Synchronization in Complex Networks
by Coupled Parametrically Excited Oscillatorswith Parameter Mismatch
Kosuke Oi, Yoko Uwate and Yoshifumi Nishio
Dept. of Electrical and Electronic Engineering, Tokushima University
2-1 Minami-Josanjima, Tokushima 770–8506, Japan
Email: {ooi,uwate,nishio}@ee.tokushima-u.ac.jp
pieces of the hubs, and we have only added small mismatch(dispersion) in whole.
In this study, in order to more particular investigation ofsynchronization in complex network, we focus on relationship between the structural metrics in subset of the nodes
including larger mismatch and synchronization in complex
network. In order to research this relationship, we investigate
synchronization in complex network by changing structuralmetrics (degree and path length) corresponding to the subset of
the nodes adding larger mismatch than other nodes. First, weinvestigate the synchronization probability in the network bychanging the degree of the three nodes added larger mismatch.
Next, we investigate the synchronization probability in thenetwork by changing the path length among hubs, three nodes
with larger mismatch.

Abstract—In this study, we investigate synchronization incomplex network with two hubs by using parametrically excited
van der Pol oscillators with parameter mismatch. By means ofcomputer simulation, we confirm various change of synchronization probability in complex network, and observe effects
on synchronization probability by changing structural metrics(degree and path length) in subset of the nodes including larger
mismatch in complex network.

I. I NTRODUCTION
Synchronization is one of the fundamental phenomena innature and it is observed over the various fields. Studies
on synchronization phenomena of coupled oscillators areextensively carried out in various fields, physics [1], biology [2], engineering and so on. Synchronization generated
in the system can model certain synchronization of natural
rhythm phenomena. Parametric excitation circuit is one ofresonant circuits, and it is important to investigate various
nonlinear phenomena of the parametric excitation circuits forfuture engineering applications. In simple oscillator including
parametric excitation, Ref. [3] reports that the almost periodic
oscillation occurs if nonlinear inductor has saturation characteristic. Additionally the occurrence of chaos is referenced inRefs. [4] and [5].
In our research group, we have investigated synchronizationof parametrically excited van der Pol oscillators [6]. By
carrying out computer calculations for two or three subcircuits
cases, we have confirmed that various kinds of synchronization phenomena of chaos are observed. In the case of twosubcircuits, the anti-phase synchronization is observed. In
the case of three subcircuits, self-switching phenomenon ofsynchronization states is observed.
However, we have investigated the only simple networkmodels. It is important to investigate more complex network
for the broad-ranging future engineering applications. In ourprevious study, we have challenged to investigate the synchronization and clustering in more complex network modified
from “Dolphin social network” [7] by using parametrically
excited van der Pol oscillators with small mismatch [8].
We have confirmed that the network with hubs can induce
synchronization. Though, we have only investigated the effects
on synchronization offered by the location and number of

978-1-5090-1570-2/16/$31.00 ©2016 IEEEII. P ROPOSED NETWORK MODEL
Topological structures in complex networks of N nodes andE edges can be evaluated by the typical structural metrics(degree, clustering coefficient and path length). First, degree isthe number of edges on a node. Second, clustering coefficientis the number of actual links between neighbors of a node
divided by the number of possible links between those neighbors. Third, path length (L) shows the average shortest pathin the network between two nodes. This is given as follows:
1 ∑lij .
N −1
N

Li =(1)i̸=j1 ∑
L=Li .
N i=1
N

(2)

Where Li is the shortest path in the network between node iand node j.
In this research, we consider coupled parametrically excitedoscillators complex network in Fig. 1. In this system, parametrically excited van der Pol oscillators are coupled by one
resistor R. In this network, the number of nodes which isparametrically excited van der Pol oscillator is 62 and thenumber of edges which is one resister is 100. The 15th and69

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51252

22

1925

1646

1530

24

n1351

60

34

41

39
1738

4557

48

258

354

62(a) Parametrically excited van der Pol oscillator.

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7620281018423359112950351 2140L

47533749v

C4449iid3656262714613223Fig. 1. Proposed network.

58th nodes are hubs in this network. The bond number of 58th
node is 11, and the bond number of 15th node is 13.
Table 1 shows the properties of the proposed network as
shown in Fig. 1.

(b) Time-varying inductor.

TABLE IP ROPERTIES OF PROPOSED NETWORK AS SHOWN IN F IG . 1.
Average degree
Average clustering coefficient
Average path lengthFig. 2. Circuit model.

3.226
0.1454.357

1+

L = L0 γ(τ ).

