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

II. S YSTEM MODEL

Abstract—In this study, we investigate chaos propagationin coupled chaotic circuits with multi-ring combination. We
compare the different coupling combination. These models arecoupled chaotic circuits when one circuit is set to generatechaotic attractor and the other circuits are set to generate
three-periodic attractors. By using computer simulations, wehave observed that the chaotic attractor is propagated to theother circuits. The three-periodic attractors are affected fromthe chaotic attractors by changing the coupling combinationand increasing the coupling strength. Moreover, we confirm thatchaos propagation of the network without hub is faster than thenetwork with hub.

The chaotic circuit is shown in Fig. 1. This circuit consistsof a negative resistor, two inductors, a capacitor and dualdirectional diodes.
We propose the triple-ring and quad-ring combination system (see Fig. 2). Each model is used chaotic circuit coupled by
the resistors. In each model, central circuit is set to generatechaotic attractor and the other circuits are set to generate
three-periodic attractors. One ring is consisted by five circuits.
Additionally, multi-ring in this study is consisted by three orfour rings.

I. I NTRODUCTION
Propagation in the network have attracted a great deal ofattention from various fields. It is important to investigate
chaos propagation under some difficult situations for thecircuits. For example of some difficult situations, network isbriefly given external stimulation and frustration is occurred in
the network. Furthermore, viral infection and the traffic jamof the transportation network are mentioned as an example
of propagation in the network. Therefore we consider that itis necessary to investigate that behavior of unlike the others
influence the whole system. In the biology, we can preventthe unknown virus spreading if we comprehend the way ofviral infections. Additionally, it is applicable to the fields of
medical science and biology and so on. However, there are notmany studies of large-scale network of continuous-time real
physical systems such as electrical circuits. [1]-[5]In the previous studies, chaos propagation have been investigated in ladder, ring or simple network system of coupledchaotic circuits [6]-[8]. In this simple network, the threeperiodic attractors are affected from the chaotic attractors when
the coupling strength and the number of edge are increasing.
Moreover, we have observed that the process to chaos ischanged in each model.
In this study, we investigate chaos propagation in coupledchaotic circuits with multi-ring combination. We propose
a multi-ring combination system using of 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 by increasing the coupling strength.
Moreover, by changing the method of connecting the edges,
we investigate chaos propagation in the entire system.

978-1-5090-1570-2/16/$31.00 ©2016 IEEE

i2

i1
L1

L2

Cnv

C-rvd

Fig. 1. Chaotic circuit.

7

Pn

Three-periodic attractors (αp )CnChaotic attractor (αc )7216112836131611513

1541010836

19491415

1415(a)Triple-ring(b)Quad-ring

Fig. 2. Multi-ring combination.

The circuit equations of this circuit are described as follows:

65APCCAS 2016


di

L1 = v + ri

dt




di
L2 = v − vddtdv
 C dt = −i1 − i2 .

III. T RIPLE RING
A. System patterns(1)

In this section, we explain the method of connecting theedges from ring to ring in triple ring model. We define thenumber of edge from central circuit to ring or from ring toring is fixed to only one. In this triple ring model, we use
16 circuits. Central circuit is set to generate chaotic attractor
and the other 15 circuits are set to generate three-periodic
attractors. Furthermore, we change the method of connecting
the edges. Figure 3 shows all system patterns in triple ringmodel under this condition.

The characteristic of nonlinear resistance is described asfollows:()rdV
Vvd =i2 +− i2 −.
(2)
2rdrd
By changing the variables and parameters,
√
√
CL1 C

V x n , i2 =V yn , v = V z n i1 =
L
L
12
√
√

C
L1
L1 Cα=r, β=, δ = rd,
LLL

122
√


√

1L1
 γ=, t = L1 C2 τ,R C

7

16

1128

3

61316

11(3)

1459104

5910
1415

14

15(a).hub-0-A(b).hub-0-B

77(4)

216

112
83

613161158

3

61

where α represents the chaos degree. f (yi ) is described asfollows:()11
1f (yn ) =yn +− yn −.
(5)
2
δδ

13

8

3

6

1

The normalized circuit equations are given as follows:dxn
= αxn + zn

dτ

 dyn
= zn − f (yn )
dτ



dzn


= −xn − βyn .
dτ

72

13

1
4

5
910(c).hub-1

910141541415(d).hub-2Fig. 3. Triple-ring combination.

