2017 RISP International Workshop on Nonlinear Circuits,Communications and Signal Processing (NCSP'17)
Guam, USA, February 28th to March 3rd, 2017

Synchronization Phenomena of Coupled Two Rings of Chaotic Circuits
Katsuki Nsakashima, Kazuki Ueta, Yoko Uwate and Nishio Yoshifumi†Dept. of Electrical and Electronic Engineering, Tokushima University,
2-1 Minamijosanjima, Tokushima, 770-8506 Japan
Email: {nakashima, kazuki, uwate, nishio}@ee.tokushima-u.ac.jp

2. System model

Abstract

Nonlinear phenomena of coupled chaotic circuits are drawThe chaotic circuit is shown in Fig. 1 and the system modeling attention from many researchers. In this study, we inves- is shown in Fig. 2. This chaotic circuit consists of two inductigate the synchronization phenomena of coupled two rings tors L1 and L2 , one capacitor C , negative resistor −r andof chaotic circuits. One ring chaotic circuit generates chaotic two diodes.
attractors and the other ring generates three-periodic attractors. In addition, we observe the synchronization phenomena
by changing the coupling strength.

1. Introduction

Synchronization phenomena have been found in variousﬁelds of natural world [1] – [3]. Especially, there are a lotof relationships of biological systems. Also, synchronization
phenomena have a relationship with the human body. For
example, cells of the human body are synchronized. Therefore, the vibration of same timing produces big vibration.
According to synchronization phenomena, small power produces very big power.
Recently, complex networks have attracted attention andtopology of complex networks is studied for inﬂuence on thesystem. Also, synchronization phenomena of chaotic circuits
are studied from various viewpoints. Synchronization phenomena of chaotic circuits are the same as the vibration of thenatural world. Before now simple system has already beenstudied. For example, it is only ring structure, only ladderstructure, only star structure and so on [4] – [6]. However,many researchers have not been studied about more complex systems. Therefore, we propose the coupled two ringsof chaotic circuits as the minimum scale of complex systems.
In this study, we investigate the synchronization phenomena of coupled two rings of chaotic circuits. We couple threechaotic circuits on the ring structure and we propose a system model that the two rings are coupled via a resistor. One
ring chaotic circuit generates chaotic attractors and the otherring generates three-periodic attractors. Then, we focus onsynchronization phenomena of the chaotic circuits.

Figure 1: Chaotic circuit.




















              

Figure 2: System model.

The circuit equations of chaotic circuits are given as follows:- 325 -⎧di1⎪⎪
L1= v + ri1 ,⎪⎪dt⎪⎪⎪⎪⎨di2
L2= v − vd ,dt⎪⎪⎪⎪⎪dv⎪⎪⎪⎩ C dt = −i1 − i2 .

z1z2x1x2z4x3z5x4z6x5x6

Figure 3: Attractor (γ = 0.001).

(1)

The current-voltage characteristics of nonlinear resistor are
given as follows:- -
rd -V -- -V -(2)vd =i2 + - − -i2 − - .
2 rdrdz3

z1z2z2z3z3z2z4z3z4z5z5z6z6z4

By changing the parameters as follows:;√⎧
CL1 C⎪⎪
i1 =V x n , i2 =V yn , v = V z n ,
⎪⎪
LL
⎪
12⎪⎪;
⎨√
C
L1 CL1,
β=,
δ=r,
α=rd⎪
⎪
LL
L
122⎪⎪⎪⎪⎪1⎩γ = , t = L1 C2 τ.
R

Figure 4: Different waveform (γ = 0.001).

Figure 5 shows attractor of each chaotic circuit and Fig. 6shows
the voltage of different waveform when we set the cou(3)pling strength as γ = 0.1. In case of γ = 0.1, CC1, CC2
and CC3 are not synchronized because this ring circuits generate chaotic attractors. In this case, we observe the chaotic
propagation. Therefore, CC4, CC5 and CC6 of three-periodic
attractors change to chaotic behavior.
The normalized equations of chaos circuits are given as follows:zz
z
z
z
z
⎧dxi
⎪⎪
x
x
x
x
x
x
= αxi + zi ,
⎪⎪dτ
⎪
⎪
⎪
dyi
⎪
⎪
Figure 5: Attractor (γ = 0.1).
⎨= zi − f (yi ),dτ(4)6⎪⎪dzizz⎪⎪γij (zi − zj ),= −xi − βyi −⎪⎪⎪zz⎪ dτi,j=1⎪⎩zz
(i, j = 1, 2, · · ·, 6).
Where γ is the coupling strength. f (yi ) is described asfollows:- -1 -1 -- -1 -(5)yi + - − -yi − - .
f (yi ) =2 δ
δ

zz
zz
zz
zz

We deﬁne αc to generate the chaotic attractor, and αp is deﬁned to generate the three-periodic attractors.

Figure 6: Different waveform (γ = 0.1).

