IEEE Workshop on Nonlinear Circuit Networks
December 7-8, 2018

Synchronization Phenomena ofSymmetric and Asymmetric Structures
Using Coupled Chaotic Circuits
Katsuki Nakashima, Yoko Uwate and Yoshifumi Nishio
Dept. of Electrical and Electronic Eng., Tokushima University
2-1 Minami-Josanjima, Tokushima 770–8506, Japan
Email:{nakashima, uwate, nishio}@ee.tokushima-u.ac.jp
Abstract—Nonlinear phenomena of coupled chaotic circuits
are drawing attention from many researchers. In this study,
we investigate the synchronization phenomena of coupled twosymmetrical structures and the influence of the network topology
when we use chaotic circuits. One structure generates chaotic
attractors and the other structure generates three-periodic attractors. Moreover, we observe the synchronization phenomena ofnetworks with asymmetrical structures and we compare networks
with symmetrical structures to networks with asymmetricalstructures.

I. I NTRODUCTION
Synchronization phenomena have been found in various
fields of natural world [1]-[3]. Especially, there are a lot ofrelationships of animate things. Also, synchronization phenomena have a relationship with the human body. For example,cells of the human body are synchronized. Therefore, theoscillation of same timing produces big oscillation. According
to synchronization phenomena, small power produces very bigpower.
Recently, complex networks have attracted attention andtopology of complex networks is studied for influence on thesystem [4], [5]. Also, synchronization phenomena of chaotic
circuits are studied from various viewpoints. For example,
it is studied in biology, engineering, medical science andso on. The oscillation that chaos occurs is the same as theoscillation of the natural world. In the chaotic circuit, aphenomenon by the name of chaotic synchronization has been
confirmed. It has been applied to chaotic control, chaotic communication and so on. Before now, network topology attracts
attention how to behave for coupled chaotic circuits because
the investigation of the influence for network topology andapplication examples are less low. Currently, simple system
has already been studied. For example, it is only ring structure,
only ladder structure, only star structure and so on [6]-[8].
However, many researchers have not been studied about morecomplex systems. Therefore, we propose the coupled twosymmetrical and asymmetrical structures of chaotic circuits
and we investigate the influence of network topology.
In this study, we investigate the synchronization phenomenaof networks with symmetrical and asymmetrical structures
when we use chaotic circuits. First, we propose system models that the two symmetrical structures are coupled via aresistor. One structure generates chaotic attractors and theother structure generates three-periodic attractors. Moreover,
we observe the synchronization phenomena of networks withasymmetrical structures and we compare networks with symmetrical structures to networks with asymmetrical structures.
In addition, we focus on the synchronization rate of eachmodel.
II. C IRCUIT MODEL
The chaotic circuit is shown in Fig. 1. This chaotic circuit
is Nishio-Inaba circuit [9]. This chaotic circuit consists of twoinductors L1 and L2 , one capacitor C , negative resistor −r
and two diodes.

CCFig. 1.

Chaotic circuit.

The circuit equations of chaotic circuits are given as follows:di1
 L1 dt = v + ri1 ,

di2L2= v − vd ,(1)dtdv
 C dt = −i1 − i2 .

The current-voltage characteristics of nonlinear resistor are
given as follows:()rdVVvd =(2)i2 +− i2 −.
2rdrd
By chaning the parameters as follows:- 23 -√
√

C
L1 C


i1 =V x n , i2 =V yn , v = V z n ,L1L2

√

√
C
L1 C
L1α=r, β=, δ = rd,LLL122√ γ = 1 , t = L C τ.
1 2Rγγγ1γ(3)γ2γ2γ

γ2γ1γ2γγ2γγγ2γγ2γ1γγ2γ2γ1γ2γ2γ2γ

γ2γ2γ2γ2γ2γ2γ2γ

γ2(a') 1-4 Coupled Model.

γ2γ

γ2γ2(a) 4-4 Coupled Model.

The normalized equations of chaos circuits are given as
follows:
dxi
= αxi + zi ,dτdyi
= zi − f (yi ),dτ(4)N∑dzi= −xi − βyi −γij (zi − zj ),dτi,j=1
(i, j = 1, 2, · · ·, N ).

γ2(b) 5-5 Coupled Model.

(b') 1-5 Coupled Model.

System models.

Fig. 3.

IV. S IMULATION RESULTS
Where γ is the coupling strength and f (yi ) is described asfollows:()
1
1
1f (yi ) =yi +− yi −.
(5)
2
δ
δ
We define αc to generate the chaotic attractor and αp isdefined to generate the three-periodic attractors. N is thenumber of coupled chaotic circuits. We use the attractor as
shown in Fig. 2. For the computar simulation, we set theparameters of the system as αc = 0.460, αp = 0.412, β = 3.0
and δ = 470.0.

