Investigation of State Transition Phenomena inCross-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

Abstract— Studies on chaos synchronization in coupled chaoticcircuits are extensively carried out in various fields. In thisstudy, two simple chaotic circuits cross-coupled by inductors
are investigated. Interesting state transition phenomenon aroundchaos synchronization is observed by computer simulations andcircuit experiments.

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]-[9]. We consider that it isvery important to investigate the phenomena related withchaos synchronization to realize future engineering application
utilizing chaos. We have reported an interesting state transition
phenomenon observed in simple coupled chaotic circuits [10].
In this study, we investigate the state transition phenomena
in detail by computer simulations and circuit experiments.
II. C IRCUIT M ODEL

Figure 1 shows the circuit model [10]. In the circuit, two
Shinriki-Mori chaotic circuits [11][12] are cross-coupled via
inductors L2 .
By using the following variables and the parameters,⎧
L1 i11
v11v12⎪⎪x1 =, y1 =, z1 =,
⎪
⎪
⎪
CVVV
⎪
⎪
 2⎪
⎪
v21v22L1 i21⎪⎪⎪x2 =, y2 =, z2 =,⎪⎪CVVV2⎪⎪⎨
L1 i12L1 i22(1), w2 =,w1 =
⎪CVC
⎪
22 V⎪
⎪⎪
⎪
⎪
⎪ α = C2 , β = L1 G, γ = L1 g,⎪⎪⎪
C1C2C2⎪⎪⎪
⎪L1⎪⎩ δ=, t = L1 C2 τ,L2
the normalized circuit equations are given as follows.
⎧ẋ1 = z1
⎪
⎪
⎪
⎪
⎪
⎪
⎪ ẋ2 = z2⎪⎪⎪
⎪ ẏ1 = α{γy1 − w1 − βf (y1 − z1 )}⎪⎪⎪⎪⎪⎨ ẏ2 = α{γy2 − w2 − βf (y2 − z2 )}⎪ż1 = β f (y1 − z1 ) + w2 − x1⎪⎪⎪⎪⎪⎪ż2 = β f (y2 − z2 ) + w1 − x2⎪⎪⎪⎪⎪⎪ẇ1 = δ(y1 − z2 )⎪⎪⎪⎩ẇ2 = δ(y2 − z1 )(2)

where f are the nonlinear functions corresponding to the v −i characteristics of the nonlinear resistors consisting of thediodes and are assumed to be described by the following 3segment piecewise-linear functions:⎧(y1 − z1 > 1)⎪⎨ y1 − z1 − 1f (y1 − z1 ) =(3)0(|y1 − z1 | ≤ 1)⎪⎩(y1 − z1 < −1)
y1 − z 1 + 1

Fig. 1.

⎧⎪⎨ y2 − z2 − 10f (y2 − z2 ) =⎪⎩y2 − z 2 + 1

Circuit model.

978-1-4244-1684-4/08/$25.00 ©2008 IEEE2394(y2 − z2 > 1)(|y2 − z2 | ≤ 1)(y2 − z2 < −1).

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III. S TATE T RANSITION P HENOMENON
From the circuit in Fig. 1, we could observe interesting statetransition phenomenon [10]. An example of the phenomenon
is shown in Figs. 2 and 3. Figure 2 shows computer calculated
results of Eq. (2) with the Runge-Kutta method. While, Fig. 3
shows the corresponding circuit experimental result. The twocircuits exhibited chaos but almost synchronized in in-phasein the sense that the attractor was almost in the quadrant Ior III on the y1 − y2 plane. The behaviors of the circuits arevery interesting because the solutions on the yi − zi planes
seem to be attracted to the ﬁxed points located at around(yi , zi )=(±1.2, 0). However, after converging to the ﬁxed
points the solution abruptly moves toward the other ﬁxed point.
When one circuit switches to/from the positive region from/tothe negative region in this way, the other follows the transition
after a few instants.

(a)(c)

2.5Fig. 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.