In the proposed multi-ring system, the circuits are connected toonly adjacent circuits by the resistors. The normalized circuit
equations of the system are given as follows:dxn
= αxn + zn

dτ


 dyn = zn − f (yn )
dτ
(6)
∑

dzn


= −xn − βyn −γ(zi − z − j)dτj∈ Sn(n = 1, 2, · · ·, N ).

We divide the coupling combination into four patterns(see Fig. 3). Each pattern is different of the connection edgeor the number of hubs. We define hub as the circuit whichconnected by four edges. In our proposed system, there aresome hubs in each system. The hub-1 model has only one
hub such as 2nd circuit (see Fig. 3(c)). The the 2nd circuit isconnected from 1st, 3rd, 6th, 7th circuit. The hub-2 modelhas two hubs such as the 2nd circuit and the 7th circuit(see Fig. 3(d)). The 2nd circuit is connected from 1st, 3rd,6th, 7th circuit and the 7th circuits connected from 2nd, 8th,11th and 12th circuit.
Additionally, we divide the hub-0 model into two patterns
according to distance from central circuit to the farthest circuit.
For example, the shortest route is passed from 1st, 2nd, 3rd,4th, 9th, 8th, 7th, 12th, 13th to 14th in the hub-0-A model(see Fig. 4(a)). The hub-0-A model is passed 9 circuits. Incontrast, the shortest route is passed from 1st, 2nd, 3rd, 8th,

In Eq. (6), N is the number of coupled chaotic circuitsand γ is the coupling strength. We define αc to generate
the chaotic attractor and αp is defined to generate the threeperiodic attractors. For the computer simulations, we calculate
Eq. (6) using the fourth-order Runge-Kutta method with thestep size h = 0.01. In this study, we set the parameters of thesystem as αc = 0.460, αp = 0.412, β = 3.0 and δ = 470.0.

978-1-5090-1570-2/16/$31.00 ©2016 IEEE

66

APCCAS 2016

7th, 12th, 13th to 14th in the hub-0-B model (see Fig. 4(b)).
The hub-0-B model is passed 7 circuits.

1611772283613161113

1158

36

chaotic circuit (see Fig. 6). The 1st circuit is chaotic attractor
and from 2nd to 16th circuits are three-periodic attractor.
Moreover, from the result of the minimum couplingstrength, we investigate the chaos propagation. When we fixed
the coupling strength as γ = 0.0060, all circuits are propagatedthe chaotic attractor of 1st chaotic circuit in the hub-0-A andthe hub-0-B model (see Fig. 7(a), (b)). On the other hand, allcircuits are not propagated chaotic attractor in the hub-1 andthe hub-2 model (see Fig. 7 (c), (d)). In the model with hub,when we fixed the coupling strength as γ = 0.0060, somecircuit are not propagated the chaotic attractor. For example,
the 10th and 11th circuits are not propagated in hub-1 model.
we consider that the the result is affected by initial value. If theinitial value change, the 10th and 11th circuits are propagatedand some other circuits are not propagated in hub-1 model.

45109101415

491415(b).hub-0-B(a).hub-0-A

Fig. 4. Route from central circuit.

IV. Q UAD RING

B. Minimum coupling strength

A. System patterns
We investigate quad ring model with hub or without hub.
In quad ring model, we use 21 circuits. Central circuit is set
to generate chaotic attractor and the other 20 circuits are set
to generate three-periodic attractors. We define the numberof edge from central circuit to ring or from ring to ring isfixed to only one. In quad-ring model, we define hub as thecircuit which connected by four edges. There are some hubsin each system. We divide the coupling combination into 4
patterns(see Fig. 8). Each pattern is different of the connectionedge and the number of hub. For example, the hub-3 modelhas 3 hubs such as the 2nd, 7th and 12th circuit.