We investigate the synchronization phenomena in eachmodel and the influence of topology. Figure 4 shows attractorof each chaotic circuit, Fig. 5 shows lissajous figure and Fig. 6shows the volage of different waveform. CC5 to CC8 generatechaotic attractor in each model. Also, between CC4 and CC5
are synchronized. CC6-CC5, CC6-CC7 and CC6-CC8 areasynchronous. CC7-CC5 and CC7-CC8 are chaotic synchronization in model (a). Therefore, in CC7-CC5 and CC7-CC8,
synchronous and asynchronous states changes irregularly inthe simulation time.

zzz1z2z3x1xz4x2x3x4z4z5x5z8x7x6z7z6x8z8(b) Three-periodic attractor.

x4Fig. 2.

z7z6(a) 4-4 Coupled Model.

x(a) Chaotic attractor.

z5Attractor.

x5x7x6x8(a’) 1-4 Coupled Model.

z1III. S YSTEM MODELS
Figure 3 shows the proposed system models. We propose
the system models that the two symmetrical structures arecoupled by a resistor in Fig. 3(a) and (b). In addition, wepropose the system models that the two asymmetric structures
are coupled by a resistor in Fig. 3(a’) and (b’). In Fig. 3(a)and (a’), we set CC1 to CC4 as chaotic solution and CC5 toCC8 as three-periodic solutions. In Fig. 3(b) and (b’), we set
CC1 to CC5 as chaotic solution and CC6 to CC10 as threeperiodic solutions. In this study, we set the coupling strength
γ as 0.2, the coupling strength γ1 between the topologies as0.1 and γ2 as 0.01 in all models. The coupling strength γ andγ2 are the synchronized value with a single structure.

- 24 -Fig. 4.

Attractors.

z4z2z2z3z4z2z4z1z3z4z6z5z7z5z8z5z4z7z6z8(a) 4-4 Coupled Model.

z6z5z5z5z7z7(a’) 1-4 Coupled Model.
Fig. 5.

z7z5z6

Lissajous figures.

z5z8z8z7

z1

z1ー z2z2ー z4z4 ー z1z2 ー z3z3ー z4z4ー z5z6ー z5z5ー z7z7ー z6z5 ー z8z8 ー z7z2z5z2z3z3z2z5z1z3z5z7z6z8z6z6z6z8z7z6z7z9z4z5z4z5z10z6z10z9z10z8z9z10(b) 5-5 Coupled Model.
z5z6z8z6z8z6z7z6z9z10z10z8(b’) 1-5 Coupled Model.
Fig. 8.

z1ー z2z2ー z5z5 ー z1z2 ー z3z3ー z5z3ー z4z4ー z5z5ー z6z7ー z6z6 ー z8z8 ー z7z6 ー z9z6ー z10z9 ー z10z10ー z8(a) 4-4 Coupled Model.

z4ー z5z6ー z5z5ー z7z7ー z6z5 ー z8z8 ー z7(a’) 1-4 Coupled Model.
Different waveform.

Fig. 6.

Lissajous figures.

Figure 7 shows attractor of each chaotic circuit, Fig. 8
shows lissajous figure and Fig. 9 shows the volage of different waveform. In both model (b) and model (b’), CC6
to CC10 generate chaotic attractor. Also, between CC5 andCC6 are synchronized. In model (b), CC6-CC7, CC6-CC8,CC6-CC9 and CC6-CC10 are asynchronous. CC7-CC8, CC8CC9 and CC9-CC10 are chaotic synchronization. Therefore,in CC7-CC8, CC8-CC9 and CC9-CC10, synchronous andasynchronous states changes irregularly in the simulation time.

(b) 5-5 Coupled Model.

z5ー z6z7ー z6z6 ー z8z8 ー z7z6 ー z9z6ー z10z9 ー z10z10ー z8(b’) 1-5 Coupled Model.

z1z2z3z4z5z7z6z9z8z10

Fig. 9.
x1x2x3x4x5x7x6x8x9

Different waveform.

x10(b) 5-5 Coupled Model.
z5z7

z6x5x6z9z8x7x8z10x9(b’) 1-5 Coupled Model.
Fig. 7.

Attractors.

x10Then, we compare with model (a) and model (a’), model (b)and model (b’). From the simulation results, we observe
the influence of the network. First, network with coupled
symmetrical structures has a strong influence on one circuit.
However, network with coupled asymmetric structure has theinfluence on all of the circuits in topology.
Finally, we investigate the synchronization rate. We comparewith symmetrical structures (model (a) and model (b)) and- 25 -asymmetrical structures (model (a’) and model (b’)). In thisstudy, definition of synchronization is given as follows:|Zi − Zj | < 0.1(i, j = 1, 2, · · ·, N ).

(6)

Figure 10 shows definition of synchronization. We define
the inside of the red line as synchronization.

|zi-zj|time
Fig. 10.

Definition of synchronization.

Tables 1 and 2 show the synchronization rate. From theseresults, we can confirm different synchronization phenomena
in different topology. When the synchronization rate between
topologies are increased, the synchronization rate of CC6CC5, CC5-CC7 and CC5-CC8 are increased in Table 1 andthe synchronization rate of CC7-CC6, CC6-CC8, CC6-CC9and CC6-CC10 are increased in Table 2. It is not to replacethe synchronized structure with a single structure.
TABLE IS YNCHRONIZATION RATE (4-4,1-4 COUPLED MODEL ).

TABLE IIS YNCHRONIZATION RATE (5-5,1-5 COUPLED MODEL ).

V. C ONCLUSIONSIn this study, we have proposed system models using twosymmetrical structures and asymmetrical structures that arecoupled by a resistor. We have investigated the synchronization
phenomena by changing the topology and the influence of thetopology of the system. By the computer simulations, we haveobserved synchronization phenomena in each model. As aresult, network with coupled symmetrical structures (model (a)and model (b)) has a strong influence on one circuit thatare coupled between the structures. However, network withcoupled asymmetric structures (model (a’) and model (b’))
has the influence on all of the circuits in topology. Therefore,
I consider that it is not to replace the synchronized structurewith a single structure.
In the future works, we will investigate the synchronization
phenomena by changing the topology. Moreover, we willobserve the influence for the topology of the network.
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