2015 International Symposium on Nonlinear Theory and its Applications
NOLTA2015, Kowloon, Hong Kong, China, December 1-4, 2015

Synchronization and Chaos Propagation of Coupled Chaotic Circuitsin Symmetric-Asymmetric Systems
Shogo Tamada, Kenta Ago, Yoko Uwate and Yoshifumi Nishio†Departmaent of Electrical and Electronic Enginnering, Tokushima University
2-1 Minamijosanjima, Tokushima, 770-8506, Japan
Email: {tamada, kentago, uwate, nishio}@ee.tokushima-u.ac.jp
Abstract—In this study, we investigate synchronizationand chaos propagation of 5 coupled chaotic circuits in various systems. We propose a ladder system model that thecentral circuit generates chaotic attractor and the other circuits generate three-periodic attractor. We observe thatchaotic attractor of the central circuit propagates to all circuits. By measuring the phase diﬀerence between the circuits, we investigate synchronization in the entire system.
Moreover, we compare the phase diﬀerence between symmetric and asymmetric systems in the cases of adding thecoupling resistor from the ladder system.

CnFigure 1: Chaotic circuit.

1. Introduction
Synchronization of chaotic systems are good models to
describe various higher-dimensional nonlinear phenomenain the field of natural science. Therefore, synchronizationof coupled chaotic circuits has been interested by many researchers [1]-[4]. In particular, it is important to investigate
synchronization phenomena of coupled circuits under somediﬃcult situations for the circuits. In our research group,
synchronization and chaos propagation have been reported
in the ring of coupled chaotic circuits [5][6]. However,these research were considered about the only one ring system.
In this study, synchronization and chaos propagation of
coupled chaotic circuits in various systems are researched.
We propose a ladder system model of 5 chaotic circuits
coupled by the resistors. In this model, the central circuit generates chaotic attractor and the other circuits generate the three-periodic attractors. First, we show synchronization and chaos propagation in the ladder system. By
measuring the phase diﬀerence among all adjacent circuits,
we investigate synchronization in the entire system. Moreover, the symmetric and asymmetric systems obtained fromadding the coupling resistor from the ladder system, arestudied.
2. System Model
Figure 1 shows the chaotic circuit. This circuit consistsof a negative resistor, two inductors, a capacitor and dualdirectional diodes. We propose the ladder system model
as shown in Fig. 2. Each circuit is coupled by a resistor R

Figure 2: Proposed ladder system model.

which is corresponding the edge in this system. The number of the circuits in the system is set to 5. The central circuit (C3 ) generates chaotic attractor and the other circuits
generate three-periodic attractors.
The circuit equations of this circuit are described as follows:
di1

L1
= v + ridt
 L2 di2 = v − vd(1)
dtdv
C= −i1 − i2 , dt
where vd is the characteristic of the nonlinear resistor consisting of the diodes, is described as follows:)(VVrd
i2 +− i2 −.
(2)
vd =2rdrd- 97 -

By using the variables and parameters:√√
CL1 C

i=Vx,
i=Vyn , v = Vzn ,1n2LL12√√L1CL1 C
,
β=,
δ=r,
α=rd
LLL
122√√
1L1d g=, t = L1C2 τ, ” · ” =,R Cdτx４x５(a)z１x３x２x１z３z２z４x４

z５x５(3)z１(b)the normalized circuit equations are given as follows:ẋ = αx + z








ẏ = z − f (y)(4)






 ż = −x − βy,where α represents the chaos degree. f (y) can be expressed
as follows:()δ11f (y) =y+ − y−.
(5)
2
δ
δ
In the proposed ladder system, the circuits are connected
to only adjacent circuits by the resistors. The normalized
circuit equations of the system are given as follows:x˙n = αxn + zn










 y˙n = zn − f (yn )



∑




z
˙
=−x−βy−g(zn − zm ),
nnn



x３x２x１(6)m∈snwhere n represents the circuit number up to 5 in this study.
S n is the set of circuits which are directly connected to Cn .
g represents the coupling strength corresponding the coupling resistor R. For the computer simulations, we calculate
Eq. (6) using the fourth-order Runge-Kutta method withthe step size h = 0.01.

z３

z２

z４z３(c)z１(b2)

x３

x４

x５

z３z２

z１

z４

z３z２(c2)

z５z４

z３z２(c1)z４(b1)
x２

x１

z５z３

z２z４(c3)z５(c4)

Figure 3: Attractors and phase diﬀerences in the ladder system, αc = 0.460, α p = 0.412, β = 3.0 and δ = 470.0. (a)g = 0, (b) g = 0.005, (b1) 30.00◦ , (b2) 41.93◦ and (c)g = 0.010, (c1) 12.88◦ , (c2) 18.12◦ , (c3) 22.68◦ , (c4)
11.02◦ .

three-periodic attractors are aﬀected from chaotic attractor
and all circuits close to the synchronous state. In order
to investigate the synchronous state, we measure the phase
diﬀerence between the adjacent circuits.
Figure 4 shows the relation between the phase diﬀerence
and the coupling strength. In Fig. 4, the phase diﬀerenceshows the average among all adjacent circuits. If all circuits
are not synchronized, the phase diﬀerence shows 90◦ . We
confirm that the phase diﬀerence is smaller and close to 0◦
by increasing the coupling strength. Namely, all circuits
reach the synchronous state by chaos propagation.

