2014 International Symposium on Nonlinear Theory and its Applications
NOLTA2014, Luzern, Switzerland, September 14-18, 2014

Influence of Local Bridge on a Complex Network of Coupled Chaotic Circuits
Kenta Ago, Yoko Uwate and Yoshifumi Nishio
Department of Electrical and Electronic Engineering, Tokushima University
2–1 Minami-Josanjima, Tokushima 770–8506, Japan
Email: {kentago,uwate,nishio}@ee.tokushima-u.ac.jp
Abstract—In this study, we investigate the influence oflocal bridge on a complex network of 25 coupled chaoticcircuits. From synchronization phenomena of coupledchaotic circuits, we show that synchronization of local
bridge is easy to break down. By means of computer simulations, the network switches to global synchronization andpartial synchronization. In order to analyze synchronization, we define asynchronous probability during a certain
time interval. Moreover, we statistically analyze the sojourn time of synchronization of each edge including localbridge.

2. Network Model
Figure 1(a) shows the chaotic circuit which is three–dimensional autonomous circuit proposed by Shinriki etal. [4][5]. This circuit is composed by an inductor, anegative resistance, two condensers and dual–directional
diodes. This circuit generates asymmetric attractor as
shown in Fig. 1(b). A proposed network model of 25 coupled chaotic circuits with local bridge is shown in Fig. 2. In
this study, chaotic circuits (CCn) are applied to each nodeof the network and each edge corresponds to resistors R. In
this model, local bridges are 1-25, 8-9, 14-15, 15-16, and22-23.

v2n

v1n

1. Introduction

idnComplex networks have attracted a great deal of attention from various fields since the discovery of “smallworld” network [1] and “scale-free” network [2]. In particular, how network topological structure influences its dynamical behaviors, is currently becoming a topic of great
important. On the other hand, synchronization phenomenaon the networks of coupled chaotic systems are very interested. However, there are not many studies of large-scale
network of continuous-time real physical systems such as
electrical circuits. Additionally we focus on synchronization phenomena of coupled chaotic circuit network with
community structure based on social theory.
In sociology, there is a famous theory called “Thestrength of weak ties” by Granovetter [3]. This is the theory that weak networks (weak ties) are important more thanstrong networks (strong ties). Because strong networks areeasy to isolate by centripetal force for homogeneity andaﬃnity. Therefore weak ties are essential for information
propagation and so on. Weak ties connect strong networks
with each other as the bridge. In large-scale network, thebridging function may be provided locally. This kind of thebridge is called “local bridge”.
In this study, synchronization phenomena on 25 coupledchaotic circuit network with local bridge are investigated.
We show that synchronization of local bridge is easy tobreak down. By means of computer simulations, the network switches to global synchronization and partial synchronization. In order to analyze synchronization, we define asynchronous probability during a certain time interval. Moreover, we statistically analyze the sojourn time ofsynchronization of each edge including local bridge.

c2

c1-g

CCnyinLz(a) Chaotic circuit.

(b) Attractor(α = 0.4, β = 20, γ = 0.5).

Figure 1: Chaotic circuit and attractor.
CC17RRCC12CC10RRR CC19CC16R

RR
CC20CC8R

CC2CC5CC7R

R

CC21CC22R

R

R

R

R

R

CC13R

R
R

R

R

R

CC11CC9CC15R

CC18R

R
CC14R

R

R

R
CC6R

CC4R

CC1CC23R

R

R

R

R

CC25

R
R

R

CC24

CC3Figure 2: Network model.
First, we approximate the i − v characteristics of the nonlinear resistors consisting of the diodes by the following
three-segment piecewise-linear function as follows:
Gd (v1n − v2n − V) (v1n − v2n > V)
0(|v1n − v2n | ≤ V)
(1)
idn = 
 G (v − v + V) (v − v < −V).
d 1n2n1n2n- 731 -By using the parameters and the variables as follows:
√

C2


V xn , v1n = Vyn , v2n = Vzn
in =


L



√

dC2
(2)t = LC2 τ, “ · ” =, α=


dτ
C1



√
√
√



LL1L


Gd , γ =g, δ =,
 β=C2C2R C2the normalized circuit equations are given as follows:x˙n = zn∑(3)y˙=αγy−αf(y−z)−αδ(yn − yk )nnnnk∈Sn z˙ = f (y − z ) − x ,nnnn

where n = 1, 2, 3, ..., 25 and S n is set of nodes which areconnected to CCn. The nonlinear function f () corresponds
to the i − v characteristics of the nonlinear resistors consisting of the diodes and are described as follows:β(yn − zn − 1) (yn − zn > 1)
0(|yn − zn | ≤ 1) (4)f (yn − zn ) =  β(y − z + 1) (y − z < −1).
nn

n

1-2
1-3
1-4
2-4
3-4
4-5
4-6
5-6
5-7
5-8
6-7
7-8
8-9
9-10
9-11
10-11
10-12
11-12
11-13
12-13
12-14
13-14
14-15
15-16
16-17
16-18
16-19
16-20
17-18
18-19
19-20
19-21
19-22
20-21
20-22
21-22
22-23
23-24
23-25
24-25
1-25
→τ

Figure 3: Phase diﬀerence waveform (δ = 1.0).

n

3. Synchronization States
In this study, we fix the same parameters as α = 0.4,β = 20, γ = 0.5 and δ on all circuits. Each circuit isgiven diﬀerent initial values each other. We show the dynamics of synchronization of the coupled chaotic circuits.
In Fig. 3, the vertical axes are the diﬀerences between thevoltage (corresponding to v1 ) of the two chaotic circuits.
Namely, if the two chaotic circuits synchronize, the valueof the graph should be almost zero like 19-20. We can confirm that synchronizations of local bridges (8-9, 14-15, 1516, 22-23 and 1-25) are easy to break down compared withothers. In this study, we define two synchronization statesof “global synchronization” and “partial synchronization”.
Figure 3 shows that the network switches to two synchronization states. Additionally, partial synchronization is almost occurred from local bridge.

synchronized during a certain time interval. We confirm
that the network easy to become global synchronization
state by increasing the coupling strength δ. In particular,
around δ = 1.6〜1.7 are rapidly become to high global synchronization probability.

