IEEE Workshop on Nonlinear Circuit Networks
December 9-10, 2016

Influence of Regional Change in Synchronizationof Complex Networks in Coupled
Parametrically Excited Oscillators
Kosuke Oi

Yoko Uwate

Yoshifumi Nishio

Dept. Electrical and Electronic Eng.
Tokushima University
Email: ooi@ee.tokushima-u.ac.jp

Dept. Electrical and Electronic Eng.
Tokushima University
Email: uwate@ee.tokushima-u.ac.jp

Dept. Electrical and Electronic Eng.
Tokushima University
Email: nishio@ee.tokushima-u.ac.jp

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 a subset of the nodes with larger
mismatch in complex networks.

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 for
future 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 two
subcircuits, 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 ofpieces of the hubs, and we have only added small mismatch(dispersion) in whole.
In this study, in order to investigate the influence of regionalchange in synchronization of complex network, we focus on
relationship between the structural metrics in a subset of thenodes is included larger mismatch and synchronization incomplex network. In order to research this relationship, weinvestigate synchronization in complex network by changing
structural metrics (degree and path length) corresponding to a
subset of the nodes is added 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, anumber of nodes with larger mismatch1 .
II. P ROPOSED NETWORK MODEL
Topological structures in complex networks can be evaluated by the typical structural metrics (degree, clustering
coefficient and path length). First, degree is the number ofedges on a node. Second, clustering coefficient is the numberof actual links between neighbors of a node divided by the
number of possible links between those neighbors. Third, path
length is the shortest path in the network between two nodes.
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 and58th nodes are hubs in this network. The bond number of 58th
node is 12, and the bond number of 15th node is 13.Table 1
shows the properties of the proposed network as shown in
Fig. 1.
1 This research is obtained by adding more particular investigation toward
path length in the paper is accepted in IEEE Asia Pacific Conference on
Circuits and Systems (APCCAS’16)- 75 -

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2 /

Fig. 3.

Function relating to parametric excitation.

Proposed network.

By changing the variables and the parameters,

√
√
√
g1


L
Cτ,
v
=
xn , ω = ω0 L0 C
t
=

0
n


g3
√
√
√
√


g1 C
L0
1 L


yn , ϵ = g1
, δ=,
 in =g3 L0
C
R C

TABLE IP ROPERTIES OF PROPOSED NETWORK AS SHOWN IN F IG . 1.
Average degree
Average clustering coefficient
Average path length

3.226
0.145
4.357

III. S YSTEM MODEL
The circuit model of van der Pol oscillator under parametric
excitation is shown in Fig. 2 (a). The circuit includes a timevarying inductor L whose characteristics are given as thefollowing equation.
L = L0 γ(τ ).

(1)

The chaotic oscillation is expressed in Fig. 2 (b). γ(τ ) isexpressed in a rectangular wave as shown in Fig. 3, andits amplitude and angular frequency are termed α and ω,
respectively. By changing the value of α, the amplitude ofparametric excitation can be changed.
Y1
i

id
C

/

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Fig. 1.

n

0
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v

L

X1
(a) Parametrically excited van der Pol oscillator.
Fig. 2.

(b) Chaotic oscillation (α = 0.950).

Circuit model.

The v − i characteristics of the nonlinear resistor are approximated by the following equation.

id = −g1 vk + g3 vk .

(2)

(3)

the normalized circuit equations are given by the following
equations.

∑
dxn


(xn − xk )= ε(xn − x3n ) − yn − δ
 dτ
k∈Sn
(4)

dyn
1


=
xn ,
dτ
γ(τ )
where n = 1, 2, 3, ..., 62. Sn is the set of nodes which are
directly connected to the node n.
IV. I NVESTIGATION OF THE SYNCHRONIZATIONPROBABILITY

A. Synchronization probability
In this research, we investigate the synchronization bymeans of computer simulation. We calculate xi by using
Runge-Kutta method. Simulation time is τ = 4000π with astep size 0.002π. We fix the circuit parameters as ε = 1.00
and ω = 1.00 for all circuits. Each circuit is given different
initial values for computer simulations. In this simulation, allof the nodes involve small mismatch mn in α (in Fig. 2) whichis corresponding to the amplitude of the function relating toparametric excitation within the compass of −0.001 ≤ mn ≤0.001.
Additionally, in order to analyze synchronous state, we
define the synchronization by the following equation.
|xn − xk | < 0.10

(k ∈ Sn ).

(5)

The synchronization probability is the rate of the synchronized edges among the number of all of the edges (100). Whenthe two oscillators (nodes) at the ends of an edge become
synchronous state, we define this edge as synchronized edge.

