IEICE TRANS. FUNDAMENTALS, VOL.E84–A, NO.9 SEPTEMBER 2001

2288

PAPER

Analysis of Chaotic Phenomena in Two RC Phase Shift
Oscillators Coupled by a Diode
Yasuteru HOSOKAWA†,††a) , Student Member, Yoshifumi NISHIO†∗ ,and Akio USHIDA† , Regular Members

SUMMARY In this paper, a simple chaotic circuit using two
RC phase shift oscillators and a diode is proposed and analyzed.
By using a simpler model of the original circuit, the mechanism ofgenerating chaos is explained and the exact solutions are derived.
The exact expression of the Poincaré map and its Jacobian matrix
make it possible to conﬁrm the generation of chaos using the
Lyapunov exponents and to investigate the related bifurcation
phenomena.
chaos, RC phase shift oscillator, diode, torus,
key words:
piecewise-linear analysis

1.

Introduction

There have been many investigations on various kindsof applications of chaos. For instance, chaos communication systems [1], [2], chaos neural networks [3], [4],spatio-temporal behavior in coupled chaotic circuits
systems [5], [6] and so on. In order to design chaos
communication systems using chaos synchronization, it
is necessary to make almost identical chaotic circuits.
Only IC realization of the circuit would make it possible. Further, realization of large-scale of chaotic circuits
networks would be possible only on IC chip. Therefore,
it is considered to be important to study on IC chip design of chaotic circuits [7]–[10]. One of the most important subject on such studies is how small is the circuit,
namely how many transistors are needed to construct
one chaotic circuit. Until now, OTA is used in almost
these studies [11] for realization of negative resistor,
gyrator and so on. Hysteresis or saturation characteristics of OTA is also exploited as a part of nonlinear
elements. However, an OTA is usually realized by using several numbers of transistors. As a result, chaotic
circuits including OTAs are not so small.
In this paper, we design a new chaotic circuit using
two RC phase shift oscillators and a diode and analyze
the circuit using a simpler model. This circuit consistsof only some diodes, transistors, resistors and capacManuscript received December 18, 2000.
The authors are with the Faculty of Engineering, Tokushima University, Tokushima-shi, 770-8506 Japan.
††
The author is with the Faculty of Management and Information Science, Shikoku University, Tokushima-shi, 7711192 Japan.
∗
Presently, the author is with the Department of Communication Systems, Swiss Federal Institute of Technology
Lausanne (EPFL), Switzerland.
a) E-mail: hosokawa@ee.tokushima-u.ac.jp†

itors, namely we do not have to realize any elementsin the circuit by some circuits including OTAs. We
show the simple circuit exhibits chaos by circuit experiments. In order to make clear the mechanism ofgenerating chaos and to investigate related bifurcation,
we derive a simpler model from the original circuit. In
the simpler model transistors are modeled by linearized
ones and a coupling diode is modeled by two-segment
piecewise-linear characteristics. This makes us possible
to explain the generation of chaos and to derive exact
solutions of the circuit. It is also possible to derive theexact expression of the Poincaré map and its Jacobianmatrix. We can calculate the Lyapunov exponent using
the Jacobian matrix.
In the Sect. 2 the circuit model and circuit experimental results are shown. In the Sect. 3 we derive thesimpler model from the original circuit and show thecomputer calculated results using exact solutions of thesimpler model. We can explain the mechanism of generating chaos. In the Sect. 4 the Poincaré map is deﬁned and the Lyapunov exponent is calculated. They
conﬁrm the generation of chaos and show the detailed
bifurcation scenario. Some concluding remarks is presented in the Sect. 5.
2.

