1ICSES 2006 I NTERNATIONAL C ONFERENCE ON S IGNALS AND E LECTRONIC S YSTEMSS EPTEMBER 17-20, 2006, Ł ÓD Ź , P OLAND

Synchronization of Chaotic Circuits withAsymmetric Coupling by Nonlinear MutualInductors

Rysunek 1:

Yuta KOMATSU, Yoko UWATE and Yoshifumi NISHIO
Department of Electrical and Electronic Engineering, Tokushima University
2-1 Minami-Josanjima, Tokushima, Japan
Email: {komatsu, uwate, nishio}@ee.tokushima-u.ac.jp
Abstract— In this research, synchronization phenomena observed from simple chaotic circuits with asymmetric couplingby nonlinear mutual inductors are investigated. A simple threedimensional autonomous circuit is considered as a chaotic
subcircuit. By carrying out computer calculations for somecases, various kinds of synchronization phenomena of chaos areobserved.

II. C IRCUIT M ODEL

I. I NTRODUCTION
Many nonlinear dynamical systems in various fields havebeen confirmed to exhibit chaotic oscillations. Recently applications of chaos to engineering systems are expected suchas chaos noise generators, control of chaos, synchronizationof chaos, and so on. In those applications, we are especiallyinterested in synchronization of chaos. Synchronization andthe related bifurcation in chaotic systems are good models
to describe various high-dimensional nonlinear phenomena in
the field of natural science and many excellent studies on synchronization of chaos have been reported. Now mechanismsof chaotic phenomena generated in low-dimensional systems
have been elucidated theoretically, and complex phenomena
observed from higher dimensional circuits represented bycoupled plural chaotic circuits attract attentions [1]-[5].
In our previous research, quasi-synchronization phenomena
observed from simple chaotic circuits coupled by linear mutualinductors were investigated [6]. We could observe various
kinds of synchronization phenomena of chaos by carrying out
computer calculations for two or three subcircuits cases. In thetwo subcircuits case, in-phase and anti-phase synchronization
were observed. Moreover in-phase and three-phase synchronization were observed in the three subcircuit case.
In this research, synchronization phenomena observed from
simple chaotic circuits with asymmetric coupling by nonlinear
mutual inductors are investigated. A simple three-dimensional
autonomous circuit is considered as a chaotic subcircuit.
This subcircuit is a symmetric version of the chaotic circuit
proposed by Inaba et al. [7]. They used an ideal piecewise
linear model of diodes in [7], but in this research the i − vcharacteristics of the nonlinear resistor consisting of diodes
are approximated by a smooth function. This is more real
than piecewise linear approximation in the sense that every
real elements in the natural field are not piecewise linear.
By carrying out computer calculations, various kinds ofsynchronization phenomena of chaos are observed.

71Fig. 1.

Circuit model.

Figure 1 shows the circuit model. In the circuit, three identical chaotic circuits are coupled asymmetrically by nonlinear
mutual inductors. Each chaotic subcircuit consists of threememory elements, one linear negative resistor and one nonlinear resistor, which is realized by connecting some diodes, andis one of the simplest autonomous chaotic circuits. First, weapproximate the i − v characteristics of the nonlinear resistor
by the following function.
√(1)
vd (ik ) = 9 rd ik .
Further, the φ − i characteristics of the nonlinear inductor inICSES 2006 I NTERNATIONAL C ONFERENCE ON S IGNALS AND E LECTRONIC S YSTEMS

whereŁ ÓD Ź , P OLAND

αα  |xk + δ| − |xk − δ|
xk + 1 −.
γγ2√f (yk ) = 9 yk .

Xk =(5)

III. C OMPUTER C ALCULATED R ESULTS

Fig. 2.

In this section, we perform computer calculations for somecases. First, we consider the case where parameter m1 is muchlarger than parameter m2 . In this case, one subcircuit is noteasy to interact with the other subcircuits. Next, the case whereparameter m1 is almost equal to parameter m2 is considered.
All subcircuits dominantly interact in this case. The parametervalues corresponding to the inductors are fixed as α = 20.0, γ= 10.0 and (4) is calculated by using the Runge-Kutta methodwith step size ∆t = 0.001.

φ − i charactristics of the nonlinear inductor.