3. Simulation Result
phase difference[° ]In this study, we fix the circuit parameters of the systemas αc = 0.460, α p = 0.412, β = 3.0 and δ = 470.0. First,
we investigate synchronization and chaos propagation in
the ladder system. Moreover, we consider synchronizationof the symmetric and asymmetric systems in the cases ofadding the edge from the ladder system.
3.1. Ladder System
Figure 3 shows some examples of the computer simulation results. Figure 3(a) shows the state when all circuits
are not connected. We can observe the state that the chaos
are propagated to the only adjacent circuits from the central chaotic circuit in a range of the coupling strength g(see Fig. 3(b)). By increasing the coupling strength g, allcoupling strength g

Figure 4: Relation between the phase diﬀerence and thecoupling strength in the ladder system.

- 98 -phase difference[° ]Table 1: Phase diﬀerences in all symmetric systems.
Edge Phase diﬀerence [◦ ]
5-A
17.89
5-B
16.15
6-A
15.02
6-B
9.57
6-C
15.04
7-A
9.35
7-B
15.32
7-C
10.05
7-D
15.32
8-A
9.39
8-B
15.64
8-C
9.41
9-A
9.27
9-B
9.67

number of edges
Figure 6: Relation between the average phase diﬀerence
and the number of edges in the symmetric systems.

the edges. However, the phase diﬀerence in 6-B is smaller
than 7-B, 7-C, 7-D and 8-B in Table 1. For this reason,
we consider that the phase diﬀerence is aﬀected from thecoupling way of each circuit.
3.3. Asymmetric System

Figure 5: Symmetric system patterns.

3.2. Symmetric System
From this section, we consider the symmetric and asymmetric systems in the cases of adding the edge from theladder system. We focus on the number of the edge and thesymmetric and asymmetric systems. In the case of the ladder system, the number of edge is 4. On the other hand,when the number of edge is 10, the system shows fullcoupled system. We consider the system in the intermediate number of the edge between 4 and 10.
In this section, the symmetric systems are considered.
Figure 5 shows the illustration of the conceivable all symmetric systems. Table 1 shows the phase diﬀerences of all
symmetric systems in the case of g = 0.01 and each systempattern corresponds to Fig. 5. The phase diﬀerence shows
the average among all adjacent circuits in Table 1. FromTable 1, the phase diﬀerence is decreased by increasing thenumber of edges. Figure 6 shows the relation between theaverage phase diﬀerence of the symmetric systems and thenumber of edges. The average phase diﬀerence close to0◦ by adding the edges. Namely, the entire system easy toreach the synchronous state by increasing the number ofIn this section, the asymmetric systems are considered.
Figure 7 shows the illustration of the all conceivable asymmetric systems. Table 2 shows the phase diﬀerences of all
symmetric systems in the case of g = 0.01 and each systempattern corresponds to Fig. 7. The phase diﬀerence shows
the average among all adjacent circuits in Table 2. Figure
8 shows the relation between the average phase diﬀerenceof the asymmetric systems and the number of edges. From
Table 2 and Fig. 8, in the asymmetric systems, the averagephase diﬀerence is not decreased by increasing the numberof the edges like the symmetric systems. We consider that
this result is aﬀected from the asymmetry of the system.
Figure 9 shows the comparison of the phase diﬀerence among the ladder system, the symmetric systems, theasymmetric systems and the full-coupled system. FromFig. 9, the entire system easy to reach the synchronous state
by increasing the number of the edges. Additionally, we
consider that synchronization and chaos propagation can- 99 -

5-A'

5-B'

6-A'

6-B'

6-C'

7-A'

7-B'

7-C'

8-A'

8-B'

8-C'

9-A'

9-B'

Figure 7: Asymmetric system patterns.

7-D'16.18phase difference[° ]Table 2: Phase diﬀerences in all asymmetric systems.
Edge Phase diﬀerence [◦ ]
5-A’
12.25
5-B’
16.28
6-A’
17.39
6-B’
13.26
6-C’
15.83
7-A’
11.43
7-B’
15.46
7-C’
11.43
7-D’
9.71
8-A’
9.52
8-B’
10.02
8-C’
11.65
9-A’
9.51
9-B’
11.62

9.46

(laddder system)

number of edges(full-coupled)

Figure 9: Comparison of the phase diﬀerence among theladder system, the symmetric systems, the asymmetric systems and the full-coupled system.

phase difference[° ]propagation speed” in the proposed system. Considering
the other types of chaotic circuits and the other couplingsystems are also important subjects for us.
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.

number of edges
Figure 8: Relation between the average phase diﬀerence
and the number of the edges in the asymmetric systems.

be achieved when the circuit generating chaotic attractor isthe hub in the system.
4. Conclusion
In this study, we have researched about synchronization and chaos propagation of 5 coupled chaotic circuitsin the symmetric and asymmetric systems. First, we proposed the ladder system model that the central circuit generates chaotic attractor and the other circuits generate threeperiodic attractor. We have observed that the three-periodicattractors are aﬀected from the chaos of the central circuit,and all circuits are synchronized by increasing the couplingstrength in the ladder system. Moreover, we compared thephase diﬀerence between the symmetric and asymmetricsystems in the cases of adding the coupling resistor fromthe ladder system. As a result, the entire system is easy toreach a state of synchronization when the central chaotic
circuit is connected to many circuits.
For the future works, we would like to investigate “chaos[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.
[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.
[5] Y. Uwate and Y. Nishio, “Collision between Chaoticand Periodic Attractors in a Ring of Coupled
Chaotic Circuits” Proceedings of International Conference on Nonlinear Dynamics of Electronic Systems (NDES’12), pp. 66-69, Jul. 2012.
[6] Y. Uwate and Y. Nishio, “Chaos Propagation in aRing of Coupled Circuits Generating Chaotic andThree-Periodic Attractors” Proceedings of IEEE Asia
Pacific Conference on Circuits and Systems (APCCAS’12), pp. 643-646, Dec. 2012.

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