2009 International Workshop on Nonlinear Circuits and Signal Processing
NCSP'09, Waikiki, Hawaii, March 1-3, 2009

Wave Propagation in a Ladder of Cross-Coupled Chaotic Circuits
Yumiko Uchitani and Yoshifumi Nishio
Tokushima University
2-1 Minami-Josanjima, Tokushima, Japan
Phone:+81-88-656-7470, FAX:+81-88-656-7471
Email: {uchitani, nishio}@ee.tokushima-u.ac.jp

basic circuit model. In this model, two simple autonomouschaotic circuits [7][8] are cross-coupled via inductors L2 .

Abstract
A large number of coupled oscillators are useful as models fora wide variety of systems in natural fields. In this study, wave
propagation generated in a ladder of cross-coupled chaotic
circuits are investigated. For the case of 20 circuits, interesting wave propagation phenomena of phase states are found.
Computer simulations show that this coupled system produces several wave propagation.
1. Introduction
Synchronization phenomena in complex systems are very
good models to describe various higher-dimensional nonlinear phenomena in the field of natural science. Studies on synchronization phenomena of coupled chaotic circuits are extensively carried out in various fields [1]. We consider that it
is very important to investigate the phenomena related withchaos synchronization to realize future engineering application utilizing chaos. Endo et al. have reported a details oftheoretical analysis and circuit experiments about some coupled oscillators as a ladder [2]. Also, wave propagation phenomenon observed from coupled chaotic circuits is also reported [3]. Yamauchi et al. have discovered interesting wave
propagation phenomena of phase states in a ladder of many
simple oscillators coupled by inductors [4].
In our past studies, two simple chaotic circuits crosscoupled by inductors are investigated. As a result, we confirmed that the circuits generated many different synchronization states.
In this study, we investigate wave propagation phenomena observed in a ladder of cross-coupled chaotic circuits byinductors. By computer simulations for the case of 20 circuits, we can find interesting wave propagation phenomenaof phase states.

Figure 1: Basic circuit model.
Typical examples of the observed phenomena are shown inFig. 2 and Fig. 3. By computer simulations and circuit experiments, we confirmed that the circuits generated many different synchronization states, by changing initial condition.
3. Ladder of Cross-Coupled Chaotic Circuits
Figure 4 shows the circuit model investigated in this study.
In the circuit, n chaotic circuits are cross-coupled via inductors L2 as a ladder. The total number of the connections of ncoupled system is expressed as n × 2 − 2 = 2(n − 1). The
normalized circuit equations are given as follows.

2. Basic Circuit [5][6]In this section, we review the phenomena observed from
simple two cross-coupled chaotic circuits. Figure 1 shows the- 93 -[First Oscillator]ẋ1 = z1ẏ1 = α{γy1 − w1 − βf (y1 − z1 )}ż
1 = β f (y1 − z1 ) + u2 − x1ẇ1 = δ(y1 − z2 − εw1 )

(1)

2.5

2.0

y1

0-2.0
2.0

00

y2-2.0

-2.5

0

2.5

τ

0

2.5

500

2.0

y1(a)

(b)

(a)

(b)

0-2.0
2.0

00

y2-2.5

0

-2.0

2.5

τ

0

2.5

500

2.0

y1

0
-2.0
2.0

0
y2-2.50(a)2.50-2.0τ

0500(b)Figure 2:
Some examples of different synchronization
states. (computer simulation results). α = 2.0, β = 4.0, γ = 0.1,and δ = 0.0014. (a) Attractor on y1 − y2 plane. (b) Timewaveform.
[Middle Oscillators] ẋk = zk ẏk = α{γyk − wk − uk − βf (yk − zk )}żk = β f (yk − zk ) + wk−1 + uk+1 − xkẇk = δ(yk − zk+1 − εwk )u̇k = δ(yk − zk−1 − εuk )(k = 2, 3, 4..., n − 1)
[Last Oscillator]
ẋn = znẏn = α{γyn − un − βf (yn − zn )}żn = β f (yn − zn ) + wn−1 − xnu̇n = δ(yn − zn−1 − εun )whererr
L1 i1k
L1 i2k
L1 i3k, wk =, uk =,xk =C2 V
C2 V
C2 V
pv2kv1k, zk =, t = L1 C2 τ,yk =VVrr
C2L1L1α=, β=G, γ =g,C1C2C2r
dC2L1, ε=R, “·”=δ=L2L1dτ(k = 1, 2, 3, ..., n).
rFigure 3: Two examples of different synchronization states. (circuit experimental results). L1 = 10.56mH, L2 = 1.28H, C1 =33.3nF,C2 =49.5nF and g=515mS. (a) Attractor on v11 −v12 plane. Horizontal and vertical: 5 V/div. (b) Time waveform v11 and v21 . Horizontal
0.5 ms/div and vertical: 5 V/div.

yk − zk − 1 (yk − zk > 1)0(|yk − zk | ≤ 1)
f (yk − zk ) =y−z+1(yk − zk < −1)kk(k = 1, 2, 3, ..., n).

