2006 RISP International Workshop
on Nonlinear Circuits and Signal Processing (NCSP'06)
Waikiki Beach Marriott, Honolulu,Hawaii, USA, March 3-5, 2006

Synchronization in Chaotic Circuits Coupled by Mutual Inductors
Yuta KOMATSU† , Yoko UWATE† and Yoshifumi NISHIO††Department of Electrical and Electronic Engineering
Tokushima University
2-1 Minami-Josanjima, Tokushima, Japan
Phone:+81-88-656-7470, Fax:+81-88-656-7471
Email: {komatsu, uwate, nishio}@ee.tokushima-u.ac.jp

Abstract
In this research, synchronization phenomena observed fromsimple chaotic circuits coupled by mutual inductors are investigated. A simple three-dimensional autonomous circuit
is considered as a chaotic subcircuit. By carrying out circuit
experiments and computer calculations for two or three subcircuits cases, various kinds of synchronization phenomenaof chaos are observed.
1. Introduction
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]-[3].
In this research, synchronization phenomena observed fromsimple chaotic circuits coupled by mutual inductors are investigated. A simple three-dimensional autonomous circuit isconsidered as a chaotic subcircuit. This subcircuit is a symmetric version of the chaotic circuit proposed by Inaba et al.
[4]. They used ideal piecewise linear model of diodes in [4],but in this research the i − v characteristics of the nonlinear
resistor consisting of diodes are approximated by a smoothfunction. This is more real than piecewise linear approximation in the sense that every real elements in the natural fieldare not piecewise linear. By carrying out computer calculations for two or three subcircuits cases, various kinds of synchronization phenomena of chaos are observed. In the two
subcircuits case, in-phase and anti-phase synchronization areobserved. Moreover in-phase and three-phase synchronization are observed in the three subcircuit case.

Figure 1: Circuit model.
2. Circuit Model
Fig. 1 shows the circuit model. In the circuit, N identical chaotic circuits are coupled by mutual inductors. Eachchaotic subcircuit is a symmetric version of the circuit model
proposed by Inaba et al. [4]. It consists of three memory elements, one linear negative resistor and one nonlinear resistor,
which is realized by connecting some diodes, and is one of thesimplest autonomous chaotic circuits. First, we approximate
the i − v characteristics of the nonlinear resistor consisting ofdiodes by the following function.
√vd (ik ) = 9 rd ik .
(1)
By changing the variables and parameters,rp
Ct = (L1 − M ) C τ , a =,L1 − M√b = 8 rd a, Ik = a b xk , ik = a b yk , vk = b zk ,- 293 -“·”=L1 − MMd, α=, β = r a, γ =,dτ
L2L1(2)(a)(a)(b)(b)(c)

(c)

(d)

(d)

(e)

(e)

(1)

(2)

(3)

(4)

(1)

(2)

(3)

(4)

Figure 2: In-phase synchronization of two chaotic circuits. γ =Figure 3: Anti-phase synchronization of two chaotic circuits. γ =0.01 (M = 2.63mH). (a) β = 0.24 (r = 585Ω). (b) β = 0.27 (r =708Ω). (c) β = 0.29 (r = 753Ω). (d) β = 0.293 (r = 758Ω). (e)β= 0.3 (r = 771Ω). (1) x1 vs. x2 . (2) x1 vs. z1 . (3) x2 vs. z2 . (4)Circuit experimental results. I1 vs. I2 . 10 V/div.

0.4 (M = 102mH). (a) β = 0.24 (r = 571Ω). (b) β = 0.25 (r = 646Ω).
(c) β = 0.265 (r = 655Ω). (d) β = 0.273 (r = 689Ω). (e) β = 0.28
(r = 706Ω). (1) x1 vs. x2 . (2) x1 vs. z1 . (3) x2 vs. z2 . (4) Circuit
experimental results. I1 vs. I2 . 10 V/div.

the circuit equations are normalized and described as

Computer calculated results and the corresponding circuit
experimental results are shown in Figs. 2 and 3. Please notethat in-phase and anti-phase synchronizations coexist, but
subcircuits are easy to be synchronized at in-phase for lower
γ and at anti-phase for higher γ. Fig. 2(a) shows in-phasesynchronization of one-periodic attractors. As the parameterβ increases, one-periodic attractor bifurcates to chaotic attractor via period-doubling route keeping in-phase synchronization. When the attractor is chaotic, two subcircuits are
not synchronized completely, but are almost synchronized asshown in Figs. 2(d) and (e). We call the situation as quasisynchronization of chaos.
While, Fig. 3 shows anti-phase synchronization. The
anti-phase synchronization undergoes complicated bifurcation route explained as follows. One-periodic attractor withsymmetry on the x1 − x2 plane in Fig. 3(a) bifurcates to two
one-periodic attractors with asymmetry as Fig. 3(b). Eachasymmetric attractor bifurcates to torus via Hopf bifurcation
as Fig. 3(c) and to chaos via torus breakdown as Fig. 3(d).
Two asymmetric chaos collide each other and one chaotic attractor with symmetry is generated via symmetry-recoveringcrisis as Fig. 3(e). Namely, the anti-phase synchronization exhibits symmetry breaking and recovering and torus via Hopfbifurcation.

ẋk = β(xk + yk ) − zk−NXγ{β(xj + yj ) − zj }1 + (N − 1)γ j=1(3)ẏk = α{β(xk + yk ) − zk − f (yk )}żk = xk + ykwhere(k = 1, 2, · · · , N )√f (yk ) = 9 yk .

