IEEE Workshop on Nonlinear Circuit Networks
December 11-12, 2015

Chaos Propagation and Synchronization in Coupled
Chaotic Circuits with Star-and-Ring Combination
Takahiro Chikazawa, Shogo Tamada, Yoko Uwate and Yoshifumi Nishio
Dept. of Electrical and Electronic Engineering, Tokushima University
2-1 Minami-Josanjima, Tokushima 770–8506, Japan
Email: {chikazawa,tamada,uwate,nishio}@ee.tokushima-u.ac.jpi2

i1

Abstract—In this study, we investigate chaos propagation andsynchronization phenomena in coupled chaotic circuits with starand-ring combination when one circuit is set to generate chaotic
attractor and the other circuits are set to generate three-periodic
attractors. Moreover, we observe how to propagate chaos andsynchronization phenomena by increasing the coupling strength.

L1

L2-r

Civ

Cvd

I. I NTRODUCTION
Fig. 1. Chaotic circuit.

Synchronization is one of the fundamental phenomena innature and one of the typical nonlinear phenomena. Therefore,
synchronization of coupled chaotic circuits has been interested
by many researchers not only engineering but also the physicaland biological fields [1]-[4]. In particular, it is importantto investigate synchronization phenomena of coupled circuits
under some difficult situations for the circuits. Additionally,
it is applicable to the fields of medical science and biology
and so on. As previous studies, synchronization and chaos
propagation have been investigated in the ring or ladder ofcoupled chaotic circuits [5]-[7]. However, these studies wereconsidered about the only one ring or ladder system.
In this study, we investigate chaos propagation and synchronization phenomena in coupled chaotic circuits with star-andring Combination. We propose a star-and-ring combination
system using of 6 chaotic circuits coupled by the resistors.
In this model, one circuit is set to generate chaotic attractor and the other circuits are set to generate three-periodic
attractors. First, we observe how to chaos propagation byincreasing the coupling strength. Moreover, by measuring thephase difference among all adjacent circuits, we investigate
synchronization in the entire system.

2ndPγp

3rdγc

Pγcγp

P
C 6thγcγpγcP
5thγpP

Three-periodic attractors (αp )CChaotic attractor (αc )P
4thFig. 2. Star-and-ring combination.

The circuit equations of this circuit are described as follows:di
L1 = v + ridtdiL2 = v − vddtdv
 C dt = −i1 − i2 ,II. S YSTEM MODEL
The chaotic circuit is shown in Fig. 1. This circuit consistsof a negative resistor, two inductors, a capacitor and dualdirectional diodes. We propose a star-and-ring combination
system model in Fig. 2. In this system, 6th circuit generates
chaotic attractor and the other circuits generate three-periodic
attractors. In order to investigate the chaos propagation, weuse two parameters of the coupling strength. Each circuit iscoupled by the resistor.

γpγc

1st(1)

The characteristic of nonlinear resistance is described asfollows:
rdVVvd =i2 +− i2 −.
(2)
2rdrd- 63 -z1

By changing the variables and parameters,r

√
C
L1 C


V xn , i2 =V yn , v = V zn
 i1 =

LL12r√
CL1L1 Cα=r, β=, δ = rd,
LLL
122
r

p

 γ = 1 L1 , t = L1 C2 τ,R C

z2z3z4z5z6(a)γp=0.0007

x1

z1x2

z2x3

z3x4

z4x5

z5x6z6(b)γp=0.0010x1(3)z1x2z2x3

z3x4

z4x5z5x6z6(c)γp=0.0015

x1
1stx3
3rdx4
4thx5
5thx6
6th

Fig. 4. Chaos propagation (γc = 0.0007).

The normalized circuit equations are given as follows: dxn dτ = αxn + zn dyn= zn − f (yn )dτN

X
dzn

=−x−βy−γij (zn − zn+1 )nn dτ
n=1
(n = 1, 2, · · ·, N ).

x2
2nd(4)In Eq. (4), N is the number of coupled chaotic circuits andγ is the coupling strength. f (yi ) is described as follows:11
1yn +− yn −.
(5)f (yn ) =2δδ

Figure 5 shows the relation between the phase difference
and the coupling strength when the coupling strength γc isonly conected from 6th to 5th circuit. At this time, we fix thecoupling strength as γc = 0.0007, and we increase couplingstrength γp . The phase difference shows the average among
all adjacence circuits. If all circuits are not synchronized, thephase difference shows 60◦ . We can confirm that the phasedifference is smaller and comes close to 0◦ by increasing thecoupling strength.
As a result, three-periodic attractors are affected fromchaotic attractor by increasing the coupling strength γp .

