Synchronization of Small Oscillations inCross-Coupled Chaotic Circuits
Yumiko Uchitani and Yoshifumi Nishio
Department of Electrical and Electronic Engineering, Tokushima University
Email: {uchitani, nishio}@ee.tokushima-u.ac.jp

Abstract— In this study, we investigate synchronization statesof small oscillations observed from simple two chaotic circuits
cross-coupled by inductors by both computer simulations andcircuit experiments. We confirm that there are many different
synchronization states coexist.

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 consider that it isvery important to investigate the phenomena related withchaos synchronization to realize future engineering application
utilizing chaos.
In our past studies, two simple chaotic circuits cross-coupled
by inductors are investigated. As a result, we could observe
interesting state transition phenomena [3][4]. In particular,
we noticed that small oscillations between transitions frompositive region to negative region tend to be synchronized inanti-phase in spite of the synchronization of the transitions.
In this study, we investigate different synchronization statescorresponding to anti-phase synchronizations of small oscillations between the transitions. We can see that the quadraturephase synchronization in [4] is one of many different synchronization states. The computer simulation results are verified
by real circuit experiments and we also carry out computer
simulations for Chua’s circuit in order to confirm some kindsof universality of the phenomenon.
II. C IRCUIT M ODEL
Figure 1 shows the circuit model [3]. In the circuit, twoShinriki-Mori chaotic circuits [5][6] are cross-coupled via
inductors L2 .
By using the following variables and the parameters,⎧⎪
L1 i1k
L1 i2k⎪⎪ xk =, wk =,⎪⎪C2 VC2 V⎪⎪⎪
v1kv2k⎪⎪⎪⎨ yk = V , zk = V , t = L1 C2 τ,(1)⎪L1L1C2⎪⎪, β=G, γ =g,α=⎪⎪C1C2C2⎪⎪⎪⎪dL⎪⎪⎩ δ = 1 , “·”=L2dτ978-1-4244-3828-0/09/$25.00 ©2009 IEEEFig. 1.

Circuit model.

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:⎧(yk − zk > 1)
⎨ yk − zk − 10(|yk − zk | ≤ 1)
f (yk − zk ) =(3)⎩yk − zk + 1(yk − zk < −1)

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2.5

III. S TATE T RANSITION P HENOMENON0

From the circuit in Fig. 1, we could observe interestingstate transition phenomenon [3][4]. A typical example ofthe observed phenomena is shown in Fig. 2. Figure 2(a) is
computer simulated results obtained by integrating Eq. (2) with
the Runge-Kutta method and Fig. 2(b) is the corresponding
circuit experimental results. In this state, the two circuits
exhibited chaos but almost synchronized in in-phase in thesense that the attractor was almost in the quadrant I or III on
the y1 vs y2 (or v11 vs v21 ) plane. The behaviors of the circuits
are very interesting because the solutions on the yi vs 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. Whenone circuit switches to/from the positive region from/to thenegative region in this way, the other follows the transition
after a few instants.

-2.5

0

2.5 -2.5

0

2.5

2.0

y1 0
-2.0
2.0

y2 0
-2.0
0

τ

1200

τ

1200

(1a)

(1b)
2.5

0

-2.5

0

2.5 -2.5

0

2.5

2.0

y1 0-2.0
2.0

2.5

y2 0
-2.0
0

0

(2a)
-2.5

02.5 -2.5(a1)02.5(a2)

2.0

y1 0-2.0
2.0

y2 0-2.0
2.0(a3)

τ1200(2b)State transition phenomenon around (1) anti-phase synchronization and (2) quadrature-phase synchronization. (a) Computercalculated results. α = 2.5, β = 4.0, γ = 0.1, and δ =0.0014. (b) Circuit experimental results. L1 = 9.93mH, L2 = 1.2H,
C1 =32.8nF, C2 =49.5nF, and g=495mS.

Fig. 3.

(b1)

(b2)

(b3)

Fig. 2. State transition phenomenon around in-phase synchronization.
(a) Computer calculated results. α = 2.5, β = 4.0, γ = 0.1, andδ = 0.0014. (b) Circuit experimental results. L1 = 9.93mH, L2 =800mH, C1 =32.8nF, and C2 =49.5nF, and g=683mS. (a1) y1 vs z1 .
(a2) y1 vs y2 . (a3) Time waveform. (b1) v11 vs v12 . (b2) v11 vs v21 .
(b3) Time waveform v11 and v21 .