4. Statistical Analysis
In this section, we fix a certain time interval as (τ= 10, 000 and step = 0.01τ) and we statistically analyze
the synchronization phenomena observed from 25 coupledchaotic circuits with local bridge. First, we show the coupling strength dependency of global synchronization. Inorder to analyze synchronization state, we define the synchronization as following equation,|yn − yk | < 0.01 (k ∈ S n ).

(5)

Figure 4 shows the coupling strength dependency of globalsynchronization. We checked whether if the all edges are

Figure 4: The coupling strength dependency of global synchronization.
From this point forward, we choose the couplingstrength as δ = 1.0 and we analyze synchronization focusing edge more statistically. Figure 5 shows that sorted
all edges in order from highest to lowest of asynchronous
probability and the number of switching between synchronization and asynchronous each edge. Local bridges are
the top five of asynchronous probability among all edges.
Additionally, local bridges are large number of switching
between synchronization and asynchronous compared with- 732 -other edges. Figure 6 shows the distribution of the numberof synchronized edges during a certain time interval where
the 41 of synchronized edge shows global synchronizationand the 0 of edge shows fully asynchronous. In this parameters, we consider that various partial synchronizations
exist on the network. In Fig. 6, it is interested that the distribution of the 41 edges is the second highest.
Next, node 15 is located in between two local bridges(14-15 and 15-16). Therefore the investigating of the statesof special node 15 in this network, is important. Figure
7 shows the distribution of the states of node 15 during acertain time interval. In Fig. 7, for example, state D showsthat nodes 14 and 15 are synchronized however nodes 15and 16 are not synchronized. In four states, state A is thehighest distribution. Therefore node 15 is easy to isolate
however state B is the second highest distribution that isinterested.

1 2 3 4 5 61-25

1 2 3 4 5 6

1-4

1 2 3 4 5 6

1 2 3 4 5 6

1 2 3 4 5 6

8-9
14-15
15-16(a) Local bridges.

1 2 3 4 5 6

1 2 3 4 5 6

1 2 3 4 5 65-7
10-12
19-22(b) Various edges.

1 2 3 4 5 622-23

1 2 3 4 5 623-25

Figure 8: Distribution of the sojourn time τ of synchronization.

Figure 7: Distribution of the states of node 15.
Moreover, we focus on the sojourn time of synchronization of each edge including local bridge. Figure 8 shows
the distribution of the sojourn time of synchronization. The
slots in the horizontal axes of the figure denote the ranges ofthe sojourn time in Tab. 1. From Fig. 8(a), the graphs of local bridges is very similar and the ratio of slot 1 is predominantly high. Namely, synchronizations of local bridges disappear immediately. On the other hand, from Fig. 8(b),graphs of various edges show that the sojourn time of thesynchronization is longer than local bridges. In particular,
the sojourn time of synchronization of edge 23-25 that islocated in between two local bridges (22-23 and 1-25), issimilar to the local bridges. From this result, edge 23-25 isconsidered quasi-local bridge.

Slot
12
3

Table 1: Ranges of slots in Fig. 8.
Sojourn time (τ) Slot Sojourn time (τ)τ<143 ≤τ<41 ≤τ< 254 ≤τ<52 ≤τ< 36τ ≥5to break down. Additionally, we confirmed that two synchronization states of global synchronization and partialsynchronization. Moreover, we statistically analyzed thatthe sojourn time of synchronizations of local bridges areshorter than other edges. Namely, local bridge almost behaves asynchronously. These phenomena show that local
bridge is the weak ties for promoting information propagation. In order to understand the phenomena correctly, moredetailed investigation considering cluster should be carriedout in our future works.
References[1] D. J. Watts and S. H. Strogatz, “Collective dynamics of smallworld”, Nature, vol. 393, pp. 440-442, 1998.
[2] A. L. Barabasi and R. Albert, “Emergence of scaling in random networks”, Science, vol. 286, pp. 509-512, 1999.
[3] M. S. Granovetter, “The Strength of Weak Ties”, Amerian Journalof Sociology, vol. 78, pp. 1360-1380, 1973.
[4] M. Shinriki, M. Yamamoto and S. Mori, “Multimode Oscillations in
a Modified van der Pol Oscillator Containing a Positive NonlinearConductance”, Proc. IEEE, vol. 69, pp. 394-395, 1981.

5. Conclusion
In this study, we have proposed the network of 25 coupled chaotic circuits with local bridge. By means of computer simulations, synchronization of local bridge is easy[5] N. Inaba, T. Saito and S. Mori, “Chaotic Phenomena in a Circuitwith a Negative Resistance and an Ideal Switch of Diodes”, Trans. ofIEICE, vol. E70, no. 8, pp. 744754, 1987.

- 733 -

Figure 5: Asynchronous probability and the number of switching between synchronization and asynchronous (δ = 1.0).

Figure 6: Distribution of the number of synchronized edges during a certain time interval (δ = 1.0).

- 734 -