The coupling strength which connected with 6th is γc , and theothers are γp . We define αc to generate the chaotic attractor,
and αp is defined to generate the three-periodic attractors.
For the computer simulations, we calculate Eq. (4) using
the fourth-order Runge-Kutta method with the step size h =0.01. In this study, we set the parameters of the system as
αc = 0.460, αp = 0.412, β = 3.0 and δ = 470.0.

III. S IMULATION RESULTS
A. Chaos propagation
Figure 3 shows the initial state when all circuits are notconnected. We observe three-periodic attractors from 1st to5th circuits and chaotic attractor in 6th circuit.
Figure 4 shows chaos propagation attractors when thecoupling strength γc is only conected from 6th to 5th circuit.
At this time, we fix the coupling strength as γc = 0.0007,
and we increase coupling strength γp . The 5th circuit is
only propagated the chaotic attractor of 6th chaotic circuit(see. Fig. 4(a)). The chaotic attractor of 5th circuit propagatesto both side of the neighbor circuits (see. Fig. 4(b)). The statesof all circuits become the chaotic attractors (see. Fig. (c)).
z1z2z3z4z5z6(a)γp=0.0000

x1
1stx2
2ndx3
3rdx4
4thx5
5thFig. 3. Chaos propagation (γc = 0.0000).

x6
6thRelation between the phase difference and the couplingstrength in star-and-ring combination.
Fig. 5.

B. Synchronization phenomena
In this section, we confirm the variation of the phasedifference, when we change the number of edges between
6th circuit and the other circuits. Figure 6 shows all systempatterns in this star-and-ring combination by changing combination. For example, this model is connected to all circuits(see. Fig. 6(g)) and this model is only connected from 6th to
5th circuits (see. Fig. 6(a)). When we fix the coupling strength
as γc = 0.007 and γp = 0.005, the phase difference shows theaverage among all adjacence circuits in Table 1.
In order to investigate the phase difference, we consider
that the entire system easy to reach the synchronous state byincreasing the number of edge.

- 64 -2nd

2ndPP

1stPCPPPPC

5thPP

4thPP
1stPPPP(c).2-bP

4th

5thP

4th
2ndP
3rd

1stPP3rdP

C 6th6th(e).3-bP

5th2ndP
5thP

3rd

3rdC 6th(b).2-aPP
4th

1st

2ndC(d).3-aP

1st6th

P
5thP

4th

2nd3rdP[7] Y. Uwate, T. Ott and Y. Nishio,“Clustering Phenomena in Coupled
Chaotic Circuits with Different Coupling Strength” Proceedings ofEuropean Conference on Circuit Theory and Design (ECCTD’13),
DOI:10.1109/ECCTD.2013.6662220 (4 pages), Sep. 2013.

P
3rd

C 6th(a).1

1st

1st

6th

P
5th

3rd

2ndCP

4th(f).46thP
5thP(g).54thFig. 6. System patterns.

TABLE IP HASE DIFFERENCES IN ALL SYSTEMS .
patterns(a).1(b).2-a(c).2-b(d).3-a(e).3-b(f).4(g).5

phase difference[◦ ]
56.11
18.54
32.59
37.98
21.60
11.85
9.06

average phase difference[◦ ]
56.11
25.57
29.79
11.85
9.06

IV. C ONCLUSIONSIn this study, we have investigated chaos propagation andsynchronization phenomena in coupled chaotic circuits as
our proposed system. By the computer simulations, We haveobserved that the chaotic attractor is propagated to the othercircuits. The three-periodic attractors are affected from thechaotic attractors when the coupling strength increase. Moreover we consider that the phase difference close synchronous
state by increasing the number of edge.
For the future works, we develop the network model into
cubic and more complex. Considering the other network ofchaotic circuits is important subjects for us.
R EFERENCES[1] N. F. Rullckov and M. M. Sushchik, “Robustness of Synchronized
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.
[4] G. Abramson, V.M. Kenkre and A.R. Bishop, “Analytic Solutions forNonlinear Waves in Coupled Reacting Systems,” Physica A: vol. 305,
no. 3-4, pp. 427-436, 2002.
[5] Y. Uwate and Y. Nishio, “Collision between Chaotic and Periodic
Attractors in a Ring of Coupled Chaotic Circuits” Proceedings ofInternational Conference on Nonlinear Dynamics of Electronic Systems(NDES’12), pp. 66-69, Jul. 2012.
[6] Y. Uwate and Y. Nishio, “Chaos Propagation in a Ring of Coupled Circuits Generating Chaotic and Three-Periodic Attractors” Proceedings ofIEEE Asia Pacific Conference on Circuits and Systems (APCCAS’12),
pp. 643-646, Dec. 2012.

- 65 -