(4)

In the following circuit experiments, the values of the inductors and the capacitor in each chaotic subcircuit are fixed
and those values are measured as L1 = 208 mH ± 0.5%, L2= 10.22 mH ± 1.6% and C = 33.58 nF ± 1.4%. While in
the following computer calculations, the parameter value corresponding to the inductors is fixed as α = 18.0 and (3) iscalculated by using the Runge-Kutta method with step size∆t = 0.001.
3. Two Subcircuits CaseIn this section, we consider the case of N = 2, namely two
chaotic subcircuits are coupled by a mutual inductor.

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(a)

(a)

(b)

(b)

(c)

(c)

(1)

(2)

(3)

(1)

(2)

(3)

(4)

(4)

Figure 4: In-phase synchronization of three chaotic circuits (com-Figure 5: Three-phase synchronization of three chaotic circuits

puter calculated results). γ = 0.01. (a) β = 0.29. (b) β = 0.293. (c)β= 0.3. (1) x1 vs. x2 . (2) x1 vs. x3 . (3) x1 vs. z1 . (4) Time waveformfor β = 0.3.

(computer calculated results). γ = 0.4. (a) β = 0.2. (b) β = 0.25.
(c) β = 0.262. (1) x1 vs. x2 . (2) x1 vs. x3 . (3) x1 vs. z1 . (4) Time
waveform for β = 0.262.

4. Three Subcircuits Case

Moreover, we made one-parameter bifurcation diagrams of
the Poincare map as shown in Figs. 8 and 9. Fig. 8 shows thatin-phase synchronization exhibits logistic chaos. We can also
observe the generation of five-periodic window around β =0.302. It is clear that the synchronization becomes weak forlarger β value. From Fig. 9 we can confirm the bifurcationroute of three-phase synchronization, namely bifurcation of
the one-periodic solution to torus around β = 0.239, the generation of periodic solution around β = 0.241 and the generation of chaotic solution for β values more than about 0.258.

In this section, we consider the case of N = 3, namely
three chaotic subcircuits are coupled by three mutual inductors.
Computer calculated results are shown in Figs. 4 and 5. In
figures we omit attractors on the xi −zi plane for i = 2, 3, because the shape is almost same as the attractors on the x1 −z1plane. As well as two subcircuits case, in-phase synchronization in Fig. 4 is confirmed to undergo period-doubling routeto chaos. While, three-phase synchronization in Fig. 5 undergoes torus breakdown.
In order to investigate the bifurcation route in detail, weconsider the Poincare map of each synchronization mode.
The poincare section is defined as z1 = 0, x1 < 0. The projections of the Poincare map onto x1 − x2 plane of in-phase andthree-phase synchronization are shown in Figs. 6 and 7, re(a)(b)(c)
spectively. From Fig. 6 we can see that one-periodic attractor(a) bifurcates to two-periodic (b), four-periodic (c), two-bandchaos (d) and one-band chaos (e). This is well-known period
doubling route to chaos. As β increase further, chaos grows
as (f). From Fig. 7 we can see the bifurcation route of threephase synchronization via torus breakdown. One-periodic at(d)(e)(f)
tractor (a) bifurcates torus (b) via Hopf bifurcation. As β∼
increases, torus grows as (c)(d). At β = 0.258, chaos (e) ap- Figure 6: Poincare map of the in-phase synchronization. Horizonpears. As β increases further, Poincare map has thickness (f) tal: x1 . Vertical: x2 . γ = 0.01. (a) β = 0.27. (b) β = 0.285. (c) β =and area-expanding chaos is considered to be generated.
0.29. (d) β = 0.291. (e) β = 0.293. (f) β = 0.3.

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0-0.1
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(a)

(b)

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(c)

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(d)

(f)

(e)

Figure 7: Poincare map of the three-phase synchronization. Hori-0.21

0.22

0.23

0.24

0.25

0.26

Figure 9: Bifurcation diagram of the Poincare map for the threephase synchronization. γ = 0.4. Horizontal: β. Vertical: x1 .

zontal: x1 . Vertical: x2 . γ = 0.4. (a) β = 0.2. (b) β = 0.24. (c) β =0.25. (d) β = 0.255. (e) β = 0.258. (f) β = 0.262.

5. Conclusions
In this study, we investigated quasi-synchronization phenomena observed from simple chaotic circuits coupled bymutual inductors. By carrying out circuit experiments andcomputer calculations for two or three subcircuits case, weconfirmed that various quasi-synchronization phenomena ofchaos were stably observed.
In the future, we investigate phenomena observed from thecase N ≥ 4. Moreover we investigate synchronization phenomena observed from two identical chaotic circuits are coupled by a nonlinear mutual inductor.

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(a)

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References
0.5[1] Y. Nishio and A. Ushida, “Quasi-Synchronization Phenomena in Chaotic Circuits Coupled by One Resistor,”
IEICE Trans. Fundamentals, vol.43, no.6, June 1996.

0.4
0.3
0.2

[2] M. Wada, Y. Nishio, and A. Ushida, “Analysis of Bifurcation Phenomena on Two Chaotic Circuits Coupled by anInductor,” IEICE Trans. Fundamentals, vol.E80-A, no.5,
pp.869-875, May 1997.

0.1

(b)

0-0.1
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0.24

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Figure 8: Bifurcation diagram of the Poincare map for the in-phasesynchronization. γ = 0.01. (a) Horizontal: β. Vertical: x1 . (b)Horizontal: β. Vertical: x1 − x2 .

[3] M. Kojima, M. Yamauchi, Y. Nishuo and A. Ushida,“Simultaneous Asynchronous Oscillation of Chaos andTorus in Coupled Chaotic Circuits,” NOLTA, vol.2,
pp.567-570, Sep 2000.
[4] N. Inaba and S. Mori, “Chaotic Phenomena in Circuitswith a Linear Negative Resistance and an Ideal Diode,”in Proc. MWSCAS’88, Aug. 1988, pp. 211-214.

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