Circuit Set-up

Figure 1(a) shows a well-known simple RC phase shiftoscillator which consists of a third order RC ladder network and a feedback ampliﬁer. Figure 1(b) shows atypical wave form of the voltage observed from the RC
phase shift oscillator. In this oscillator, the amplitudeof the oscillation does not become so large, because it
is controlled by turn-on of the transistor. Hence, if weneed oscillation with larger amplitude, we have to modify the circuit.
Figure 2(a) shows a modiﬁed RC phase shift oscillator with level shift diodes. Figure 2(b) shows a typicalwave form of the voltage observed from the modiﬁed RC
phase shift oscillator with level shift diodes. We can see
the amplitude becomes larger by the level shift diodes.
Figure 3 shows our circuit model. In this circuit,
two RC phase shift oscillators with level shift diodes
are coupled by a diode D0 . In the beginning we carried out circuit experiments for a similar circuit without level shift diodes. But, we could not observe any

HOSOKAWA et al.: ANALYSIS OF CHAOTIC PHENOMENA IN TWO RC PHASE SHIFT OSCILLATORS COUPLED BY A DIODE

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(a)

Fig. 3RC-Transistor chaotic circuit.

(b)
Fig. 1 (a) RC phase shift oscillator. (b) Typical wave form.
Horizontal axis is 10 [µs/div]. Vertical axis is 0.5 [V/div]. Ro =2.00 [kΩ], R = 2.00 [kΩ], C = 4.70 [nF] and E = 15.0 [V].

(a)Fig. 4 Typical examples of attractors obtained from the circuitin Fig. 3. Horizontal axis is va1 (0.2 [V/div]) and vertical axis isvb1 (0.2 [V/div]). Rao = 6.47 [kΩ], Ra = 0.820 [kΩ], Rb = 2.00[kΩ] and C = 4.70 [nF]. (a) Rbo = 2.00 [kΩ], (b) Rbo = 1.88 [kΩ],(c) Rbo = 1.85 [kΩ], (d) Rbo = 1.58 [kΩ].
(b)Fig. 2 (a) RC phase shift oscillator with level shift diodes. (b)Typical wave form. Horizontal axis is 10 [µs/div]. Vertical axis
is [0.5V/div]. Ro = 2.00 [kΩ], R = 2.00 [kΩ], C = 4.70 [nF] andE = 15.0 [V].

chaotic oscillations. We considered that it was because
the amplitude of the oscillation of each RC phase shiftoscillator was too small to exploiting nonlinearity of thecoupling diode. That’s why we added level shift diodes
to the RC phase shift oscillators. As a result, we not
only observed chaotic oscillations but also understoodthat the nonlinearity of the transistors did not play any
roles of generating chaos. We explain about this in thenext section.

Figure 4 shows typical examples of attractors obtained by circuit experiments. We choose Rbo as a control parameter which is varied in 1.58 [kΩ] < Rbo <2.00 [kΩ]. Other parameter values are ﬁxed as Rao =6.47 [kΩ], Ra = 0.820 [kΩ], Rb = 2.00 [kΩ] and C =4.70 [nF], respectively. The BJT is type 2SC1815.
We can see periodic orbit (a), quasi-periodic orbit(b) and complex attractors being looked chaotic (c),(d).
This circuit consist of only two simple RC phaseshift oscillators and a diode. Namely, neither inductors nor negative resistors are included. Therefore, it isconsidered to be realized on the IC chip without complication.
However, it is too diﬃcult to analyze this circuit

IEICE TRANS. FUNDAMENTALS, VOL.E84–A, NO.9 SEPTEMBER 20012290

as it is, because this circuit includes many nonlinear
elements. Therefore, it is impossible to make clear themechanism of generating chaos. This means that we
can not maintain that chaos will be observed if we replace, for example, the transistor by another one. Because there is possibility that very slight something of
the element may play important role to generate chaos.
We do not consider that such a circuit is very useful for
future applications.
3.

