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

Synchronization Phenomena of Chaotic Circuitswith Stochastically-Changed Network Topology
Toshiaki Nishiumi† , Yoko Uwate† and Yoshifumi Nishio††Departmaent of Electrical and Electronic Enginnering, Tokushima University
2-1 Minamijosanjima, Tokushima, 770-8506, Japan
Email: {nishiumi, uwate, nishio}@ee.tokushima-u.ac.jp
Abstract—In this study, we investigate the synchronization phenomena observed from five chaotic circuits withstochastically coupling. First, we research the synchronization state changing network topology. For the simulation, we focus on synchronization ratio with various coupling topologies. Furthermore, we propose several coupling rules and investigate the synchronization by the influence of the coupling rules.
1. Introduction
Synchronization phenomena is one of the basic nonlinear phenomena and it can be observed everywhere in ourliving life [1]-[4]. For example, vibration of the cells of thehuman and begins to croak at the same time of the frog.
Recently, in particular, synchronization phenomena in coupled chaotic circuits are attracted many researchers attentions. There are many real physical phenomena exhibiting by coupled chaotic circuits, and it is important to investigate synchronization phenomena observed from coupled chaotic circuits for future engineering applications.
We consider that there are many real physical phenomena exhibiting by coupled chaotic circuits, and it is important to investigate synchronization phenomena observed
from coupled chaotic circuits for future engineering applications.
In this study, we investigate the relationships between network connection and synchronization phenomena. First, we research network topology when the coupling probability is changed. Second, we confirm basicsynchronization. Finally, we focus on the synchronization
ratio of various combination coupled chaotic circuits. Furthermore, we compare the ratio of synchronization whenrules of combination coupling are applied.
2. Circuit Model
2.1. Nishio-Inaba Chaotic Circuit
Figure 1 shows the model of the used chaotic circuit
called Nishio-Inaba circuit.
The circuit consists of a negative resistance, a nonlinear resistance consisting of two diodes, a capacitor and twoinductors. The following equation describes the approximated I-V characteristic of the nonlinear resistance , whereFigure 1: Nishio-Inaba Chaotic Circuit.

the parameter rd is the slope of the nonlinear resistance.
VVrd(|i2 + | − |i2 − |).
2rdrdvd (i2 ) =(1)The circuit dynamics is then described by the following
piecewise-linear third-order ordinary diﬀerential equation:
di1
dt
di1
L1
dt
dv
Cdt
L1

=v + ri1=v − vd (i2 )=−i1 − i2 .

(2)Equation (2) can be normalized by changing the variables according to√√C
L1 C
V x; i2 =Vy; v = Vz;
i1 =L1L2√√√
C
L1L1 C
α=r; β=δ = rd; t = L1Cτ.
L1L2L2yielding- 811 -ẋ

=αxi + ziẏż==z − f (y)−xi − βyi − βy(3)where f (y) is described as follows:f (y) =δ
11(|y + | − |y − |).
2
δδ

(4)

Next, we consider the following equations when eachchaotic circuits is coupled.
(a) Coupling state (p = 80%).

ẋẏ

= αxi + zi= zi − f (y)ż= −xi − βyi − γi jn∑(zi − z j )

(5)

j=1

(i, j = 1, 2, · · · , n)where i in the equation represents the circuit itself, and jindicates the coupling with other circuits. The parameterγi j represents the coupling strength between the circuits.
2.2. Proposed Model
Figure 2 shows the circuit model used in this study. Eachcircuit is connected by a resistor. Also, coupled state (coupling or uncoupling) is determined by stochastically one byone. We define the uncoupling probability (= p) in the coupled chaotic circuits. For example, when if the uncoupling
probability is set to p = 80%, the state of coupling and thephase diﬀerence become like Fig. 3(a) and (b). For thissimulation, we change the network topology one hundred
times when the uncoupling probability p is changed from
0% to 100%.

Figure 2: Circuit model (n = 5).

2.3. Synchronization Ratio
In this study, we research synchronization ratio withrange from p = 0% to p = 100%. In order to analyze
synchronization ratio, we define the synchronization as following equation:|z M − zN | < 0.1.

(M, N = 1, 2, · · · , 5)(6)where M is the number of circuits and N is circuit whichconnected circuit M.

