Chaos Propagation in a Ring of Coupled Circuits
Generating Chaotic and Three-Periodic Attractors
Yoko Uwate

Yoshifumi Nishio

Dept. of Electrical and Electronic Engineering,
Tokushima University
2-1 Minami-Josanjima, Tokushima, Japan
Email: uwate@ee.tokushima-u.ac.jp

Dept. of Electrical and Electronic Engineering,
Tokushima University
2-1 Minami-Josanjima, Tokushima, Japan
Email: nishio@ee.tokushima-u.ac.jp

Abstract— In this study, we investigate synchronization phenomena observed in coupled chaotic circuits as a ring topology
when one chaotic circuit is set to generate chaos and the othersare set to generate three-periodic attractors. By using computer
simulations, we observe chaos propagation which is that threeperiodic attractors are affected chaos and they are changing
between chaotic and three-periodic attractors.

II. C IRCUIT M ODELIn this study, we consider a ring of the chaotic circuits asshown in Fig. 1. In the circuit system, two adjacent chaotic
circuits are coupled by a resistor R. By using this coupling
method, two adjacent circuits tend to synchronize with inphase state.

I. I NTRODUCTIONRSynchronization and the related bifurcation of chaotic systems are good models to describe various higher-dimensional
nonlinear phenomena in the field of natural science. In particular, the breakdown of chaos synchronization has attracted
many researchers’ attentions and their mechanisms have been
gradually made clear [1]-[8]. However, a lot of phenomena
around chaos synchronization are still veiled as well as othernonlinear circuits problems. Hence, in order to understand andexploit such phenomena, it is important to discover them, tomodel them, and to investigate them.
On the other hand, it is important to investigate synchronization phenomena of coupled circuits under a difficult situation
for the circuits. For examples, there are some frustrations in thecircuit systems such as the coupling structure, frequency errors
and external stimuli. Setou et al. have observed interesting
synchronization phenomena when van der Pol oscillators withdifferent oscillation frequencies are coupled by a resistor as astar topology [9]. We have investigated a ring consisting of vander Pol oscillators with different oscillation frequencies. By
using computer simulations, oscillation death, independent oscillation and double-mode oscillation were observed [10][11].
We consider that if coupled circuits with more different characteristics are coupled, the coupled circuits feel the frustrations
more strongly.
In this study, we investigate synchronization phenomena
observed in coupled chaotic circuits as a ring topology when
one chaotic circuit is set to generate chaos and the others areset to generate three-periodic attractors. By using computer
simulations, we observe chaos propagation which is that threeperiodic attractors are affected chaos and they are changing
between chaotic and three-periodic attractors. Furthermore, theeffect of the coupling strength between the adjacent chaoticcircuits is also investigated.

978-1-4577-1729-1/12/$26.00 ©2012 IEEE.

i 11i 21v1

CR
L2

L1

i 12

i 22v2CR
L2

L1i 2NvN

C

L2

L1
vd -r

vd -r-ri 1Nvd(a) Ring of coupled chaotic circuits.
3rd

2nd
P

1st

4th
P

5thPchaosPP

6thCPP
10th
9thPPchaos
7th

8th

P Periodic attractor (βp)
C Chaotic attractor (βc)(b) Conceptual circuit model.
Fig. 1.

Coupled chaotic circuits as a ring topology.

First, the i−v characteristics of the diodes are approximatedby two-segment piecewise-linear functions asrd V  V vd (i2 ) =(1)i2 +  − i2 −  .
2rdrd
By changing the variables and parameters,CC
V xk , i2k =V yk , vk = V zk , t = L1 Cτ
i1k =L1L1
√
CL1 CL11 L1, δ = rd, β=, γ=, (2)α=rL1L2R CL2643

1st(a)

2nd

3rd

4th

5th

6th

7th

8th

9th

10th

6th

7th

8th

9th

10th

τ=0(b) τ=2e+4(c) τ=3e+5(d) τ=5e+5(a) γ = 0.002.
1st(a)

2nd

3rd

4th

5th

τ=0(b) τ=2e+4(c) τ=1e+5(d) τ=2e+5(b) γ = 0.008.
Fig. 2.

the normalized circuit equations are given as⎧dxk⎪⎪= αxk + zk⎪⎪dτ⎪⎪⎪⎪dyk⎪⎪= zk − f (yk )⎪⎨dτdzk⎪⎪= −xk − βyk − γ(zk − zk+1 )⎪⎪dτ⎪⎪⎪⎪⎪⎪⎪⎩(k = 1, 2, · · · , N )Examples of chaos propagation.

In this study, we consider that the total number of coupledchaotic circuits is 10. One chaotic circuit generates chaotic
attractor and the other circuits generate the three-periodic
attractors (see. Fig. 1 (b)).

A. Attractors

wherex(N +1) = 0,
1 1  1 f (yk ) =yk +  − yk −  .
2δδIII. S YNCHRONIZATION P HENOMENA

(3)

(4)
(5)

In these equations, γ is the coupling strength, N is thenumber of coupled chaotic circuits and α denotes the bifurcation parameter. By changing α, the obtained attractors arecontrolled. We define αc to generate the chaotic attractors, andαp is defined to generate the three-periodic attractors.
For the computer simulations, we calculate Eq. (3) using thefourth-order Runge-Kutta method with step size h = 0.005.
We set the parameters of this circuit model as follows; αc =0.382, αp = 0.413, β = 3.0 and δ = 470.0.