Fig. 2 is described as following function.
φk  1
1  |φk + Φ| − |φk − Φ|
Ik =+−.
L2
L1
L2
2
By changing the variables and parameters,s rpdC8t = L1 C τ , a = rd, “·”=,L1dτrpCyk , vk = a z k ,φk = a L 1 C x k , i k = aL1rL1CL2Φ
α=, β=r, γ=, δ= √.
L0
L1L0a L1 C

A. Case for m1  m2(2)(a)(3)(b)the circuit equations are normalized and described as
1 − m m1
1 2{β(X1 + y1 ) − z1 }ẋ1 =1 + m1 − 2m1 m2
1 − m1m1 (1 − m2 )−{β(X2 + y2 ) − z2 }1 − m1
− m2 {β(X3 + y3 ) − z3 }ẋ2 =(c)(d)

1 − m m1
1 2{β(X2 + y2 ) − z2 }1 + m1 − 2m1 m2
1 − m1m1 (1 − m2 )−{β(X1 + y1 ) − z1 }1 − m1
− m2 {β(X3 + y3 ) − z3 }ẋ3 =(e)(1)

1(1 + m1 ){β(X3 + y3 ) − z3 }1 + m1 − 2m1 m2− m1 {β(X1 + y1 ) − z1 }− m1 {β(X2 + y2 ) − z2 }(k = 1, 2, 3)

(2)

(3)

(4)

In-phase synchronization for case A. m1 = 0.2. m2 = 0.02.
δ = 0.5. (a) β = 0.15. (b) β = 0.18. (c) β = 0.181. (d) β = 0.2. (e)β = 0.23. (1) x1 vs. z1 . (2) x3 vs. z3 . (3) x1 vs. x2 . (4) x1 vs. x3 .

Fig. 3.

ẏk = α{β(Xk + yk ) − zk − f (yk )}żk = Xk + yk

We set the parameter values m1 = 0.2 and m2 = 0.02 inthis case. Subcircuit 3 is not easy to interact with subcircuits
1 and 2. Therefore subcircuit 3 becomes asynchronous to theothers.

(4)

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Figure 3 shows the in-phase synchronization and the asynchronous state. From Figs. 3(1) and (3), we can confirm thatthe attractors observed from subcircuit 1 and 2 bifurcate tochaotic attractors keeping in-phase synchronization (a)-(d).
While, the attractor observed from subcircuit 3 bifurcates tochaotic attractor via period-doubling route Figs. 3(2)(a)-(e). InICSES 2006 I NTERNATIONAL C ONFERENCE ON S IGNALS AND E LECTRONIC S YSTEMSŁ ÓD Ź , P OLAND

(a)
(a)
(b)

(c)

(b)

(d)
(1)

(2)(3)(4)

Fig. 5. Anti-phase synchronization for case A. m1 = 0.2. m2 = 0.02. δ =0.5. (a) β = 0.02. (b) β = 0.18. (c) β = 0.2. (d) β = 0.23. (1) x1 vs. z1 . (2)x3 vs. z3 . (3) x1 vs. x2 . (4) x1 vs. x3 .

(c)Fig. 4. Bifurcation diagram for the in-phase synchronization. m 1 = 0.2. m2= 0.02. δ = 0.5. (a) Horizontal: β. Vertical: x1 . (b) Horizontal: β. Vertical:
x3 . (c) Horizontal: β. Vertical: x1 − x2 .

order to investigate the bifurcation route in detail, we madeone-parameter bifurcation diagrams. The poincare section isdefined as z1 = 0, x1 < 0. Figure 4 shows the bifurcation
diagram for in-phase synchronization. From these figures, wecan confirm the bifurcation route via period-doubling. We canalso confirm that breakdown of chaos synchronization aroundβ = 0.223 from Fig. 4(c).
Next, we also investigate the anti-phase synchronization bychanging the initial values. Figures 5(a) and (b) shows theanti-phase synchronization for subcircuit 1 and 2. While, theattractors observed from subcircuit 3 become asynchronous
state. Figure 6 shows bifurcation diagram for anti-phasesynchronization. As parameter β increases, the anti-phasesynchronization becomes unstable and the in-phase synchronization is observed around β = 0.2 as Fig. 5(3)(c) and Fig. 6.
B. Case for m1 ≈ m2
In this case, we set the parameter values m1 = 0.2
and m2 = 0.19. Therefore, subcircuit 3 is easy to interact
subcircuit 1 and 2.
First, we investigate the in-phase synchronization. Figure 7
shows that all subcircuits are synchronized at in-phase. How-73

