We present
experimental results which show the existence of ragged synchronizability, i.e., the existence of multiple
disconnected synchronization regions in the coupling parameter space. This phenomenon has been observed in
an electronic implementation of an array of nondiagonally coupled van der Pol’s oscillators. Numerical simulations show good agreement with the experimental observations.
DOI: 10.1103/PhysRevE.78.017203
PACS number共s兲: 05.45.Xt, 05.45.Pq, 07.50.Ek
Synchronization 关1兴 has become a well-known and widely
used feature in driven periodic and chaotic oscillators. Over
the last decade the subject of network synchronization has
attracted increasing attention from different fields 关1–7兴.
Synchronization thresholds and their dependence on various
structural parameters of the network, such as the type and
strength of coupling 关2兴 are of particular interest. The introduction of the master stability function 共MSF兲 关3兴 allowed
for the establishment of a number of important results 关4,5兴.
In our previous work 关6兴, we presented an example of
nondiagonally coupled array of Duffing oscillators, in which
multiple disconnected synchronous regions of coupling
strength occur. The term nondiagonal coupling means that
the network nodes are linked with others via nondiagonal
components of linking 共output兲 function 关see Eq. 共5兲兴. We
have also observed the appearance or disappearance of such
synchronous windows in coupling parameter space, when the
number of oscillators in the array or topology of connections
between them changes. This phenomenon has been called the
ragged synchronizability 共RSA兲. The existence of RSA has
been numerically confirmed in 关7兴.
In this Brief Report we give the experimental evidence of
the existence of RSA. We consider the dynamics of an array
of coupled van der Pol’s 共VdP兲 oscillators which has been
implemented as an electronic circuit. Our numerical studies
are supported by a simple electronic experiment. Our experimental results are in satisfactory agreement with numerical
simulations.
In our experimental and numerical studies, the VdP oscillator
ż2 = d共1 − x22兲z2 − x2 + cos共兲 + 共2x1 + x3 − 3x2兲,
共2d兲
ẋ3 = z3 ,
共2e兲
ż3 = d共1 − x23兲z3 − x3 + cos共兲 + 共x2 − x3兲,
共2f兲
where is a constant coupling coefficient.
In numerical analysis we assume d = 0.401, = 1.207, and
consider as a control parameter. In experiments we use an
electronic implementation of this array shown in Fig. 2.
The dynamics of the considered array can be described in
a block form,
ẋ = F共x兲 + 共G 丢 H兲x,
where x = 共x1 , x2 , x3兲 苸 R6, F共x兲 = 关f共x1兲 , f共x2兲 , f共x3兲兴, G is the
connectivity matrix, i.e., the Laplacian matrix representing
the topology of connections between the network nodes,
G=
冤
−2
2
2
−3
1
0
1
−1
H=
ż = d共1 − x 兲z − x + cos共兲,
共1兲
where d and are constant, has been taken as an array node.
represents the frequency of the external excitation. Consider an open array of three coupled VdP oscillators shown
in Fig. 1. The evolution of oscillators coupled in this array is
given by
ẋ1 = z1 ,
共2a兲
ż1 = d共1 − x21兲z1 − x1 + cos共兲 + 共2x2 − 2x1兲,
共2b兲
ẋ2 = z2 ,
共2c兲
1539-3755/2008/78共1兲/017203共4兲
0
冥
,
共4兲
丢 is a direct 共Kronecker兲 product of two matrices and
H : R2 → R2 is an output function of each oscillator’s variables that is used in the coupling 共it is the same for all
nodes兲. The connection of oscillators shown in Eqs. 共2a兲–共2f兲
can be classified as a case of pure 共diagonal components are
equal to zero兲 nondiagonal coupling due to the form of output function
ẋ = z,
2
共3兲
冋 册
0 0
1 0
.
共5兲
Here, a subject of our interest are the ranges of the coupling
coefficient where the so-called complete synchronization,
i.e., full coincidence of phases 共frequencies兲 and amplitudes
of responses of coupled systems 关1兴, occurs. The complete
synchronization requires an ideal identity of these systems,
FIG. 1. The model of an open array of van der Pol’s oscillators
共VdPO兲.