Linearized ModelIn this section, in order to make clear the mechanism ofgenerating chaos and to investigate related bifurcation,
we derive a simple linearized model of the circuit modelin Fig. 3.
First of all, the transistors with two level-shift
diodes in the circuit are assumed to operate as ideal
linear elements as shown in Fig. 5, namely
1(vBE − VB )Rt
iC = βF iB

iB =

(1)

where βF is the forward current gain of the device. Thisapproximation is signiﬁed forward active region only.
Next, the v −i characteristics of the diode coupling
two RC phase oscillators are approximated by the following two-segment piecewise-linear function shown in
Fig. 6, 1 (v − V ) (v > V )
d
th
d
th
Rd
(2)
id =

0
(vd ≤ Vth ).
By using the above simpliﬁed transistors anddiode, the circuit equations are described as follows:

dva1
Ra C
=
dt




Rao
− 2 va1 + va2
Rao + Ra

Rao Ra βF
Rao Ra βF
va3 −VB(Rao + Ra ) Rt
(Rao + Ra ) Rt
Ra−
E + Ra id
Rao + Radva2
= va1 − 2va2 + va3
Ra C
dt


dva3
Ra
Ra
= va2 −Ra C
+ 1 va3 −
VB
dt
Rt
Rt


Rbo
dvb1
=
Rb C
− 2 vb1 + vb2
dt
Rbo + Rb−

Rbo Rb βF
Rbo Rb βF
vb3 −VB(Rbo + Rb ) Rt
(Rbo + Rb ) Rt
Rb−
E − Rb id
Rbo + Rbdvb2
= vb1 − 2vb2 + vb3
Rb C
dt


Rb
dvb3
Rb
= vb2 −
Rb C
+ 1 vb3 −
VB .
dt
Rt
Rt
−

(3)

By changing the variables and parameters,
vak = Vth xak , vbk = Vth xbk , (k = 1, 2, 3),
αa =

Rao
Rbo
, αb =,
Rao + Ra
Rbo + Rb

βa =

Ra E
Rb E
Ra
, βb =, γ=,
Rao VthRbo Vth
Rt

δ=

VB
Ra
Ra
, ε=
, ζ=
, t = Ra Cτ,
Vth
Rb
Rd

(4)

the normalized circuit equations are described by thefollowing six-dimensional piecewise-linear diﬀerential
equations.
ẋa1 = (αa − 2)xa1 + xa2 − αa γβF xa3+ αa γβF δ + αa βa + ζyd
ẋa2 = xa1 − 2xa2 + xa3
Fig. 5

BJT model.

ẋa3 = xa2 − (γ + 1)xa3 + γδẋb1 = ε(αb − 2)xb1 + εxb2 − αb γβF xb3+ αb γβF δ + αb βb ε − ζydẋb2 = ε(xb1 − 2xb2 + xb3 )ẋb3 = εxb2 − (γ + ε)xb3 + γδFig. 6

Diode model.

(5)where yd is a piecewise-linear function corresponding

HOSOKAWA et al.: ANALYSIS OF CHAOTIC PHENOMENA IN TWO RC PHASE SHIFT OSCILLATORS COUPLED BY A DIODE

2291

eλ11 τeλ12 τ


λ13 τ


e
,

fa3 (τ ) = 
λ14 τ


 σ τe e 1 cos ω1 τ 
eσ1 τ sin ω1 τto the v − i characteristics of the coupling diode and is
described as
xb1 − xa1 − 1 for xb1 − xa1 > 1
(6)
yd =
0
for xb1 − xa1 ≤ 1.
Since the circuit equations (5) are piecewise-linear,
exact solutions in each linear region can be derived. At
ﬁrst, we deﬁne two piecewise-linear regions as follows.
R1 : xb1 − xa1 > 1
R0 : xb1 − xa1 ≤ 1.

