RISP

NCSP’16

International Workshop on
Nonlinear Circuits, Communications and Signal Processing

Honolulu, Hawaii, USA, March 6-9, 2016

Chaos Propagation and Synchronizationin Coupled Chaotic Circuits as Double Ring Combination
Takahiro Chikazawa, Shogo Tamada, Yoko Uwate, and Yoshifumi Nishio†Dept. of Electrical and Electronic Engineering, Tokushima University,
2-1 Minamijosanjima, Tokushima, 770-8506 Japan
Email: {chikazawa, tamada, uwate, nishio}@ee.tokushima-u.ac.jp

Abstract

2. System model

In this study, we compare the differences between double andsingle ring model. Moreover, we investigate chaos propagation and how to propagate chaos by increasing the couplingstrength. These modeles are coupled chaotic circuits when
one circuit is set to generate chaotic attractor and the othercircuits are set to generate three-periodic attractors.

The chaotic circuit is shown in Fig. 1. This circuit consistsof a negative resistor, two inductors, a capacitor and dualdirectional diodes. We propose two system models. The double ring combination system, chaotic circuits are connected
to only 5 circuits. Moreover, other 5 circuits are connectedwith the outside (see. Fig. 2). On the other hand, the single
ring combination system, chaotic circuits are connected to direct all circuits. In this system, the circuit in the center of thesystem generates chaotic attractor and the other circuits generate three-periodic attractors. We use the 11 chaotic circuits
coupled by the resistors in each system.

1. Introduction
Synchronization is one of the fundamental phenomena innature and one of the typical nonlinear phenomena. Synchronization observed on large-scale coupled chaotic systems
have been attracted attention in various science field because
it can be regarded as models of real physical system. Chaos
synchronization has been interested by many researchers not
only engineering but also the physical and biological fields[1]∼ [3]. In particular, it is important to investigate synchronization phenomena of coupled circuits under some difficult
situations for the circuits. Additionally, it is applicable to thefields of medical science and biology and so on. As previousstudies, synchronization and chaos propagation have been investigated in the ring or star-and ring of coupled chaotic circuits [4]∼ [5].
In this study, we investigate chaos propagation and synchronization phenomena in coupled chaotic circuits with double or single ring combination. We use the 11 chaotic circuits
coupled by the resistors in each system. The double ringcombination system, chaotic circuits are connected to only
5 circuits. Moreover, other 5 circuits are connected with theoutside. On the other hand, the single ring combination system, chaotic circuits are connected to direct all circuits. In
each model, one circuit is set to generate chaotic attractor andthe other circuits are set to generate three-periodic attractors.
First, we observe chaos propagation by increasing the coupling strength. Moreover, by measuring the phase difference
among all adjacent circuits in each combinations, we investigate synchronization in the entire system.

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i1
L1-r

L2
C

Civ

vd

Figure 1: Chaotic circuit.

Figure 2: Double ring model.

In Eq. (1), N is the number of coupled chaotic circuits andγ is the coupling strength. f (yi ) is described as follows:11
1f (yn ) =yn +− yn −.
(5)2δδWe define αc to generate the chaotic attractor, and αp is defined to generate the three-periodic attractors.

Figure 3: Single ring model.
3. Simulation ResultThe circuit equations of this circuit are described as follows:di
L1 = v + ri

dt




di
L2 = v − vddt





dv



 C dt = −i1 − i2 .

(1)

We observe chaos propagation by increasing the couplingstrength. By measuring the phase difference among all adjacent circuits in each combinations, we investigate synchronization in the entire system.
Moreover, we change the number of edges between 1st circuit and the other circuits. Chaotic ciricuit connects to allcircuits in each system model (see. Fig. 4). Chaotic ciricuit
connects to onlly one circuit in each system model (see. Fig.
5).

The characteristic of nonlinear resistance is described as follows:
rdvd =2V
Vi2 +− i2 −rdrd
.

(2)

By changing the variables and parameters,r√C
L1 C
i1 =V xn , i2 =V yn , v = V zn
L1L2
r√C
L1L1 C
α=r, β=, δ = rd,
LLL122rp γ = 1 L1 , t = L1 C2 τ,R C

Figure 4: System model.

(3)

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

Figure 5: System model.