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©2008 The American Physical Society
PHYSICAL REVIEW E 78, 017203 共2008兲
BRIEF REPORTS
FIG. 2. An electronic implementation of an open array of VdP oscillators shown in Fig. 1.
i.e., they are given with the same ordinary differential equations with identical system’s parameters. In order to estimate
the synchronization thresholds of coupling parameter, we apply the idea of the MSF 关3兴. Under this approach, the synchronizability of a network of oscillators can be quantified
with eigenvalues ␥k of connectivity matrix G, k = 0 , 1 , 2. In
the case under consideration, matrix G has three real eigenvalues ␥0 = 0, ␥1 = −1.27, ␥2 = −4.73, so this is a variant of
diffusive real coupling 关4兴 共in the general case ␥k can be a
complex number兲. After the block diagonalization of the
variational equation of Eq. 共3兲 there appear three separated
blocks ˙ k = 关Df + ␥kDH兴k, where Df and DH are Jacobi
matrices of the node system and linking function, respectively 共k = 1 , 2 , 3兲. For ␥0 = 0 we have linearized the equation
of the node system 关Eq. 共1兲兴 which is corresponding to the
mode longitudinal to invariant synchronization manifold x1
= x2 = x3. The remaining two eigenvalues ␥1,2 represent two
different transverse modes of perturbation from synchronous
state 关3,4兴.
Assuming that ␥ represents an arbitrary value of ␥k and
symbolizes an arbitrary transverse mode, we can define the
generic variational equation for any node system
˙ = 关Df + ␥DH兴 .
共6兲
Substituting the analyzed system 关Eq. 共1兲兴 in Eq. 共6兲 we
obtain
˙ = ,
˙ = d共1 − x2兲 − 2dxz − + ␥ .
共7兲
Thus, the generic variational equation 关Eq. 共7兲兴 describes
an evolution of any perturbation in the directions transversal
to the final synchronous state, that dynamics is governed by
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PHYSICAL REVIEW E 78, 017203 共2008兲
BRIEF REPORTS
FIG. 3. The largest transversal Lyapunov exponent T, calculated for generic variational equation 共7兲, versus product ␥;
= 1.207, d = 0.401.
Eq. 共1兲. Now, we can define the MSF for the considered case
as a bifurcational diagram of the largest transversal
Lyapunov exponent T, calculated for the generic variational
equation 关Eq. 共7兲兴, versus product ␥. The MSF graph for
the presented example 关Eqs. 共1兲 and 共7兲兴 is depicted in Fig. 3.
If products ␥1,2 corresponding to both transversal eigenmodes 共in the general case any number of them兲 can be
found in the ranges of negative transversal Lyapunov exponent, then the synchronous state is stable for the analyzed
configuration of couplings.
In order to confirm our numerical simulations experimentally we have built a setup which is schematically depicted in
Fig. 2. Each VdP oscillator has been implemented as the
circuit 关8兴 共shown in black frame in Fig. 2兲 composed of two
capacitors C1 and C2, seven resistors R共1 − 7兲, and two multiplicators AD-633JN which introduce nonlinearity. Multiplicators have the following characteristic: W = 共1 / Vc兲共X1
− X2兲共Y 1 − Y 2兲 + Z, where X1, X2, Y 1, and Y 2 are the input
signals, W is an output signal, and Vc = 10 V is a characteristic voltage. The input Em cos t, where amplitude Em and
frequency are constant, represents external excitation. The
additional resistors R8 and R have been used to realize the
coupling. In our implementation we used out of shelf elements: R1 = 9920关⍀兴, R2 = 999关⍀兴, R3 = 501关⍀兴, R4
= 100关⍀兴, R5 = 10 000关⍀兴, R6 = 10 000关⍀兴, R7 = 16 150关⍀兴,
R = 180 000关⍀兴,
C1 = 10关nF兴,
C2 = 10关nF兴.
