2007 IEEE Workshop on Nonlinear Circuit Networks
NCN'07, Tokushima, Japan, Dec 7--8, 2007

Investigation of Synchronization Phenomena in
Cross-Coupled Chaotic Circuits
Yumiko Uchitani and Yoshifumi Nishio
Department of Electrical and Electronic Engineering, Tokushima University
2-1 Minami-Josanjima, Tokushima, 770-8506 JAPAN
Email: {uchitani, nishio}@ee.tokushima-u.ac.jp
I. I NTRODUCTION
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][2]. We have reported aninteresting state transition phenomenon observed in simple
coupled chaotic circuits [3].
In this study, we investigate the state transition phenomena
in detail by computer simulations and circuit experiments.
II. C IRCUIT M ODELẋk = zk ẏk = α{γyk − wk − βf (yk − zk )}żk = β f (yk − zk ) + wk+1 − xkẇk = δ(yk − zk+1 )(k = 1, 2),(2)where f are nonlinear functions corresponding to the v − icharacteristics of the nonlinear resistors and are described asfollows.
yk − z k − 1(yk − zk > 1)
 0(|yk − zk | ≤ 1)
f (yk − zk ) =(3) yk − z k + 1(yk − zk < −1)(k = 1, 2).
III. S TATE T RANSITION P HENOMENON

From the circuit in Fig. 1, we could observe interestingstate transition phenomenon. An example of the phenomenon
is shown in Fig. 2.
2.50-2.50

2.5 -2.50

2.5

2.0

y1 0-2.0
2.0y2 0-2.00

Fig. 1.

Circuit model.

(a)

τ1000

τ

1000

2.5

Figure 1 shows the circuit, two Shinriki-Mori chaotic circuits [4] are cross-coupled via inductors L2 . By using thefollowing variables and the parameters,
rr

C2C2


ik1 =V xk , ik2 =V wk ,

L1L1




 vk1 = V yk , vk2 = V zk ,rr(1)
C2L1L1α=, β=G, γ =g,
C1C2C2pdL δ = 1 , t = L1 C2 τ, “·”=(k = 1, 2)L2dτ
the normalized circuit equations are given as follows.

- 43 -0-2.50

2.5 -2.5

02.5

2.0

y1 0-2.0
2.0

y2 0-2.00(b)Fig. 2.
State transition phenomenon for different coupling parameters(computer calculated result). α = 1.5, β = 5.0, γ = 0.2, and δ= (a) 0.001,(b) 0.003.

IV. D ETAILED INVESTIGATION

V. A NTI - PHASE STATE TRANSITIONIn this study, we investigate the above-mentioned statetransition phenomena in detail. First, we vary the coupling
parameter δ and observe the phenomena. The results areshown in Fig. 2. We can see that the sojourn time between
the transitions becomes shorter as increasing the coupling
parameter δ.
The corresponding circuit experimental results are shown inFig. 3.

Next, similar state transition can be observed around antiphase synchronization as shown in Fig. 5. Similar to the inphase, the sojourn time between the transitions decreases as
the coupling δ increases.
2.50-2.50

2.5 -2.50

2.5

2.0

y1 0-2.0
2.0

y2 0-2.00(a)

(b)

τ1000

Fig. 5. Anti-phase state transition (computer calculated result). α = 2.0,
β = 4.0, γ = 0.1, and δ = 0.0014.

VI. Q UADRATURE - PHASE STATE TRANSITION
We also observed an interesting state transition aroundquadrature-phase, namely 90 degrees as shown in Fig. 6.
The corresponding circuit experimental results are shown inFig. 11. In this state, the solution moves to the quadrant I,II, III, and IV on the y1 − y2 plane in order as shown in thefigures.
(c)

2.5

Fig. 3.
State transition phenomenon around in-phase synchronization(circuit experimental result). L1 = 9.93mH, L2 = 800mH, C1 =32.8nF, andC2 =49.5nF, and g=683mS. (a) Attractor on v11 − v12 plane. Horizontal andvertical: 1 V/div. (b) Attractor on v11 − v21 plane. Horizontal and vertical: 1V/div. (c) Time waveform v11 and v21 . Horizontal 1.0 ms/div and vertical:
2 V/div.

When the amplitude of the oscillation of the LC resonatorapproaches to zero, the state transition occurs; namely yabruptly moves toward to the other fixed point quickly. Though
the mechanism of this transition cannot be explained very well,
we consider that this is because the coupling inductor can beregarded as a short circuit when the oscillation stops. In otherwords, the coupling inductor plays a role of a kind of switch
depending on the oscillation amplitude of the LC resonator.
Figure4 shows the magnification of the time waveform ofy. We can see that the switching timing of y1 and y2 aresynchronized in in-phase, however, small oscillations between
the transitions are synchronized in anti-phase.

0-2.502.5 -2.50

2.5

2.0y1 0-2.0
2.0

y2 0-2.00τ

1000

Fig. 6. Quadrature-phase state transition (computer calculated result). α =2.0, β = 4.0, γ = 0.1,and δ = 0.0014.

2.0

y1 0(a)-2.0
2.0

y2 0-2.00τ

250

Fig. 4. Magnification of the time waveform around transition. α = 1.5,
β = 5.0, γ = 0.2, and δ = 0.003.

- 44 -(b)Fig. 7. Quadrature-phase state transition (circuit experimental result). L 1 =9.93mH, L2 = 1.2H, C1 =32.8nF, C2 =49.5nF, and g=495mS. (a) Attractor onv11 − v21 plane. Horizontal and vertical: 1 V/div. (b) Time waveform v11
and v21 . Horizontal 1.0 ms/div and vertical: 2 V/div.

VII. C ONCLUSIONSIn this study, we have investigated interesting state transition
phenomenon observed from two Shinriki-Mori chaotic circuits
cross-coupled by inductors.
Investigating the coexistence of the states and statisticalanalysis of the observed phenomena are our important future
work as well as more detailed explanation of the mechanismof its generation.
ACKNOWLEDGMENTS
This work was partly supported by Yazaki Memorial Foundation for Science and Technology.
R EFERENCES[1] N. Platt, E.A. Spiegel and C. Tresser, “On-Off Intermittency:
A Mechanism for Bursting,” Phys. Rev. Lett., vol. 70, no. 3,pp. 279-282, 1993.
[2] 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.
[3] Y. Uchitani, R. Imabayashi and Y. Nishio, “State Transition
Phenomenon in Cross-Coupled Chaotic Circuits,” Proceedingsof NOLTA’07, pp. 397-400, 2007.
[4] N. Inaba, T. Saito and S. Mori, “Chaotic Phenomena in a Circuitwith a Negative Resistance and an Ideal Switch of Diodes,”Trans. of IEICE, vol. E70, no. 8, pp. 744–754, 1987.

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