RISP

NCSP’16

International Workshop on
Nonlinear Circuits, Communications and Signal Processing

Honolulu, Hawaii, USA, March 6-9, 2016

Synchronization in Complex Networks with Dispersion
by Coupled Parametrically Excited Oscillators
Kosuke Oi, Yoko Uwate and Yoshifumi Nishio
Tokushima University
2-1 Minami-Josanjima, Tokushima 770–8506, Japan
Phone:+81-88-656-7470
E-mail: {ooi, uwate, nishio}@ee.tokushima-u.ac.jp
Abstract
In this study, we investigate synchronization in complex network with hubs and dispersion by using parametrically excited van der Pol oscillators. By means of computer simulation, we confirm various synchronous states and clustering
phenomena and observe the effect of hubs in complex network.
1. Introduction
Synchronization is one of the fundamental phenomena innature and it is observed over the various fields. Studies on
synchronization phenomena of coupled oscillators are extensively carried out in various fields, physics [1], biology [2],engineering and so on. The coupled van der Pol oscillator is
one of typical coupled oscillators, and synchronization generated in the system can model certain synchronization of natural rhythm phenomena. Parametric excitation circuit is oneof resonant 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 in Refs. [4] and [5].
In our research group, we have investigated synchronization of 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 oftwo subcircuits, the anti-phase synchronization is observed.
In the case of three subcircuits, self-switching phenomenonof synchronization 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 mod-ified from “Dolphin social network” [7] by using parametrically excited van der Pol oscillators with dispersion [8]. We
have confirmed that the network with hubs can induce synchronization. Though, we have only investigated the case of
densely-packed with hubs.
In this study, we focus on the location of the hubs, and investigate synchronization and clustering in the complex network scattering hubs. This network constructed by reference
to the network modified from “Dolphin social network” [8].
In this network, the hubs are scattered about dual places. We
focus the effect of the hubs in network, and investigate mechanism of synchronization clustering phenomena in the complex network scattering hubs.
2. System model
The circuit model of van der Pol oscillator under parametric excitation is shown in Fig. 1.

iidnCv

LFigure 1: van der Pol oscillator under parametrically excitation
The circuit includes a time-varying inductor L whose characteristics are given as the following equation. The timevarying inductor is shown as Fig. 2.
L = L0 γ(τ ).

(1)γ(τ ) is expressed in a rectangular wave as shown in Fig. 2,and its amplitude and angular frequency are termed α and ω,respectively. By changing the value of α, the amplitude of- 33 -

Cluster 252( )

56

9/57

Figure 2: Function relating to parametric excitation76

By changing the variables and the parameters,
⎧&%
g1
⎪
⎪
t = L0 Cτ, vn =xn
⎪
⎪
g3
⎪
⎪
⎪
⎪
⎪%
⎪⎪
⎪
ω = ω 0 L0 C⎪⎪⎨
& &g1 C⎪⎪⎪ in =yn⎪⎪
g 3 L0⎪⎪⎪⎪⎪&&⎪⎪⎪LL10⎪⎩ ε = g1, δ=CR C441534518322748

4334750

35
59

5462

3182842

1144

39

1738

29

13342026
23

Figure 3: Proposed network

(2)

Count

id = −g1 vk + g3 vk .

55

21461

parametric excitation can be changed. The v − i characteristics of the nonlinear resistor are approximated by the following equation.