R8
苸 关0⍀ , 44 000⍀兴 has been taken as a control parameter. The
equivalent elements in each circuit can differ by 1% of their
nominal values.
The relation between the circuits real parameters and dimensionless parameters of Eqs. 共2a兲–共2f兲 is as follows: 20
1
1
3
= 0 , x1−3 = V1−3 ERmR2
, d = C1R1
, z1−3
= C1C2R2R7
0 ,
= ␦V1−3 EmRR23 0 ,
R2
= R8
.
Nonidentity of elements used in each
circuit introduces the mismatches of d and parameters in
Eqs. 共2b兲, 共2d兲, and 共2f兲. The estimated mismatches are
smaller than ⫾0.001.
Data acquisition is performed using a Data Acquisition
System 3200A \ 415 board connected to a computer controlled by software developed in Microstar Labs. The dynamical variables of interest in this circuit are the voltages
V1−3 of each oscillator measured in the points indicated in
Fig. 2. The first derivatives of the potentials ␦V1−3 are taken
in the point also indicated in Fig. 2.
FIG. 4. Comparison of numerical and experimental results, d
= 0.401, = 1.207; 共a兲–共c兲 desynchronizing mechanism: A projection from the MSF diagram 共a兲, via eigenvalues ␥k of connectivity
matrix G 共b兲, to the bifurcation diagram of synchronization error e
versus coupling coefficient 共c兲 共desynchronous intervals are
shown in gray, complete synchronization takes place in ranges
where e approaches zero value兲, 共d兲 synchronization error e versus
; experimental results, gray line with scatters 共marking measurement points兲, numerical results for the case of parameter mismatch;
black line, the values of d taken in Eqs. 共2b兲, 共2d兲, and 共2f兲 are,
respectively, 0.400, 0.401, and 0.402.
In our example we have considered = 1.207 and its real
equivalent = 4780 Hz in experiment. Then, in the absence
of coupling each oscillator exhibits periodic behavior 共the
largest LE 1 = −0.126兲 with the period equal to the period of
excitation. Looking at the MSF diagram 共Fig. 3兲, we can
expect an appearance of the RSA of coupled VdP oscillators
关Eq. 共1兲兴, because two disconnected regions of negative
transversal Lyapunov exponent 关␥ 苸 共0 , 1 − 兲 or 共1 + , ⬁兲兴
can be observed. Consequently, at least two separated synchronous ranges of coupling parameter should be visible,
i.e., the RSA effect takes place. However, synchronous intervals of the coefficient not always are an exact reflection of
the MSF intervals, where the transversal Lyapunov exponent
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PHYSICAL REVIEW E 78, 017203 共2008兲
BRIEF REPORTS
is positive. The RSA mechanism is explained in Figs.
4共a兲–4共c兲, where a projection from the MSF diagram 关Fig.
4共a兲兴, via eigenvalues of connectivity matrix G 关Fig. 4共b兲兴, to
the bifurcation diagram of synchronization error
3
e = 兺 兩共x1 − xi兲兩,
共8兲
i=2
versus coupling strength 关Fig. 4共c兲兴 is shown. Complete
synchronization takes place in the ranges where e approaches zero value 关Fig. 4共c兲兴. We can observe third syn2
1
2
2
, 1−
兲 and second desynchronous 共1−
, 1+
兲
chronous 共1+
intervals in comparison with only two synchronous and one
desynchronous MSF ranges, respectively. ‘‘Additional’’ desynchronous interval appears because the mode 2 共associated
with eigenvalue ␥2兲 crosses the desynchronous MSF interval
共1 − , 1 + 兲 while the mode 1 共associated with ␥1兲 is still located in the first synchronous MSF interval 共0 , 1 − 兲, see Fig.