In this section, we investigate the minimum couplingstrength when three-periodic attractors are changed to chaos
attractor. When we increasing the coupling strength, we compare the each model. The simulation results are summarized
in Fig 5. We define the attractor as a chaos attractor, if
the attractor becames over three-periodic attractor. First, we
compare the model with hub and without hub. The couplingstrength of the model without hub is smaller than the modelwith hub. Next, we compare the hub-0-A model and the hub0-B model. The distance form central circuit to the farthest
circuit is different in each model. The Coupling strength of
the hub-0-B model is smaller than the hub-0-B model.
Accordingly, we confirm that the model without hub is thatchaos attaractor propagates faster than the model with hub.
From these results, we consider that the hub weakens weight ofchaos from central circuit. Because the hub circuit connects to
many circuit than other circuit, we consider that the influenceof chaos is slacked off by hub circuit.

Coupling strength(γ)

B. Minimum coupling strength
In this section, we investigate the minimum couplingstrength when three-periodic attractors are changed to chaosattractor. When we changing the coupling strength, we compare the each model. The simulation results are summarized
in Fig 9. We define the attractor as a chaos attractor, if theattractor becames over three-periodic attractor. The minimum
coupling strength of hub-0 model is smaller than the modelwith hub. Additionally, as the number of the hub increase, sominimum coupling strength is larger.
From the result, when we increasing the number of hub,chaos propagation of the model with many hubs is moredifficult than the model without hub.
V. C ONCLUSIONSIn this study, we have investigated chaos propagation incoupled chaotic circuits as our proposed system. By the computer simulations, we have observed that the chaotic attractor
is propagated to the other circuits. The three-periodic attractors
are affected from the chaotic attractors. Moreover, we confirm
that the model with hub weakens weight of chaos from centralcircuit. From the result, chaos propagation has been changed
over time from three-periodic attractor to chaotic attractor orfrom chaotic attractor to three-periodic attractor.
For the future works, we define the convergence time fromthree-periodic attractors to chaos in order to investigate theobtained phenomena of multi-ring system in detail.

System patterns

Fig. 5. Minimum coupling strength (triple-ring).

C. Chaos propagation
Figures 6 and 7 show chaos propagation attractors in thedouble ring combination system by increasing the couplingstrength γc . At this time, we fix the coupling strength as γ =0.0000, all circuits is not propagated the chaotic attractor of 1st978-1-5090-1570-2/16/$31.00 ©2016 IEEE67APCCAS 2016

z1z2

z3x1

z4x2

z5x3

z7

z6x4x5z8

z9x7x6z10x8x9z11

z12x11x10

z14

z13

z15x13x12

z16x14x15

z15

z16x16

Fig. 6. Chaos propagation (γ = 0.0000).

z1z2

z3

z4

z5

z6

z7

z8z9z10z11z14z13

z12(a).5x1

z1x2z2x3

z3x4

z4x5z5x6x7z7z6x8z8x9z9x11x10z10z11x13x12z12x15x14z14z13z15x16

z16(d).7-ax1

z1x2z2x3z3x4z4x5

z5x7x6z7z6x8z8x9z9x11x10z10z11x14x13x12z12z14z13x16x15

z15

z16

(f).7-cx1

z1x2z2x3

z3x4

z4x5

z5x7x6

z7

z6x8

z8x9

z9x11x10

z10

z11x14x13x12z12z14z13x16x15

z15

z16(i).9x1x2x3x4x5x7x6x8x9x10x11x12x13x14x15x16161177228361316111131415(a)hub-0(b)hub-1

7722836131611158910141511134591016814536Coupling strength(γ)

Fig. 7. Chaos propagation (γ = 0.0060).

System patterns
361459109101415Fig. 9. Minimum coupling strength (quad-ring).
41415(c)hub-2International Conference on Nonlinear Dynamics of Electronic Systems(NDES’12), pp. 66-69, Jul. 2012.
[7] 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),
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[8] 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.

(d)hub-3

Fig. 8. Quad-ring combination.

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