2014 RISP International Workshop on Nonlinear Circuits,Communications and Signal Processing
NCSP'14, Honolulu, Hawaii, USA. February 28 - March 3, 2014

Synchronization Phenomena in Chaotic Circuitwith Stochastically Coupling
Toshiaki Nishiumi, Yoko Uwate and Yoshifumi Nishio
Dept. of Electrical and Electronic Engineering, Tokushima University,
2-1 Minami-Josanjima, Tokushima, 770-8506, Japan
Phone: +81-88-656-7470, FAX:+81-88-656-7471
E-mail: {toshiaki,uwate,nishi}@ee.tokushima-u.ac.jp

Abstract

2. Circuit ModelIn this study, we investigate the synchronization phenomena
observed from two chaotic circuits with stochastically coupling. First, we research the basic synchronization phenomena. Second, we confirm on synchronization ratio of twochaotic circuit when the coupling probability is changed. Finally, we focus on synchronization ratio of chaotic circuit
when the five coupling probability is changed. Then we compare the difference in synchronization ratio between two andfive coupled chaotic circuits.

2.1 Nishio-Inaba Chaotic Circuit
Figure 1 shows the model of the used chaotic circuit called
Nishio-Inaba circuit, investigated in [6]-[7].
The approximated I-V characteristic of the nonlinear resistance is given in Fig 2.

1. Introduction
Synchronization phenomena is one of the basic nonlinear
phenomena and it can be observed everywhere in our living
life [1]-[5]. For example, vibration of the cells of the humanand 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 areFigure 1: Nishio-Inaba Chaotic Circuit
many real physical phenomena exhibiting by coupled chaotic
circuits, and it is important to investigate synchronization
phenomena observed from coupled chaotic circuits for futureengineering 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
focus on synchronization ratio of two chaotic circuits when
the coupling probability is changed. For this simulation, wechange the bifurcation parameters in chaotic circuit and observe a change of synchronization ratio. Finally we focus onsynchronization ratio of five chaotic circuits when the coupling probability is changed. Then, we compare difference Figure 2: The I-V characteristic of the nonlinear resistanceof synchronization with two chaotic circuits and five chaotic in Fig. 1.
circuits.

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The circuit consists of a negative resistance, a nonlinear 2.2 Propose Model
resistance consisting of two diodes, a capacitor and two inFigure 3 shows the circuit model used in this study. Eachductors. The following equation describes the approximatedcircuitis connected by a resistor. Also, coupled state (couI-V characteristic of the nonlinear resistance , where the papling
oruncoupling) is determined by stochastically. We derameter rd is the slope of the nonlinear resistance.
fine the uncoupling probability (= p) in chaotic circuit. Forexample, the two chaotic circuits are coupled, if the uncourdVV
vd (i2 ) =(|i2 + | − |i2 − |).
(1) pling probability is p = 0%. Whereas, the two chaotic cir2rdrdcuits are uncoupling, if the uncoupling probability is p =The circuit dynamics is then described by the following 100%.
piecewise-linear third-order ordinary differential equation:
In this study, we investigate the ratio of synchronization.
In order to analyze synchronization ratio, we define the synchronization as following equation,
di1
L1= v + ri1dt
di1
L1
= v − vd (i2 )dt|zM − zN | < 0.1.
(6)
di1
L1
= v − vd (i2 ).
(2)dt
where M is the number of circuits and N is circuit whichconnected
circuit M .
Equation (2) can be normalized by changing the variables
according to√√C
L1 C
i1 =V x; i2 =V y; v = V z;
L1L2
√√
√
CL1 C
L1Figure 3: Circuit model (n = 2).
α=r; β=δ = rd; . t = L1 Cτ.
L1
L2
L2yielding
3. Synchronization Phenomenaẋ = αxi + ziẏż= z − f (y)= −xi − βyi − βy3.1 Basic Phenomena(3)

where f (y) is described as follows:f (y) =δ11(|y + | − |y − |).
2δδ

(4)

Next, we consider the following equations when eachchaotic circuits is coupled globally.
ẋẏ

In this section, we compare the synchronization phenomena, in the case of p = 0% and p = 100%. First, we set
parameter as α= 0.46 , β= 3.0, γ= 470 and δ. In this study,
we focus on the parameter δ. Figures 4 (a) and 4 (b) show thesimulation result of the phase difference. From these results,
we confirm synchronization phenomena state, in the case ofp = 0% and p = 100%.
3.2 Synchronization Phenomena with Stochastically
Coupling (n = 2)