(b) Phase diﬀerent (p = 80%).
Figure 3: Simulation results (p = 80%).

3. Synchronization Phenomena
3.1. Basic Synchronization
In this section, we confirm the synchronization ratio
when network topology is changed at every τ=100, 1500.
When τ is 1500, the synchronization just settle down insynchronous or asynchronous. Figure 4 shows the simulation result of the synchronization ratio. The horizontal
axis is an indicator of the uncoupling probability. The vertical axis is an indicator of the synchronization ratio of average when the network topology is changed one hundred
times every parameter p. From the result, the ratio of asynchronous increases with the uncoupling probability. Then,
there is similar result as far as 30% for 100%. However,there is diﬀerent result as far as 30% for 100%. Namely,
if the network topology is changed before the synchronization just settle down in synchronous or asynchronous,
The ratio of synchronization predisposition toward asynchronous.
3.2. Synchronization Phenomena is Based on Rule ofCombination Coupling
In this section, we focus on synchronization phenomena
with combination coupling. Now, we define three rules depending on combination coupling. We call their rules ruleA, rule B and rule C. Table 1(a) and (b) show the corresponded coupling rule A and B. For example rule A, if CC1- CC2 determines uncoupling, CC2 - CC5 is surely uncoupling. In the case of rule B, if CC1 -CC2 determines uncou-- 812 -Table 1: Uncoupling combination.

(a) Uncoupling combination (Rule A).

Figure 4: Simulation result (Basic synchronization).

pling, CC2 - CC3, CC3 - CC4, CC4 - CC5 and CC5 - CC1
are surely uncoupling. In the case of rule C, if any couplingis determined uncoupling, every other couplings is surely
uncoupling. Then, we investigate the synchronization ratiowith changing network topology one hundred times when
the network topology is changed at every τ= 1500. Figure 5 shows the simulation result. From the result, we obtain a diﬀerent result when chaos synchronization is basedon rule of combination coupling. Namely, the number ofcorresponding couping is larger, the synchronization ratio
tends to high proportion.
In order to obtain details of the Fig. 5, we investigate
the synchronization ratio when the network topology ischanged several times. Figures 6(a) and 7 show the synchronization ratio when the network topology is changed
ten times. Also, Figure 6(b) shows the coupling state corresponded Fig. 6(a). From the result, synchronization ratio
is wildly diﬀerent depending on coupling or uncoupling. In
addition to that, there are diﬀerent result even if there aresame number of couplings.

(b) Uncoupling combination (Rule B).

4. Conclusions
In this study, we investigate the chaos synchronizationin stochastic system with various coupling topologies. For
this investigation, we observe the diﬀerence in synchronization ratio with various coupled chaotic circuits. By
using the computer simulations, we have observed the interesting synchronization phenomena.
In the future work, we would like to examine more complex networks. Additionally, we would like to investigate
the mechanism of observed synchronization phenomena indetail.
References[1] N.F. Rullckov and M.M. Sushchik, “Robustness ofSynchronized Chaotic Oscillations,” Int. J. Bifurcation and Chaos, vol. 7, no. 3, pp. 625-643, 1997.
[2] M. Wada, Y. Nishio and A. Ushida, “Analysis of Bifurcation Phenomena in Two Chaotic Circuits Coupled by an Inductor,” IEICE Trans. Fundamentals,
vol. E80-A, no. 5, pp. 869-875, 1997.
[3] Y. Nishio and A. Ushida, “Chaotic Wandering and its
Analysis in Simple Coupled Chaotic Circuits,” IEICE
Trans. Fundamentals, vol. E85-A, no. 1, pp. 248-255,
2002.

Figure 5: Simulation result (Synchronization phenomena
is based on rule of combination coupling).

[4] G. Abramson, V.M. Kenkre and A.R. Bishop, “Analytic Solutions for Nonlinear Waves in Coupled Reacting Systems,” Physica A: vol. 305, no. 3-4, pp.
427-436, 2002.

- 813 -(a) Ratio of synchronization (Pattern A, 50%).

Figure 7: Ratio of synchronization (Pattern C, 70%).

(b) State of topology (Pattern A, 50%).
Figure 6: Ratio of synchronization (Pattern A, 50%).

- 814 -