2.0

y1 0-2.0
2.0

By changing initial conditions, similar transition phenomena can be observed around anti-phase synchronization andquadrature-phase synchronization as shown in Fig. 3.
Figure 4 shows the magnification of the time waveform ofy. We can see that the switching timing of y1 and y2 are almost
synchronized in in-phase, however, small oscillations between
the transitions are synchronized in anti-phase.
IV. I NVESTIGATION OF SMALL OSCILLATIONSIn this study, we pay our attentions on the synchronizationof the small oscillations between the transitions. If all the statescorresponding to the anti-phase synchronizations of the small
oscillations can be stable, we can observe as many different
synchronization states as the number of the small oscillations
between the transitions instead of just three (in-phase, antiphase, quadrature-phase) synchronizations.
Figures 5 shows some examples of different synchronization
states obtained for the same parameter values. In Fig. 5(a), they2 0-2.00τ 250

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

Fig. 4.

first peak of y2 after its transition from negative to positive issynchronized to the second bottom of y1 . The first peak of y2of Fig. 5(b) and (c) are synchronized to the 9th and the 15thbottoms of y1 , respectively.
As we expected, we could observe many different types ofthe synchronization states, although we do not say that wecan always observe exactly same number of the states as thenumber of the small oscillations between the transition.
Figure 6 shows two examples of the different synchronization states observed from real circuit experiments.

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2.5

2.0

y10
-2.0
2.00
y20
-2.0

(1)

-2.50

2.5

2.5

5000

τ

500

τ

500

0
-2.0
2.00
y200
-2.0

2.5

(2)2.52.0y10-2.0
2.00y2(3)

τ2.0

y1

-2.50-2.5

2.50-2.00(a)(b)

Some examples of different synchronization states. (computer simulation results). α = 2.0, β = 4.0, γ = 0.1, and δ = 0.0014.
(1) First peak of y2 is synchronized to the second bottom of y1 . (2) Ninth bottom of y1 . (3) Fifteenth bottom of y1 . (a) Attractor on y1 vs y2
plane. (b) Timewaveform.
Fig. 5.

V. C HUA’ S CIRCUIT CASE
In order to show that the synchronization in the previous
section is not special only for the chaotic circuit in Fig. 1, wecarried out similar computer simulations for the well-known
Chua’s circuit (Figs. 7 and 8).
The circuit equations can be written after an appropriate
nondimensional variables and parameters as follows:⎧ẋ1 = −βz1
⎪
⎪⎪
⎪
ẋ
⎪
2 = −βz2⎪⎪⎪ẏ⎪
1 = α{−y1 + z1 − w1 − f (y1 )}⎪⎨ẏ2 = α{−y2 + z2 − w2 − f (y2 )}ż1 = x1 + y1 − z1 + w2⎪⎪⎪⎪ż2 = x2 + y2 − z2 + w1⎪⎪⎪⎪ẇ1 = βδ(y1 − z2 )⎪⎪⎩ẇ2 = βδ(y2 − z1 )VI. C ONCLUSIONSIn this study, we have investigated different synchronization
states corresponding to anti-phase synchronizations of small
oscillations between the transitions observed from simple
chaotic circuits cross-coupled by inductors. By computer simulations and circuit experiments, we confirmed that the circuits
generated many different synchronization states characterized
by the number of the small oscillations between the transitions.
R EFERENCES(4)

where f are the nonlinear functions corresponding to the v − icharacteristics of the Chua diode and can be described by 3segment piecewise-linear functions.
Figures 9 shows some examples of computer calculatedresults. We could observe similar synchronization phenomena
from the coupled Chua’s circuits, namely several different synchronization states characterized by anti-phase synchronizationof the small oscillations between the transitions.

[1] G. Abramson,V.M. Kenkre and A.R. Bishop, “Analytic Solutions
for Nonlinear Waves in Coupled Reacting Systems,” Physica A,
vol. 305, no. 3-4, pp. 427-436, 2002.
[2] 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.
[3] Y. Uchitani, R. Imabayashi and Y. Nishio, “State Transition
Phenomenon in Cross-Coupled Chaotic Circuits,” Proc. ofNOLTA’07, pp. 397-400, Sep. 2007.
[4] Y. Uchitani and Y. Nishio, “Investigation of State Transition Phenomena in Cross-Coupled Chaotic Circuits,” Proc. of ISCAS’08,pp. 2394-2397, May. 2008.
[5] 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.
[6] 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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3.0

3.0

y1

00

-3.0
3.0

y2-3.0-3.0

3.0000

τ

600

τ

60

τ

60

(1)
3.0

3.0

y10
-3.0
3.000y2-3.0-3.0

3.0(2)
3.0

3.0y100-3.0
3.0

y2-3.03.0(3)0-3.00(a)(b)

Fig. 9. Synchronization obtained from coupled Chua’s circuits. (computer simulation results). α = 15.6, β = 50.0, δ = 0.0003, δ = 0.0014,
m0 , m1 . (1) First peak of y2 is synchronized to the fourth bottom of y1 . (2) Eighth bottom of y1 . (3) Twelfth bottom of y1 . (a) Attractor
on y1 vs y2 plane. (b) Timewaveform.

GiiR
C0v0CvL(a)(b)

Fig. 7.

Chua’s circuit.

iR(a)(b)EFig. 6. Two examples of different synchronization states. (circuitexperimental results). L1 = 10.56mH, L2 = 1.28H, C1 =33.3nF,
C2 =49.5nF and g=515mS. (a) Attractor on v11 vs v12 plane. Horizontal and vertical: 5 V/div. (b) Time waveform v11 and v21 . Horizontal
0.5 ms/div and vertical: 5 V/div.

m0m1

Fig. 8.

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v0-Ev − i characteristics of Chua diode.