(8)

The eigenvalues in R1 are obtained numerically fromthe 6th order eigenequation of Eq. (5). On the other
hand, λa , σa ± jωa can be derived from λ − (αa − 2)−1
αa γβF 


=0
−1
λ+2
−1
(9)



0
−1
λ+γ+1 and λb , σb ± jωb can be derived from λ − ε(αb − 2)−εαb γβF 

=0

−ε
λ + 2ε
−ε



0
−ε
λ+γ+ε (10)

by virtue of the decoupling.
Next we deﬁne the equilibrium points in R1 and
R0 as




E1a1
E0a1
 E1a2  E0a2  E1a3 
 E(11)E1 = 
 and E0 =  0a3  , E1b1  E0b1 


 E
 E
1b2
0b2
E1b3
E0b3
respectively. These values are calculated by putting theright side of Eq. (5) to be equal to zero.
Then, we can describe the exact solutions in eachlinear region as follows.
R1 :Diode is ON.
x(τ ) − E1 = F(τ ) · F−1 (0) · (x(0) − E1 ),
xa1 (τ )fa1 (τ ) xa2 (τ )  fa2 (τ )  xa3 (τ )  fa3 (τ ) x(τ ) =  , F(τ ) = , xb1 (τ )  fb1 (τ )  x (τ )  f (τ ) b2b2
xb3 (τ )fb3 (τ )dfa3 (τ )+ (γ + 1)fa3 (τ ),dτfa1 (τ ) =dfa2 (τ )+ 2fa2 (τ ) − fa3 (τ ).
dτ(7)Namely, R1 corresponds to the region where the coupling diode is ON, while R0 corresponds to OFF. Notethat in R0 the circuit equations are completely decoupled into two three-dimensional equations.
We calculate the eigenvalues in each region from
Eq. (5). The eigenvalues in each region are describedas follows.
R1 : λ11 , λ12 , λ13 , λ14 , σ1 ± jω1
R0 : λa , σa ± jωa , λb , σb ± jωb .

fa2 (τ ) =(12)

We omit the description of fbk (k=1,2,3), because it iscomplicated enough to write here in spite of the solutions of linear algebraic equations.
R0 :Diode is OFF.
xa (τ ) − E0a = Ga (τ ) · Ga −1 (0) · (xa (0) − E0a ),xb (τ ) − E0b = Gb (τ ) · Gb −1 (0) · (xb (0) − E0b ),
xa1 (τ )xb1 (τ )xa (τ ) =  xa2 (τ )  , xb (τ ) =  xb2 (τ )  ,xa3 (τ )xb3 (τ )




E0a1
E0b1
E0a =  E0a2  , E0b =  E0b2  ,
E0a3
E0b3




ga1 (τ )gb1 (τ )Ga (τ ) =  ga2 (τ )  , Gb (τ ) =  gb2 (τ )  ,ga3 (τ )gb3 (τ )eλa τ
ga3 (τ ) =  eσa τ cos ωa τ  ,eσa τ sin ωa τ
ga2 (τ ) =dga3 (τ )+ (γ + 1)ga3 (τ ),dτdga2 (τ )+ 2ga2 (τ ) − ga3 (τ ),
dτ


eλb τ
gb3 (τ ) =  eσb τ cos ωb τ  ,eσb τ sin ωb τ

ga1 (τ ) =gb2 (τ ) =1 dgb3 (τ ) γ + 1+
gb3 (τ ),ε dτεgb1 (τ ) =1 dgb2 (τ )+ 2gb2 (τ ) − gb3 (τ ).
ε dτ

(13)

Figure 7(1) shows computer calculated results ofthe exact solutions in Eqs. (12) and (13). We choose αbas the control parameter and other parameters are ﬁxed
as αa = 0.80, βa = 5.71, βb = 3.21, γ = 0.267, δ = 3.00,

IEICE TRANS. FUNDAMENTALS, VOL.E84–A, NO.9 SEPTEMBER 2001

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Fig. 7 Computer calculated results. (1) Attractors. (2) Poincaré map. αa = 0.80,
βa = 5.71, βb = 3.21, γ = 0.267, δ = 3.00, ε = 0.670, ζ = 60.0, and βF = 100. (a) αb =0.700, (b) αb = 0.750, (c) αb = 0.815, (d) αb = 0.817, (e) αb = 0.825, (f) αb = 0.832,(g) αb = 0.848, (h) αb = 0.860, (i) αb = 0.865, (j) αb = 0.880, (k) αb = 0.902.