2
1
4共b兲. Then in the narrow range 共1+
, 1−
兲, two modes are in
synchronous MSF interval so that one can observe an “additional window” of synchronization in the interval. The
second desynchronous interval corresponds to mode 1 desynchronizing bifurcation. Finally, the steady synchronous
state is achieved due to increasing coupling strength at
= 0.6. The ragged synchronizability manifests in alternately
appearing windows of synchronization and desynchronization. In the above description some special notation for
ranges has been introduced. Such a notation brings information of which mode 共first or second兲 desynchronizing bifurcation 共superscripts兲 takes place during the transition from
the synchronous to the desynchronous regime and which
edge of the desynchronous interval of MSF 共1− and 1+ in
subscripts correspond to lower and higher edges, respectively兲 is associated with the given boundary value of the
coupling coefficient. In Fig. 4共d兲 the results of experimental
investigation of the synchronization process in the analyzed
circuit are demonstrated. The plot of experimentally generated synchronization error 关reduced to nondimensional form
and calculated with the use of Eq. 共8兲兴 versus coupling
strength is shown in gray with scatters. Obviously, in the
case of real VdP oscillators the perfect complete synchroni-
关1兴 H. Fujisaka and T. Yamada, Prog. Theor. Phys. 69, 32 共1983兲;
V. S. Afraimovich, N. N. Verichev, and M. Rabinovich, Radiofiz. 28, 1050 共1985兲; I. Blekhman, Synchronization in Science and Technology 共ASME Press, New York, 1988兲; L. M.
Pecora and T. L. Carroll, Phys. Rev. Lett. 64, 821 共1990兲; A.
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共1990兲.
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共1983兲; A. Pikovsky, Z. Phys. B: Condens. Matter 55, 149
共1984兲; A. Stefanski, J. Wojewoda, T. Kapitaniak, and S. Yanchuk, Phys. Rev. E 70, 026217 共2004兲; S. Dmitriev, M. Shirokov, and S. O. Starkov, IEEE Trans. Circuits Syst., I: Fundam. Theory Appl. 44, 918 共1997兲.
zation cannot be achieved due to unavoidable parameter mismatch. However, in such a case the imperfect complete synchronization can be observed, i.e., the correlation of
amplitudes and phases of the system’s responses is not ideal,
but a synchronization error remains relatively small during
the time evolution. One can notice a qualitative coincidence
between numerical simulations and experiment comparing
Fig. 4共c兲 with Fig. 4共d兲, i.e., regions of the imperfect complete synchronization tendency in a real circuit correspond
well to the complete synchronization ranges in a numerical
model. In the last stage of our research the influence of parameters mismatch on the synchronization error e has been
analyzed numerically. We have estimated a slight disparity of
the values of d in all three VdP oscillators with measuring
their real parameters. Next, such an approximated mismatch
has been realized in the considered model 关Eqs. 共2a兲–共2f兲兴
关the values of d taken in Eqs. 共2b兲, 共2d兲, and 共2f兲 are, respectively, 0.400, 0.401, and 0.402兴. The synchronization error
simulated numerically for this model is represented with a
black line in Fig. 4共d兲. Its good visible agreement with experimental result shows that a slight difference of coupled
oscillators does not destroy their synchronization tendency,
i.e., the imperfect complete synchronization takes place.
To summarize, comparing analytical 共obtained by MSF
approach兲 关Fig. 4共b兲兴, numerical 关Fig. 4共c兲兴, and experimental results 关Fig. 4共d兲兴 we can confirm the occurrence of the
RSA in the real system of coupled oscillators. We have observed this phenomenon in an electronic implementation of
an array of VdP oscillators with nondiagonal coupling between the nodes. A good agreement between numerical simulation and experimental observations which shows that the
mechanism responsible for the appearance or disappearance
of the windows of synchronizability is the same as described
in 关6兴. It seems that the phenomenon of RSA is common for
the systems with nondiagonal coupling and not sensitive for
the small parameter mismatch, i.e., can be observed in real
experimental systems.
This work has been supported by the Polish Department
for Scientific Research 共DBN兲 under Contracts No. 4 T07A
044 28 and No. 0710/B/TO2/2007/33.
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017203-4
Tomasz Kapitaniak