Figure 2 shows chaos propagation phenomena when 6th
chaotic circuit generates chaos at the initial states. First, thechaos attractor of 6th circuit propagates to both side of theneighbor circuits. Then, three-periodic attractors change tochaos attractors and its chaos behavior shifts to the neighbor
circuits by rotation in this circuit system. Furthermore, we canconfirm that the speed of chaos propagation becomes faster by
setting larger value of the coupling strength γ.
B. Phase Difference
The phase differences between 1st and 3rd (φ13 ) / 1st and6th (φ16 ) chaotic circuits are calculated in Figs. 3, 4. In thecase of φ13 , 1st and 3rd chaotic circuits is located far from the
6th chaotic circuits generating chaos attractor. Namely, the φ13shows 0 degree (in-phase state) a certain time in simulations
when the coupling strength is fixed with γ = 0.002 (see. Fig. 3(a)). By increasing the value of γ, 1st and 3rd chaotic circuits

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are not synchronized as shown in Fig. 3 (a), (b). Finally, Fig.3(d) shows that chaos synchronization with in-phase state isobserved a certain time.
In the case of φ16 , 6th chaotic circuit generates chaotic
attractor. Then, the phase difference φ16 switches from 0 to180 degree as shown in Fig. 4 (a). Namely, 1st and 6th chaotic
circuits does not synchronize. By changing the γ, we can seethat the phase difference φ16 changes with simulation time.
180160140120
φ13 100
80
60
40200(b) γ=0.004(c) γ=0.006

180160140120100
φ16 8060
40200(b) γ=0.004

180160140120
φ13 10080
60
40200(a) γ=0.002

180160140120100
φ16 806040200180160140120100
φ16 806040200(a) γ=0.002

180160140120
φ13 100
806040200180160140120100
φ16 8060
402000500

1000

1500

2000

2500

(d) γ=0.008

Fig. 4.

500

1000

1500

2000

2500

(d) γ=0.008

Fig. 3.

Phase differences (1st and 6th chaotic circuits).

x1-x2x2-x3(c) γ=0.006

180160140120
φ13 100
80
60
40
20003000

τpx10-x13000

τp

Phase differences (1st and 3rd chaotic circuits).

(a) γ=0.002

τ(b) γ=0.004

τ(c) γ=0.006

τ(d) γ=0.008

τ

x1-x2
x2-x3

C. Difference of Voltages between Neighbors Circuits
In order to investigate the chaos propagation, we calculate
the difference of voltages between neighbors circuits. Figure 5 shows the simulation results. The horizontal axis is
the computer simulation time τ and the vertical axis shows
the difference of xk and xk+1 . In these figures, the white
region means in-phase state and the black region means that
two adjacent circuits has some amount of the phase shift.
The chaotic attractor occurs the phase shift and its phaseshift propagates to all circuits in this system. The speed ofphase shift propagation becomes faster by setting larger valueof the coupling strength γ. By increasing the value of γ,
we can see the chaos synchronization of whole circuits (see.
Fig.5 (d)).

x10-x1x1-x2x2-x3

x10-x1

x1-x2
x2-x3

D. Synchronization State
Finally, Fig. 6 shows one example of synchronization states
when the coupling strength γ is changed from 0.002 to 0.008.
The horizontal axis is the Poincare time and the vertical axis
shows the value of xk when the solution of the first circuit
crosses to the Poincare section (x1 < −1.2, z1 = 0). Namely,
if the circuits generates chaotic attractor, the curve of xk

645

x10-x1Fig. 5.

Time wave forms.

oscillates. While if the circuits generates the periodic attractor,
the curve of xk becomes constant.
From these results, we can see that the region of chaotic
behavior increases with the value of the coupling strength γ.

10987IV. C ONCLUSION

65432
10500

1000

1500

τp20002500

3000(a) γ = 0.002.

10987
6

In this study, we have investigated synchronization phenomena observed in coupled chaotic circuits as a ring topology
when one chaotic circuit is set to generate chaos and the others
are set to generate three-periodic attractors. By using computer
simulations, we have observed chaos propagation which is thatthree-periodic attractors are affected chaos and they are changing between chaotic and three-periodic attractors. Furthermore,
the effect of the coupling strength between the adjacent chaoticcircuits was also investigated.
As our future works, we develop an efficient method to analyze the obtained synchronization phenomena of the proposedsystem. Considering the other types of chaotic circuits and theother coupling methods are also important subjects for us.

5R EFERENCES

432
10500

1000

1500

τp2000

2500

3000

2000

2500

3000

2000

2500

3000(b) γ = 0.004.

1098765432
10500

1000

1500

τp(c) γ = 0.006.

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765432
10500

1000

1500

τp(d) γ = 0.008.
Fig. 6.

Synchronization state.

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