0-2.5(b)0(a)

2.5 -2.50(b)2.5

3.0

y1 0-3.0
3.0

y2 0-3.00(c)τ300Fig. 2.
State transition phenomenon around in-phase synchronization(computer calculated result). α = 1.5, β = 5.0, γ = 0.2, and δ = 0.005.
(a) Attractor on y1 − z1 plane. (b) Attractor on y1 − y2 plane. (c) Time
waveform.

IV. D ETAILED INVESTIGATIONIn this study, we investigate the above-mentioned statetransition phenomena in detail. First, we vary the coupling
parameter δ and observe the phenomena. The results are shownin Fig. 4. We can see that the sojourn time between thetransitions becomes shorter as increasing the coupling parameter δ. The corresponding circuit experimental results areshown in Fig. 5. We can say that this interesting phenomenon
can be observed from both computer calculations and circuit
experiments.
Next, Fig. 6 shows how the variables change around thetransitions. The variables x and z correspond to the current andthe voltage of the LC resonator in the original chaotic circuit.
Hence, they oscillate around zero and its amplitude decays astime goes. On the other hand, the variable y corresponds tothe voltage across the capacitor in parallel with the negative
resistor and the value tends to go toward the equilibrium points
located around ±1.2.
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.
Moreover, Fig. 7 shows the magniﬁcation of the time
waveform of y. We can see that the switching timing of y1 andy2 are synchronized in in-phase, however, small oscillations
between the transitions are synchronized in anti-phase.
V. A NTI - PHASE STATE TRANSITION
Next, similar state transition can be observed around antiphase synchronization as shown in Figs. 8 and 9. Similar to thein-phase, the sojourn time between the transitions decreases asthe coupling δ increases.
VI. Q UADRATURE - PHASE STATE TRANSITION
We also observed an interesting state transition aroundquadrature-phase, namely 90 degrees. The computer calculated
results are shown in Fig. 10 and the corresponding circuit
experimental results are shown in Fig. 11. In this state, thesolution moves to the quadrant I, II, III, and IV on the y1 − y2plane in order as shown in the ﬁgures.
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 their generations.

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2.50

-2.5

0

2.5 -2.5

0

2.5

2.0

y1 0

(1a)

-2.0
2.0

(1b)

y2 0
-2.00

τ

(a)

1000

2.5

0

-2.5

0

2.5 -2.5

0

2.5

2.0

(1c)

y1 0
-2.0
2.0

y2 0
-2.00

τ

(b)

1000

2.5

0

(2a)
-2.50

2.5 -2.50

(2b)

2.5

2.0

y1 0
-2.0
2.0

y2 0
-2.00(c)τ1000

Fig. 4.
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, (c) 0.008.

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] P. Ashwin, J. Buescu and I. Stewart, “Bubbling of Attractors
and Synchronization of Chaotic Oscillators,” Phys. Lett. A,193,pp. 126-139, 1994.
[3] E. Ott and J.C. Sommerer, “Blowout Bifurcations: the Occurrence of Riddled Basins and On-Off Intermittency,” Phys. Lett.
A,199, pp. 39-47, 1994.
[4] Y. Nishio and A. Ushida, “Spatio-Temporal Chaos in Simple
Coupled Chaotic Circuits,” IEEE Trans. Circuits Syst. I, vol. 42,
no. 10, pp. 678-686, 1995.
[5] N.F. Rul’kov and M.M. Sushchik, “Robustness of Synchronized
Chaotic Oscillations,” Int. J. Bifurcation and Chaos, vol. 7, no. 3,
pp. 625-643, 1997.
[6] M. Wada, Y. Nishio and A. Ushida, “Analysis of Bifurcation
Phenomena in Two Chaotic Circuits Coupled by an Inductor,”
IEICE Trans. Fundamentals, vol. E80-A, no. 5, pp. 869-875,
1997.
[7] 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.