4058

1033

1530

1 21492 /4651

37

Cluster 1

36

25

1660

1-22

19

24

1

0

5

121+(3)Value
The normalized circuit equations are given by the followingequations.
Figure 4: Degree distribution of proposed network⎧'dxn⎪⎪= ε(xn − x3n ) − yn − δ(xn − xk )⎨ dτk∈Sn(4)⎪1dynnodes, and horizontal axis denotes the value of degree. In⎪⎩
=xndτγ(τ )this network, the number of nodes which is parametrically
excited van der Pol oscillator is 62 and the number of edges
where n is the count of the nodes. Sn is the set of the nodes
which is one resister is 99. Other feature quantities of prowhich are directly connected to the node n.
posed network is expressed in Table 1. This network can bedivided two clusters as shown in Fig. 3. The 58th node is thehub in cluster 1, and the 15th node is the hub in cluster 2. The
Table 1: Feature quantities of proposed network
bond number of 58th node is 11, and the bond number of 15th
Average degree
3.194
node is 13.
Average clustering coefficient 0.145
Average path length
4.384
3. Simulation method

In our system, parametrically excited van der Pol oscillators are coupled by one resistor R. Figure 3 shows proposedcomplex network and Fig. 4 shows degree distribution of thisnetwork. In this figure, vertical axis denotes the number ofIn this research, we investigate the synchronization by using computer simulation. We fix the 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- 34 -(a) δ = 0.017(b) δ = 0.015(c) δ = 0.010(d) δ = 0.010

Figure 5: Clustering phenomena.

this simulation, all of the nodes involve dispersion mn in α(in Fig. 2) which is corresponding to the amplitude of thefunction relating parametric excitation within the compass of−0.01 ≤ mn ≤ 0.01.
In order to analyze synchronous state, we define the synchronization by the following equation:|xn − xk | < 0.10 (k ∈ Sn ).

(5)

We fix the count of calculation as 100,000.
4. The results of simulation
In this simulation, we investigate change of synchronous
state and observe clustering phenomena by changing coupling strength with uniformity. Figure 5 shows clustering

phenomena in the proposed complex network. In this figure,the green edges express synchronous state and the gray edges
express unsynchronous state. First of all, this network becomes full synchronous state when the value of δ is 0.022. In
this research, we decrease the coupling strength δ from 0.022
in order to observe synchronization and clustering.
Next, in the case of δ = 0.017, this network is divided two
cluster as shown in Fig. 5 (a). In addition, we investigate
clustering phenomena by decreasing the coupling strength
δ. When the case of Fig. 5. (b), the unsynchronous state
comes into existence centering around 58th node. Whencoupling strength decrease to δ = 0.010, the synchronized
nodes in the cluster 1 decrease in a marked fashion. Thereare more synchronized nodes in the cluster 2 compared tocluster 1, and 58th node has more unsynchronized edges than- 35 -15th node whichever coupling strength δ. In each case ofFig. 5 (b), (c) and (d), we can observe the synchronous stateof cluster 1 crumples up faster than cluster 2. From the above,
we can confirm that the clustering phenomena of complex
network draw influence from location of the hubs.
5. Conclusions
In this study, we have investigated synchronization andclustering phenomena in proposed complex network withhubs and dispersion by using parametrically excited van derPol oscillators.
We could observed various synchronous state and clustering phenomena, and confirmed that the clustering phenomenaof complex network draw influence from location of the hubs.
References[1] I. Belykh, M. Hasler, M. Lauret and H. Nijmeijer, “Synchronization and graph topology,” Int. J. Bifurcation andChaos, vol.15, no.11, pp.3423-3433, Nov. 2005.
[2] J. Cosp, J. Madrenas, E. Alarcon, E. Vidal and G. Villar, “Synchronization of nonlinear electronic oscillatorsfor neural computation,” IEEE Trans. Neural Networks,vol.15, no.5, pp.1315-1327, Sep. 2004.
[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, “Themethod of mapping 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 ofthe Almost Periodic Oscillation and the Generation ofChaos 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 features a large population oflong-lasting associations. Can geographic isolation explain this unique trait?”, Proc. Behavioral Ecology andSociobiology, vol.54, pp. 396-405, 2003[8] K. Oi, K. Ago, Y. Uwate, Y. Nishio, “Effect of the Hubin Complex Networks of Coupled Parametrically Excited- 36 -Oscillators with Dispersion,” Proc. NCN’15, pp. 11-14,
Dec, 2015.