(4)
3.1 Convergence TimeIn this section, we investigate convergence time fromthree-periodic attractors to chaos attractor. When all circuits
are connected, we compare single-10 model and double-5-5- 716 -model. The simulation results are summarized in Table 1.
Convergence time of single ring model is faster than double
ring model. On the other hand, when chaotic ciricuit conected
to onlly one circuit, we compare single-1 model and double1-1 model. Convergence time of double ring model is fasterthan single ring model. Accordingly, when the number ofedges is reduced, convergence time of the double ring model
becames faster than single ring model.
Table 1: convergence time to chaos.
model
1st
2nd
3rd
4th
5th
Averagesingle-10
194654
181786219974202727186565196959double-5-5269522280897290051269123276105277140single-1
703099662670697170683756663768682093

double-1-1
395966395175463205400016451242421130Figure 7: Chaos propagetion (single ring).
As a result, when the coupling strength increase, all threeperiodic attractors are propageted the chaotic attractor of 1st
chaotic circuit in each system model.

3.2 Chaos Propagation

3.3 Synchronization Phenomena

Figure 6, shows chaos propagation attractors in the double
ring combination system by increasing the coupling strength
γ. At this time, we fix the coupling strength as γ = 0.0000,
all circuits is not propageted the chaotic attractor of 1st
chaotic circuit (see. Fig. 6(a)). When the coupling strength γ
increase to γ = 0.0010, all circuits is propageted the chaotic
attractor of 1st chaotic circuit (see. Fig. 6(b)).

In this section, we investigate the relation between thephase difference and the coupling strength in two models.
The phase difference shows the average among all adjacencecircuits.
First, we investigated the phase difference when chaotic
ciricuit conected to all circuits in each system model (see.
Fig. 4). If all circuits are not synchronized, the phase difference shows 90◦ . We can confirm that the phase differenceis smaller and comes close to 0◦ by increasing the couplingstrength.
The next, we investigated the phase difference whenchaotic ciricuit conected to onlly one circuit in each systemmodel (see. Fig. 5). If all circuits are not synchronized, thephase difference shows 90◦ . We can confirm that the phasedifference is smaller and comes close to 00◦ by increasing thecoupling strength.

Figure 6: Chaos propagetion (double ring).
Figure 7 shows chaos propagation attractors in the double
ring combination system by increasing the coupling strength
γ. At this time, we fix the coupling strength as γ = 0.0000,
all circuits is not propageted the chaotic attractor of 1st
chaotic circuit (see. Fig. 7(a)). When the coupling strength γ
increase to γ = 0.0010, all circuits is propageted the chaotic
attractor of 1st chaotic circuit (see. Fig. 7(b)). Figure 5 shows Figure 8: Relation between the phase difference and the couplingchaos propagation attractors in the single ring combination strength in double or single combination.
system. In this model, all circuits is s propageted the chaotic
attractor of 1st chaotic circuit as well as double ring combination system.

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[3] 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.
[4] Y. Uwate and Y. Nishio, “Chaos Propagation in a Ringof Coupled Circuits Generating Chaotic and ThreePeriodic Attractors,” Proceedings of IEEE Asia PacificConference on Circuits and Systems (APCCAS’12), pp.
643-646, Dec. 2012.

Figure 9: Relation between the phase difference and the couplingstrength in double or single combination.

[5] T. Chikazawa, S.Tamada, Y. Uwate and Y. Nishio,“Chaos Propagation and Synchronization in CoupledChaotic Circuits with Star-and-Ring Combination,”
Proceedings of IEEE Workshop on Nonlinear Circuit
Networks (NCN’15), pp. 63-65, Dec. 2015

As a result, when the coupling strength increase, phase difference is smaller and comes close to 0◦ in each system. On
the other hand, phase difference reach the synchronous state
in double ring combination slightly than in single ring combination.

4. Conclusion
In this study, we have investigated chaos propagation andsynchronization phenomena in coupled chaotic circuits as ourproposed two system models. By the computer simulations,
We have observed that the chaotic attractor is propagated tothe other circuits. The three-periodic attractors are affectedfrom the chaotic attractors when the coupling strength increases in each system model. Moreover we consider that thephase difference close synchronous state by increasing coupling strength in each system model.
Furthermore, we propose two difference system models,
nevertheless chaos propagation and phase difference has become a similar result. In consequence, chaos propagation andphase difference are not affected by connected each circuits.
References[1] N.F. Rullckov and M.M. Sushchik, “Robustness of Synchronized Chaotic Oscillations,” Int. J. Bifurcation andChaos, 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 byan Inductor,” IEICE Trans. Fundamentals, vol. E80-A,
no. 5, pp. 869-875, 1997.

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