ε = 0.670, ζ = 60.0, and βF = 100. We can observe almost same attractors and the same bifurcation scenario
as those observed from circuit experiments. Namely,
our linearized model does not lose important featuresof the original circuit. This means that the RC phaseshift oscillators in the original circuit behave as simple
divergently oscillating parts and that only the nonlinearity of the coupling diode controls the amplitude. In

HOSOKAWA et al.: ANALYSIS OF CHAOTIC PHENOMENA IN TWO RC PHASE SHIFT OSCILLATORS COUPLED BY A DIODE

2293

other words, the RC phase oscillators play a role of expanding, while the coupling diode plays a role of folding.
These two features, expanding and folding, are known
as the essence of generating chaos. From the result we
can now maintain that our circuit model is enough to
be robust to generate chaos, because the RC phase shiftoscillators do not have to possess any special characteristics, but they have only to oscillate divergently.
This result also encourages us to establish a designing method of chaos-generating circuits in a simple
way, for example just to replace RC phase oscillatorsby another types of oscillators. Such designing methodof chaos-generating circuits would contribute to realize
engineering applications of chaos and our most important future research.
4.

Poincaré Map

In order to conﬁrm the generation of chaos and to investigate bifurcation scenario, we derive the Poincarémap.
Let us deﬁne the following subspace
S = S1 ∩ S2

(14)

where
S1 : xb1 − xa1 = 1
S2 : {ε(αb − 2)xb1 + εxb2 − αb γβF xb3+αb γβF δ + αb βb ε} − {(αa − 2)xa1+xa2 − αa γβF xa3 + αa γβF δ + αa βa }
> 0.

(15)

where τ1 is given by using the ﬁrst and fourth rows of
Eq. (17). The solution hits S again at x2 =(Xa12 , Xa22 ,Xa32 , Xa12 + 1, Xb22 , Xb32 ) when τ = τ1 + τ2 . x2 isgiven by
Xa12 − E0a1
 Xa22 − E0a2 
Xa32 − E0a3


Xa11 − E0a1
−1
= Ga (τ2 ) · Ga (0) ·  Xa21 − E0a2 
Xa31 − E0a3
Xa12 + 1 − E0b1
 Xb22 − E0b2 
Xb32 − E0b3
The subspace S1 corresponds to the boundary condition between R1 and R0 , while the subspace S2 corresponds to the condition ẋb1 − ẋa1 > 0. Namely, S
corresponds to the transitional condition from R0 to
R1 .
Let us consider the solution starting form an initialpoint on S. The solution returns back to S again after
wandering R1 and R0 as shown in Fig. 8. Hence, we
can derive the Poincaré map as follows.
T : S → S, x0 → T(x0 )follows using the exact solutions in Eqs. (12) and (13).
Suppose that the solution starting from x0 =(Xa10 ,Xa20 , Xa30 , Xa10 + 1, Xb20 , Xb30 ) when τ = 0 hits S1and enters R0 at x1 =(Xa11 , Xa21 , Xa31 , Xa11 +1, Xb21 ,Xb31 ) when τ = τ1 . In this case, x1 is given by
Xa11 − E1a1
 Xa21 − E1a2  Xa31 − E1a3  Xa11 + 1 − E1b1  X −Eb21
1b2
Xb31 − E1b3


Xa10 − E1a1
 Xa20 − E1a2  Xa30 − E1a3 = F(τ1 ) · F−1 (0) ·  (17)
 Xa10 + 1 − E1b1  X −Eb20
1b2
Xb30 − E1b3

(16)

where x0 is an initial point on S, while T(x0 ) is thepoint at which the solution starting from x0 hits S
again.
The concrete representation of T(x0 ) is given as
Xa11 + 1 − E0b1
= Gb (τ2 ) · Gb −1 (0) ·  Xb21 − E0b2 
Xb31 − E0b3


(18)
where τ2 is given by using the ﬁrst rows of two equations
in Eq. (18). Finally, we get
T(x0 ) = x2 .