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CountB. The results of simulation
In these simulations, first, we fixed the coupling strength δ
in 1.70 for this network is synchronized as completely. Next,
we investigate change of the synchronization probability byadding larger mismatch M in a number of nodes. These nodes
have a certain characteristic corresponding to the structural
metrics. In order to investigate the relationship between the
structural metrics in a subset of the nodes is included larger
mismatch M and synchronization in complex network, we
simulate the passage of the synchronization by adding andchanging M from 0.002 to 0,01. This result can be shown in
Fig. 5 : Fig. 9.
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 with the 5 types degrees(1 to 5). Figure 4 shows degree distribution of the proposed
network. In this figure, vertical axis denotes the number ofnodes, and horizontal axis denotes the value of degree.

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.
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.
Furthermore, 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 from hubs to a number ofnodes with M : In this section, we investigate in what way
the synchronization probability in complex network draw
influence from the path length from hubs to a number ofnodes with larger mismatch M . The value of path length (1 :3) expresses the shortest path from the hubs to a number ofnodes with M . In this simulation, we investigate the passage
of synchronization probability by changing the value of path
length (1 : 3) from hubs to a number of nodes with M , andadding M from 0.002 to 0.01 in a number of nodes.

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Degree distribution of the proposed network.

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Fig. 4.

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(a) Synchronization probability.

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Fig. 5.

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(a) Synchronization probability.

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Simulation result (M : 3 nodes).

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Fig. 6.

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(b) The location of thenodes with M .

23(b) The location of thenodes with M .

Simulation result.

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

In Fig. 6, we add M in three nodes. Three lines arecorresponding to the value of the path length from the hubs
to the three nodes with M , and the colors of each line iscorresponding to the colors of each node. As a result, we canconfirm the synchronization probability of each line increase
with decreasing the value of the path length from the hubs tothe three nodes with M . From this result, we can consider hubs
are dispersing the effect of larger mismatch, and performing
a function as maintaining the synchronous state. In orderto make generalization this consideration, we investigate thesynchronization probability by increasing the nodes with M .
Next, synchronization probability changes as shown in
Fig. 7 by adding M to four nodes. In this result, relative
merits between path length : 2 and path length : 3 change from- 77 -

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(b) The location of thenodes with M .

(a) Synchronization Probability.
Fig. 7.

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Simulation result (M : 4 nodes).

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(b) The location of thenodes with M .

(a) Synchronization Probability.
Fig. 8.

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M = 0.006. However, synchronization probability maintains
the highest condition when path length is 1. Similarly, byincreasing the number of nodes with M from 5 to 6, changesof the synchronization probability are obtained as shown in
Fig. 8 and Fig. 9.
From these results, we can confirm the influence of thenodes with M can be dispersed by hubs when the value ofthe path length from hubs to a number of nodes with larger
mismatch M is 1.
V. C ONCLUSION
In this study, in order to investigate the influence of regionalchange in synchronization of complex network, we focus on
the relationship between the structural metrics in a subset of
the nodes is included larger mismatch and synchronization incomplex network. In order to research this relationship, weinvestigate synchronization in complex network by changing
structural metrics (degree and path length) of the nodes corresponding to the subset 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 adding
larger mismatch. Next, we investigate the synchronization
probability in the network by changing the path length fromhubs to a number of nodes with larger mismatch M .
We could observe the various change of the synchronization
probability. Furthermore, we could consider the hub disperse
the influence of larger mismatch, and perform a function as
maintaining the synchronous state.
ACKNOWLEDGMENT
This work was partly supported by JSPS Grant-in-Aid for
Scientific Research 16K06357.
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Simulation result (M : 5 nodes).

[2] J. Cosp, J. Madrenas, E. Alarcon, E. Vidal and G. Villar, “Synchronizationof nonlinear electronic oscillators for neural computation,” IEEE Trans.
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[6] H. Kumeno, Y. Nishio, “Synchronization Phenomena in Coupled Parametrically Excited van der Pol Oscillators,” Proc. NOLTA’08, pp. 128-131,
Sep. 2008.

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Fig. 9.

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(a) Synchronization Probability.

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[5] M. Inoue, “A Method of Analysis for the Bifurcation of the Almost Periodic Oscillation and the Generation of Chaos in a Parametric Excitation
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23(b) The location of thenodes with M .

Simulation result (M : 6 nodes).

[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
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with Dispersion by Coupled Parametrically Excited Oscillators”, Proc.
NCSP’15, pp. 33-36, March, 2016.

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