Figure 7 shows attractor of each chaotic circuit and Fig. 8
shows the voltage of different waveform when we set the coupling strength as γ = 0.2. In case of γ = 0.2, we also ob3. Simulation results
serve the chaotic propagation. When CC5 and CC6 alternate
We set the parameters of the system as αc = 0.460, αp =between synchronous and asynchronous states, the cycle of0.412, β = 3.0 and δ = 470.0. We investigate synchronizasynchronous and asynchronous states of CC4 - CC5 and CC4tion phenomena by changing the coupling strength γ between- CC6 are symmetric. Then, Synchronous and asynchronous
the circuits. Figure 3 shows attractor of each chaotic circuit
states change with the simulation time.
and Fig. 4 shows the voltage of different waveform when weset the coupling strength as γ = 0.001. Then, CC5 and CC6of three-periodic attractors are synchronized.

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z1

z2x1

z3x2

z4x3z5z6x5x4

Figure 11 shows deﬁnition of synchronization. We deﬁne
the inside of the red line as synchronization.

x6|zi-zj|time

Figure 7: Attractor (γ = 0.2).
Figure 11: Deﬁnition of synchronization.

z z
z z
z z
z z
z z
z z
z zFigure 12 shows simulation results of synchronization rate.
The horizontal axis shows the coupling strength γ. The vertical axis shows the synchronization rate. From Fig. 12, thesynchronization rate of the circuits of the right side ring(CC5and CC6) are decreased as increasing γ. However, the synchronization rate of the circuits of the left side ring(CC1 andCC2) are increased. We can conﬁrm that the synchronization
rate of the circuits of the left side ring becomes higher fromthe coupling strength γ = 0.15.

Figure 8: Different waveform (γ = 0.2).

Figure 9 shows attractor of each chaotic circuit and Fig. 10shows the voltage of different waveform when we set thecoupling strength as γ = 0.23. As is the case in γ = 0.2,
we observe the chaotic propagation. Synchronous and asynchronous states change with the simulation time. Here, we
make a comparison the circuits between the left side ring andthe right side ring. From Fig. 10, we can say that the circuitsof the left side ring are more synchronized than the other ring.

z11z22x1

z33
x2

z44x3

z55
x4z66
x5

Figure 12: Synchronization rate.

x6Figure 9: Attractor (γ = 0.23).
4. Conclusion

z1z2z2z3z3z2z4z3z4z5z5z6z6z4
Figure 10: Different waveform (γ = 0.23).

Finally, we investigate the synchronization rate. We compare with the synchronization rate of chaotic solutions(CC1and CC2) and periodic solutions(CC5 and CC6). In thisstudy, deﬁnition of synchronization is given as follows:|Zi − Zj | < 0.1(i, j = 1, 2, · · ·, 6).

In this study, we have proposed a system model using tworing circuits that are coupled by a resistor. One ring chaotic
circuit generates chaotic attractors and the other ring generates three-periodic attractors. We have investigated synchronization phenomena by changing the coupling strength γ. By
the computer simulations, we have observed different waveform. As a result, we conﬁrmed that synchronous and asynchronous state of chaotic solutions changes in the simulation
time when the coupling strength γ set to 0.2 and 0.23.
In the future works, we will investigate the reason forchanging synchronous and asynchronous states in the simulation time. Also, we investigate the reason why the chaotic solutions are synchronized when the coupling strength changes.
Acknowledgment

This work was partly supported by JSPS Grant-in-Aid for(6) Challenging Exploratory Research 26540127.

- 327 -References[1] C. M. Gray, “Synchronous Oscillators in Neural Systems: Mechanisms and Functions,” J. ComputationalNeuroscience, vol. 1, pp. 11-38, Feb. 1994.
[2] G. Abramson, V. M. Kenkre and A. R. Bishop, “Analytic Solutions for Nonlinear Waves in Coupled ReactigSystems,” Physical A: Statistial Mechanics and its Applications, vol. 305, no.3-4, pp. 427-436, 2002.
[3] S. Cooper, “Rethinking Synchronization of Mammmalian Cells for Cell Cycle Analysis,” CMLS, Cell.
Mol. Life Sci. 60 (2003) 0019, 2003.
[4] Yoshifumi Nishio, Katsunori Suzuki, Shinsaku Moriand Akio Ushida ”Synchronization in Mutually Coupled Chaotic Circuits” Proceedings of European Conference on Circuit Theory and Design (ECCTD’93), vol. 1,pp. 637-642, Aug. 1993.
[5] Yoshifumi Nishio and Akio Ushida ”On Synchronization Phenomena in Coupled Chaotic Circuits Networks”
Proceedings of IEEE International Symposium on Circuits and Systems (ISCAS’96), vol. 3, pp. 92-95, May.
1996. (Invited paper)[6] M. S. Papadopoulou, I. M. Kyprianidis, I. N. Stouboulos ”Chaotic Dynamics of Coupled Nonlinear Circuitsin Ring Connection ” Chaotic Modeling and Simulation(CMSIM) 1: 177-184, 2012

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