(2c)
Fig. 5. State transition phenomenon for different coupling parameters (circuitexperimental result). L1 = 9.93mH, C1 =32.8nF, and C2 =49.5nF. (1) L2 =800mH and g=626mS. (2) L2 = 1.2H and g=495mS. (a) Attractor on v11 −v12 plane. Horizontal and vertical: 1 V/div. (b) Attractor on v11 − v21 plane.
Horizontal and vertical: 1 V/div. (c) Time waveform v11 and v21 . Horizontal
0.5 ms/div for (1) and 1.0 ms/div for (2), and vertical: 2 V/div.

[8] G. Abramson,V.M. Kenkre and A.R. Bishop, “Analytic Solutions
for Nonlinear Waves in Coupled Reacting Systems,” Physica
A: Statistical Mecanics and its Applications, vol. 305, no. 3-4,pp. 427-436, 2002.
[9] I. Belykh, M. Hasler, M. Lauret and H. Nijmeijer, “Synchronization and Graph Topology,” Int. J. Bifurcation and Chaos,vol. 15, no. 11, pp. 3423-3433, 2005.
[10] Y. Uchitani, R. Imabayashi and Y. Nishio, “State Transition
Phenomenon in Cross-Coupled Chaotic Circuits,” Proceedingsof NOLTA’07, pp. 397-400, 2007.
[11] M. Shinriki, M. Yamamoto and S. Mori, “Multimode Oscillations in a Modiﬁed van der Pol Oscillator Containing a Positive
Nonlinear Conductance,” Proc. IEEE, vol. 69, pp. 394-395,
1981.
[12] 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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2.0

x1 0-2.0
2.0

y1 0-2.0
2.0

z1 0-2.0
2.0

x2 0-2.0
2.0(a)y2 0(b)-2.0
2.0

z2 0-2.00

τ1000

Fig. 6. Time waveforms around transition. α = 1.5, β = 5.0, γ = 0.2, andδ = 0.005.

Fig. 9. Anti-phase state transition (circuit experimental result). L1 = 9.93mH,
L2 = 1.2H, C1 =32.8nF, C2 =49.5nF, and g=495mS. (a) Attractor on v11 −v21
plane. Horizontal and vertical: 1 V/div. (b) Time waveform v11 and v21 .
Horizontal 1.0 ms/div and vertical: 2 V/div.

2.0

y1 0

2.5-2.0
2.0

y2 0-2.00

τ0250Fig. 7. Magniﬁcation of the time waveform around transition. α = 1.5,
β = 5.0, γ = 0.2, and δ = 0.003.

-2.50

2.5 -2.50

2.5

2.0

y1 0-2.0
2.02.5y2 0-2.00(a)

0

τ

1000

τ

1000

2.5-2.50

2.5 -2.5

0

2.50

2.0

y1 0-2.0
2.0-2.5y2 00

2.5 -2.50

2.5

2.0-2.00

τ

(a)

y1 0

1000

-2.0
2.0

2.5

y2 0
-2.0
0(b)

0-2.5

0

2.5 -2.50

Fig. 10. Quadrature-phase state transition (computer calculated result). α =2.0, β = 4.0, γ = 0.1,and (a) δ = 0.0004, (b) δ = 0.0014.

2.5

2.0

y1 0-2.0
2.0

y2 0-2.00

τ

(b)

1000

2.50-2.502.5 -2.50

2.5(a)

2.0y1(b)0-2.0
2.0y2 0-2.00(c)τ1000

Fig. 8. Anti-phase state transition (computer calculated result). α = 2.0,
β = 4.0, γ = 0.1,and (a) δ = 0.0005, (b) δ = 0.0008, (c) δ = 0.0014.

Fig. 11. Quadrature-phase state transition (circuit experimental result). L1= 9.93mH, L2 = 1.2H, C1 =32.8nF, C2 =49.5nF, and g=495mS. (a) Attractor
on v11 − v21 plane. Horizontal and vertical: 1 V/div. (b) Time waveform v11and v21 . Horizontal 1.0 ms/div and vertical: 2 V/div.

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