ever as β increases, the in-phase synchronization becomes
unstable and breaks down (c),(d).
Next, the three-phase synchronization is observed by changing the initial values. Figure 8 shows the three-phase synchronization. From this figure, we can confirm that each
one-periodic attractor (a) bifurcates to torus (b), and as βincreases, the torus bifurcates to chaos (c) and the chaos
grows as (d). Figure 9 shows bifurcation diagram of the threephase synchronization. We can confirm that the bifurcationroute of the three-phase synchronization from Fig. 9, namely
bifurcation of the one-periodic solution to torus around β =0.203, the generation of chaotic solution for β values morethan 0.215 and the generation of periodic solution around β= 0.225.
IV. C ONCLUSIONSIn this research, we investigated quasi-synchronization phenomena observed from simple chaotic circuits with asymmetric coupling by nonlinear mutual inductors. By carrying out
the computer calculations, we confirmed that various quasisynchronization phenomena of chaos were observed.
In the future, we investigate phenomena observed from thecase which four or more subcircuit are coupled asymmetricallyby nonlinear mutual inductors.
R EFERENCES[1] Y. Nishio and A. Ushida, “Spatio-Temporal Chaos in Simple Coupled
Chaotic Circuits,” IEEE Trans. on Circuit and Systems-I, vol. 42, no. 10,
pp. 678.686, Oct. 1995.
[2] J.A.K. Suykens, P.F. Curran and L.O. Chua, “Master-Slave Synchronization Using Dynamic Output Feedback,” International Journal ofBifurcation and Chaos, vol. 7, no. 3, pp. 671-679. 1997.
[3] M. Wada, Y. Nishio and A. Ushida, “Analysis of Bifurcation Phenomena
in Two Chaotic Circuits Coupled by an Inductor,” IEICE Transactions onFundamentals, vol. E80-A, no. 5, pp. 869-875, May 1997.

ICSES 2006 I NTERNATIONAL C ONFERENCE ON S IGNALS AND E LECTRONIC S YSTEMSŁ ÓD Ź , P OLAND

(a)
(a)
(b)

(c)

(b)

(d)
(1)

(2)

(3)

(4)

(5)
(c)
Fig. 8. Three-phase synchronization for case B. m1 = 0.2. m2 = 0.19. δ =1.5. (a) β = 0.15. (b) β = 0.2. (c) β = 0.22. (d) β = 0.24. (1) x1 vs. z1 . (2)x3 vs. z3 . (3) x1 vs. x2 . (4) x1 vs. x3 . (5) Time waveform for β = 0.22.

Fig. 6. Bifurcation diagram for the anti-phase synchronization. m 1 = 0.2.
m2 = 0.02. δ = 0.5. (a) Horizontal: β. Vertical: x1 . (b) Horizontal: β. Vertical:
x3 . (c) Horizontal: β. Vertical: x1 + x2 .

(a)Fig. 9. Bifurcation diagram for the three-phase synchronization. m 1 = 0.2.
m2 = 0.19. δ = 1.5. Horizontal: β. Vertical: x1 .

(b)[4] G.O. Zhong, C.W. Wu and L.O. Chua, “Torus-Doubling Bifurcations inFour Mutually Coupled Chua’s Circuits,” IEEE Transactions on Circuits
and Systems I, vol. 45, no. 2, pp. 186-193. 1998.
[5] I.P. Marino, V. Perez-Munuzuri and M.A. Matias, “Desynchronization
Transitions in Rings of Coupled Chaotic Oscillators,” International Journal of Bifurcation and Chaos, vol. 8, no. 8, pp. 1733-1738. 1998.
[6] Y. Komatsu, Y. Uwate and Y. Nishio, “Synchronization in Chaotic Circuits
Coupled by Mutual Inductors,” Proc. of NCSP’06, pp. 293-296, Mar.
2006.
[7] N. Inaba and S. Mori, “Chaotic Phenomena in Circuits with a Linear
Negative Resistance and an Ideal Diode,” Proc. of MWSCAS’88, Aug.
1988, pp. 211-214.

(c)

(d)
(1)

(2)

(3)

(4)

Fig. 7. In-phase synchronization for case B. m1 = 0.2. m2 = 0.19. δ = 1.5.
(a) β = 0.15. (b) β = 0.23. (c) β = 0.26. (d) β = 0.27. (1) x1 vs. z1 . (2) x3
vs. z3 . (3) x1 vs. x2 . (4) x1 vs. x3 .

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