(2)

(5)

4. Wave propagation phenomenon
In this study, we observe wave propagation from the 20 circuits. Figure 5 shows examples of wave propagation phenomena change as the parameter γ corresponding to the negative(3)
resistor. By changing γ, we can see the various wave propagation patterns. We cannot understand why the wave propagation happens. Investigating the mechanism of the wave
propagation is our future research.
5. Conclusions
In this study, we investigated wave propagation phenomena
observed in a ladder of chaotic circuits. By computer simulations for the case of 20 circuits, we could find interesting
wave propagation phenomena of phase states. We confirmedthat several wave propagation patterns could be observed bychanging parameter γ. The investigation of the various size
circuits and the their experiments are our future research.

(4)
The function f is nonlinear function corresponding to the Acknowledgmentsv − i characteristics of the nonlinear resistors of the diodes This work was partly supported by Yazaki Memorial Foundaand are described as follows.
tion for Science and Technology.

- 94 -Figure 4: Ladder of cross-coupled chaotic circuits.

References[1] I.P. Marino, V. Perez-Munuzuri and M.A. Matias, “Desynchronization Transitions in Rings of Coupled Chaotic Oscillators,”
Int. J. Bifurcation and Chaos, vol. 8, no. 8, pp. 1733-1738,
1998.

[8] N. Inaba, T. Saito and S. Mori, “Chaotic Phenomena in a Circuit with a Negative Resistance and an Ideal Switch of Diodes,”Trans. of IEICE, vol. E70, no. 8, pp. 744-754, 1987.

[2] T. Endo and S. Mori, “Mode Analysis of a Multimode LadderOscillator,” IEEE Trans. Circuits and Syst., vol. 23, pp. 100113, 1976.
[3] M.J. Ogorzalek, Z. Galias, A.M. Dabrowski, and W.R.
Dabrowski, “Chaotic Waves and Spatio-Temporal Patterns inLarge Arrays of Doubly-Coupled Chua’s Circuits,” ?IEEETrans. Circuits and Syst. I, vol. 42, pp. 706-714, 1995.
[4] M. Yamauchi, M. Wada, Y. Nishio and A. Ushida, “Wave Propagation of Phase States in Oscillators, Coupled by Inductors
as a Ladder,” IEICE Trans. Fundamentals, vol. E82-A, no. 11,
pp. 2592-2598, 1999.
[5] Y. Uchitani, R. Imabayashi and Y. Nishio, “State Transition
Phenomenon in Cross-Coupled Chaotic Circuits,” Proc. ofNOLTA’07, pp. 397-400, Sep. 2007.
[6] Y. Uchitani and Y. Nishio, “Investigation of State Transition
Phenomena in Cross-Coupled Chaotic Circuits,” Proc. of ISCAS’08, pp. 2394-2397, May. 2008.
[7] M. Shinriki, M. Yamamoto and S. Mori, “Multimode Oscillations in a Modified van der Pol Oscillator Containing a Positive
Nonlinear Conductance,” Proc. of IEEE, vol. 69, pp. 394-395,1981.

- 95 -y1+y2
y2+y3
y3+y4
y4+y5
y5+y6
y6+y7
y7+y8
y8+y9
y9+y10y10+y11y11+y12y12+y13y13+y14y14+y15y15+y16y16+y17y17+y18
y18+y19y19+y20(a)τ 80000

y1+y2
y2+y3
y3+y4
y4+y5
y5+y6
y6+y7
y7+y8
y8+y9
y9+y10y10+y11y11+y12y12+y13y13+y14y14+y15y15+y16y16+y17y17+y18
y18+y19y19+y20(b)τ 80000

Figure 5: Wave propagation phenomena. α = 2.0, β = 4.0, δ = 0.0014, and ε = 0.0005. (a) γ = 0.1266. (b) γ = 0.1270.

- 96 -