(19)

Figure 7(2) shows the Poincaré map obtained by
calculating Eqs. (17), (18) and (19). We can observe
the bifurcation scenario more clearly.
The Jacobian matrix DT of the Poincaré map T
can be also derived rigorously from Eqs. (17)–(19). We
can calculate the largest Lyapunov exponent by
1 
log |DTj · ej |N →∞ Nj=1
Nµ = limFig. 8Derivation of the Poincaré map.

(20)

where ej is a normalized base.
One-parameter bifurcation diagram and the calculated largest Lyapunov exponents are shown in Fig. 9

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bifurcation phenomena.
Our future research is the development of a designing method of chaos-generating circuits based on thisstudy.
References

Fig. 9 One-parameter bifurcation diagram. Horizontal: αb .
Vertical: xa1 .

Fig. 10 Largest Lyapunov exponents. Horizontal: αb .
Vertical: µ.

and Fig. 10, respectively. Control parameter is αb andother parameters are ﬁxed as Fig. 7.
By using these results, we can say the generationof chaos is conﬁrmed numerically and we can describe
detailed bifurcation scenario as follows.
One-periodic orbit (a) bifurcates to torus (b)around αb = 0.74. For 0.74 < αb < 0.805, we canobserve several phase-locked states in torus region. For
0.805 < αb < 0.817, the largest Lyapunov exponent becomes positive occasionally though the attractor looks
like torus (c), (d). We consider that chaos via foldedtorus appears in this interval [12]. For 0.817 < αb ,chaotic attractors can be observed (f), (h), (j), (k) for
almost parameter values except two large periodic windows (e), (i). We can also conﬁrm a lot of small periodic
windows are embedded in chaotic region (g).
5.

Conclusions

In this paper, we have proposed a chaotic circuit using two RC phase shift oscillators. By using a simpler
model of the original circuit, the mechanism of generating chaos has been explained. Further, we have
conﬁrmed the generation of chaos by calculating the
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Yasuteru Hosokawa
was born in
Hyogo, Japan, on 1974. He received the
B.E. and M.E. degrees from Tokushima
University, Tokushima, Japan, in 1997,
1999, respectively. He is currently working towards Ph.D. degree at the same university. In 2001, he joined the facultyof Management and Information Scienceat Shikoku University, Tokushima Japan,
where he is currently a research assistant.
His research interest is in chaos circuit design. Mr. Hosokawa is a student member of the IEEE.

HOSOKAWA et al.: ANALYSIS OF CHAOTIC PHENOMENA IN TWO RC PHASE SHIFT OSCILLATORS COUPLED BY A DIODE

2295

Yoshifumi Nishio
received the B.E.,
M.E., and Ph.D. degrees in electrical engineering from Keio University, Yokohama
Japan, in 1988, 1990, and 1993, respectively. In 1993, he joined the Department of Electrical and Electronic Engineering at Tokushima University, Tokushima Japan, where he is currently an
Associate Professor. From May 2000 heis spending a year in the Laboratory of
Nonlinear Systems (LANOS) at the Swiss
Federal Institute of Technology Lausanne (EPFL) as a visiting
professor. His research interests include analysis and applicationof chaos in electrical circuits, analysis of synchronization in coupled oscillators circuits, development of analyzing methods fornonlinear circuits and theory and application of cellular neuralnetworks. Dr. Nishio is a member of the IEEE.

Akio Ushidareceived the B.E.
and M.E. degrees in electrical engineering from Tokushima University in 1961
and 1966, respectively, and the Ph.D. degree in electrical engineering from University of Osaka Prefecture in 1974. He wasan associate professor from 1973 to 1980
at Tokushima University. Since 1980 hehas been a Professor in the Department
of Electrical Engineering at the university.
From 1974 to 1975 he spent one year as
a visiting scholar at the Department of Electrical Engineeringand Computer Sciences at the University of California, Berkeley. His current research interests include numerical methods
and computer-aided analysis of nonlinear systems. Dr. Ushida isa member of IEEE.