(3)

id = −g1 vk + g3 vk .

(4)

By changing the variables and the parameters,
√
√
g1



 t = L0 Cτ, vn = g3 xn




√



ω = ω0 L0 C√ √g1 C
in =yn
g3 L0




√
√



L1L0
 ε = g1, δ=CR C

978-1-5090-1570-2/16/$31.00 ©2016 IEEE( )III. S YSTEM MODELThe circuit model of van der Pol oscillator under parametric
excitation is shown in Fig. 2 (a). The circuit includes atime-varying inductor L whose characteristics are given as
the following equation. The time-varying inductor is shownas Fig. 2 (b). γ(τ ) is expressed in a rectangular wave asshown in Fig. 3, and its amplitude and angular frequency aretermed α and ω, respectively. By changing the value of α, theamplitude of parametric excitation can be changed. The v − icharacteristics of the nonlinear resistor are approximated by
the following equation.

11-0/

2 /Fig. 3. Function relating to parametrically excitation.

The normalized circuit equations are given by the followingequations.
∑dxn= ε(xn − x3n ) − yn − δ(xn − xk ) dτk∈Sn(6)dyn1
=xndτγ(τ )where n = 1, 2, 3, ..., 10. Sn is the set of nodes which are
directly connected to the node n.
IV. I NVESTIGATION OF SYNCHRONIZATION
A. Simulation method
In this research, we investigate the synchronization by using
computer simulation. We fix the the circuit parameters asε = 1.00 and ω = 1.00 for all circuits. Each circuit isgiven different initial values for computer simulations. In thissimulation, all of the nodes involve small mismatch mn inα (in Fig. 2) which is corresponding to the amplitude of the(5)70APCCAS 2016

function relating parametric excitation within the compass of−0.001 ≤ mn ≤ 0.001.
Additionally, in order to analyze synchronous state, wedefine the synchronization by the following equation:

51252

223656
1925

1646|xn − xk | < 0.10(k ∈ Sn ).

153024(7)

1351

60

3444

494147533937

We fix the count of calculation as 100,000.

173845

1 2149B. The results of simulation

4057

In these simulations, first, we fixed the coupling strengthδ into 1.70 for this network is synchronized as completely.
Next, we investigate change of the synchronization probability
by adding larger mismatch M from 0.002 to 0.01 in threenodes. These nodes have a certain characteristic corresponding
to the structural metrics in order to investigate the relationshipbetween the structural metrics in subset of the nodes including
larger mismatch M and synchronization in complex network.
1) Effect of the degree including larger mismatch: In thissimulation, we investigate change of synchronous state byadding larger mismatch M from 0.002 to 0.01 in 5 types node.
These nodes have the degrees which is 1 to 5. We add the M in
the amplitude of the function relating to parametric excitation(1 + α + mn + M ) in three nodes which have the 5 types
degrees (1 to 5). Figure 4 shows degree distribution of thisnetwork. In this figure, vertical axis denotes the number ofnodes, and horizontal axis denotes the value of degree.

354624331855

7620281018423348

25859
112950352627146132

23Fig. 5. The location of the 5-color nodes with larger mismatch M .

CountFig. 6. Synchronization probability.

On the other hand, when three nodes have 1, we can confirm
the synchronization probability maintains the lowest condition.
From this result, we can confirm degree of the node including larger mismatch affects the synchronization probability.
Additionally, the synchronization probability is affected fromnot only the size of degrees but also the location of the nodes
with larger mismatch M . Following this result, we consider
it is important to investigate about the effect of the locationof the nodes with larger mismatch M on the synchronization
probability in the network.
2) Effect of the path length among nodes which have M : In
this simulation, we investigate in what way the synchronization
probability in complex network draw influence from the path
length among hubs and three nodes with larger mismatch M .
We investigate this influence in 3 cases. In the case 1, weinvestigate the passage of the synchronization probability bychanging larger mismatch M and the path length between
hubs and each of three nodes with M . In the case 2, thehubs (15th and 58th node) are not connected to any of threenodes with M . Based on the acknowledgment, we investigate
the passage of the synchronization probability by changing
larger mismatch M and the path length among each of threenodes with M . In the case 3, the hubs (15th or 58th node) areconnected to any of three nodes with M . On that basis, weValueFig. 4. Degree distribution of proposed network.