= αxi + zi= zi − f (y)In this section, we focus on changing synchronization ratiowith range from p = 0% to p = 100%. For this investigating,n
we change the bifurcation parameters in chaotic circuit and∑ż = −xi − βyi − γij(zi − zj )(5) observe a change of synchronization ratio. First, we set theparameters as α, β= 3.0, γ= 470, and δ= 0.64. Figure 5 showsj=1
the chaotic attractors in the case of α= 0.37, 0.4 and 0.46. Fig(i, j = 1, 2, · · · , n)ure 6 shows the simulation results of the synchronization ratio
where i in the equation represents the circuit itself, and j in- by changing the bifurcation parameters of the coupled chaotic
dicates the coupling with other circuits. The parameter γij circuits with stochastically coupling. The horizontal axis isrepresents the coupling strength between the circuits.
an indicator of uncoupling probability. The vertical axis is an- 182 -uncoupling(p = 100%).

coupling(p = 0%).

(b)

(a)

Figure 4: Simulation results of phase difference.

Figure 6: Simulation result (n = 2).
indicator of synchronization ratio. We confirm changing synchronization ratio of chaotic circuit when the coupling probability. From these results, the ratio of asynchronous increases
with the uncoupling probability in each parameter α. Then, 0.151. Then we compare difference in synchronization whenin the case of that the bifurcation parameter is higher, the syn- the coupling probability is changed with two and five couchronization ratio tends to low proportion.
pled chaotic circuits. Figure 9 shows the simulation result of
the synchronization phenomena. From these results, there aresimilar results as far as 10% for 30%. However, by increasing
the uncoupling probability, as the number of coupled chaotic
circuits are larger, predisposition toward synchronization.

(a) α=0.37.

(b) α=0.4.

Figure 7: circuit model (n = 5).
(c) α=0.46.

Figure 5: Chaotic attractor.

4. Conclusions

In this study, we investigate the synchronization phenomena observed from chaotic circuits with stochastically coupling. We focus on synchronization in chaotic circuits with
3.3 Synchronization Phenomena with Stochasticallystochastically. For this investigation, we change the bifurCoupling (n = 5)
cation parameters in chaotic circuit and observe a change inIn this section, we investigate the synchronization phenom- synchronization ratio. Also, we observe the difference in synena with five coupled chaotic circuits (n = 5). Figure 7 chronization phenomena with two and five coupled chaotic
shows the circuit model. Figure 8 shows the chaotic attrac- circuits. By using the computer simulations, we have obtor. We set the parameter as α= 0.46, β= 3.0, γ= 470 and δ= served the interesting synchronization phenomena.

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mechanisms and functions,” J. Computational Neuroscience, Vol. 1, pp. 11-38, 1994
[6] Y. Nishio, N. Inaba, S. Mori and T. Saito. “RigorousAnalyses of Windows in a Symmetric Circuit” IEEE
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473-487, Apr. 1990.

Figure 8: Chaotic attractor(n = 5).

[7] C. Bonatto and J. A. C. Gallas, “Periodicity Hub andNestred Spirals in the Phase Diagram of a Simple Resistive Circuit,” Phys. Rev. Lett., 101, 054101, Aug. 2008.
[8] R. Stoop, P. Benner and Y. Uwate, “Real- World Existence and Origins of the Spiral Organization of ShrimpShaped Domains,” Phys. Rev. Lett., 105, 074102, Aug.
2010.
[9] K. Okamoto, Y. Takamaru, Y. Uwate and Y. Nishio. “Synchronization of an Array of Chaotic Circuits Coupled byDifferent Coupling Strength,” Proc. NCSP’13 March 4-7,
2013[10] Y. Takamaru, Y. Uwate, T. Ott and Y. Nishio, “Clustering Phenomena of Coupled Chaotic Circuits for Large
Scale Networks” Proc. of NDE’S 2012, pp.70-73, Jul.
2012.

Figure 9: Simulation result.

In the future work, we would like to examine more complex networks. Additionally, we would like to investigate themechanism of observed synchronization phenomena in detail.

[11] Y. Takamaru, H. Kataoka, Y. Uwate and Y. Nishio, “lustering Phenomena in Complex Networks of Chaotic Circuits,” Proc. ISCAS’12 Mar. 2012.

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