Figure 5 shows the 5 types nodes with the value of mismatch
M . In this figure, the red nodes express the node of 1 degree,
the green nodes nodes express the node of 2 degrees, the purple
nodes express the node of 3 degrees, the watery nodes express
the node 4 degrees, and the blue nodes express the nodeof 5 degrees. Figure 6 shows the change of synchronization
probability when we add the mismatch as Fig. 5. The colors ofeach line is corresponding to the colors of each node in Fig. 5.
In Fig. 6, the vertical axis denotes synchronization probability,
and the horizontal axis denotes the value of mismatch (M ). 5
colors lines express degree-by-degree passage of the synchronization probability.
When three nodes have 4 degrees, we can confirm thesynchronization probability maintains the highest condition.

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investigate the passage of the synchronization probability bychanging larger mismatch M and the path length among eachof three nodes with M .
First of all, in the case 1, synchronization probabilitychanges as shown in Fig. 7. Three lines are corresponding tothe value of the path length from the hubs to the three nodes
with M . From this result, the synchronization probability ofeach line increase with decreasing the path length from thehubs to the three nodes with M .
Next, in the case 2, synchronization probability changes asshown in Fig. 8. Four lines are corresponding to the valueof the path length among each of three nodes with M . In
this figure, the passage of the synchronization probability
does not have that large a change, however we can confirm
the synchronization probability maintains the lowest condition
when the path length among each of three nodes with M is1.
Finally, in the case 3, synchronization probability changes
as shown in Fig. 9. Four lines are corresponding to the valueof the path length among each of three nodes with M . As
compared with the result of Fig. 8, the difference of eachline broaden out. Following this result, we can confirm thehubs exercise a much further effect in the synchronization
probability in the network.

Fig. 9. Synchronization probability (case 3).

between the structural metrics in subset of the nodes including
larger mismatch and synchronization in complex network. Inorder to research this relationship, we investigate synchronization in complex network by changing structural metrics(degree and path length) of the nodes corresponding to thesubset of the nodes adding larger mismatch than other nodes.
First, we investigate the synchronization probability in the
network by changing the degree of the three nodes added larger
mismatch. Next, we investigate the synchronization probability
in the network by changing the path length among hubs, threenodes with larger mismatch.
We could observe the various change of the synchronization
probability by changing structural metrics (degree and pathlength) of the nodes corresponding to the subset of the nodes
adding larger mismatch than other nodes.
R EFERENCES[1] I. Belykh, M. Hasler, M. Lauret and H. Nijmeijer, “Synchronization andgraph topology,” Int. J. Bifurcation and Chaos, vol.15, no.11, pp.34233433, Nov. 2005.
[2] J. Cosp, J. Madrenas, E. Alarcon, E. Vidal and G. Villar, “Synchronizationof nonlinear electronic oscillators for neural computation,” IEEE Trans.
Neural Networks, vol.15, no.5, pp.1315-1327, Sep. 2004.

Fig. 7. Synchronization probability (case 1).

[3] C. Hayashi, “Nonlinear Oscillations in Physical Systems,” Chap. 11,
McGraw-Hill, New York (1964).
[4] C. Hayashi, M. Abe, K. Oshima and H. Kawakami, “The method ofmapping as applied to the solution for certain types of nonlinear differential equations,” Ninth International Conference on Nonlinear Oscillations,
Kiev (Aug.-Sept.1981).
[5] M. Inoue, “A Method of Analysis for the Bifurcation of the Almost Periodic Oscillation and the Generation of Chaos in a Parametric Excitation
Circuit,” Trans. of IEICE, vol. J68-A, no. 7, pp. 621-626, 1985.
[6] H. Kumeno, Y. Nishio, “Synchronization Phenomena in Coupled Parametrically Excited van der Pol Oscillators,” Proc. NOLTA’08, pp. 128-131,
Sep. 2008.
[7] D. Lusseau, K. Schneider, O. J. Boisseau, P. Haase, E. Slooten and S. M.
Dawson, “The bottlenose dolphin community of Doubtful Sound featuresa large population of long-lasting associations. Can geographic isolation
explain this unique trait?”, Proc. Behavioral Ecology and Sociobiology,
vol.54, pp. 396-405, 2003

Fig. 8. Synchronization probability (case 2).

[8] K. Oi, K. Ago, Y. Uwate, Y. Nishio, “Synchronization in Complex Networks with Dispersion by Coupled Parametrically Excited Oscillators”,
Proc. NCSP’15, pp. 33-36, March, 2016.

V. C ONCLUSIONSIn this study, in order to more particular investigation ofsynchronization in complex network, we focus on relationship978-1-5090-1570-2/16/$31.00 ©2016 IEEE72

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