2009 IEEE Workshop on
Nonlinear Circuit Networks, Tokushima, Dec 11-12, 2009

Analysis of Chaotic Circuit Using One Nonlinear
Resistor and Two Resonators
Tomohiro Okubo

Yasuteru Hosokawa

Yoshifumi Nishio

Dept. Information Science,
Shikoku University
Email: shikoku2003002@yahoo.co.jp

Dept. Information Science,
Shikoku University
Email: hosokawa@keiei.shikoku-u.ac.jp

Dept. Electrical and Electronic Eng.,
Tokushima University
Email: nishio@ee.tokushima-u.ac.jp

Abstract— Until now, we have proposed some chaotic circuits
using our designing method. In this study, by using our designing
method, a new chaotic circuit is proposed and a generation ofchaos is conﬁrmed.

I. I NTRODUCTION
Studies of designing principles of chaotic circuits aim tounderstand chaos and to apply chaos[1]-[12]. Because electric
circuit is superior in its simplicity, repeatability and responsein the experiment, a electric circuit is a very useful tool forunderstanding chaos. On the other hand, many researchersdeveloping chaotic circuit applications use a few popularchaotic circuits. In order to develop the excellent application,a chaotic circuit which has suitable characteristics should be
used for the application. Therefore, designing methods which
can design many kind of chaotic circuits are needed.
In our previous studies, a designing method of chaotic
circuits was proposed. This method is to couple an oscillator
and a resonator with diodes. In this method, an active element
and a nonlinear element is divided. Therefore, a method using
one element which combine an active element and a nonlinear
element is considered. This idea is already mentioned[12].
However, one chaotic circuit is proposed only and this circuit
is not investigated in detail.
In this study, in order to conﬁrm a chaos generation of thecircuit, we investigate the circuit in detail.
II. C IRCUIT M ODELicFig. 1.

L1

Nonlinear element

Oscillator

Oscillator

Resonator(a) Previous study.
Fig. 2.

Resonator(b) This study.

System models of designing methods of chaotic circuits.

First of all, we derive a simple linearized model of thecircuit model in Fig. 1. Figure 3(a) shows the model ofa nonlinear element in Fig. 1. The characteristic of thisnonlinear element shows a piecewise linear function as shownin Fig. 3(b). The others of circuit elements are modeled aslinear elements.

vc

ic

i2

R1

We have proposed designing method as shown in Fig. 2(a).
In this method, one oscillator and one resonator are coupledby diodes. Namely, this system consists of two resonator,
one active element and one nonlinear passive element. By
combining one active element and one nonlinear passiveelement, new combination can be considered as shown inFig. 2(b) In order to investigate this conbination, the circuit
as shown in Fig. 1 is invesitgated.

ici4

C1

v1 R 2L2C2vc+v3vc

g 1 g2
g1+ g 2−Vth 0g1(a) Circuit model.

- 75 -−g 1

1

Chaotic circuit using one nonlinear resistor and two resonators.

Figure 1 shows a chaotic circuit using one nonlinear resistor
and two resonators. This circuit is designed based on afollowing idea.

g2

1Fig. 3.

(b) Its characteristic.

Nonlinear element model.

III. E XACT S OLUTIONS
By using the linearized circuit model, the circuit equations
are derived as follows.
dv11
C1= − v1 − i2 − ic ,dtR1di2
= v1 ,
 L1dt(1)dv31
C=−v−i−i,234cdtR2

 L2 di4 = v3 ,dtwhere,ic =0.5(g1 + g2 )Vth + 0.5(g2 − g1 )v−0.5(g1 + g2 )|v − Vth |.

By changing parameters and variables,√v3v1
L1 i2, x2 =, x3 =,x1 =VthC1 VthVth√√
L1 i41L1x4 =, α1 =,C1 Vth
R1 C1√√
1L1L1C1α2 =, β = g2, γ=,
R2 C1
C1C2dL1
g1
1t andδ=, ε= , τ = √=”·”
L2g2dtL1 C1

eσ1 τ cos ω1 τ eσ1 τ sin ω1 τ 
f2 (τ ) =  eσ2 τ cos ω2 τ  ,eσ2 τ sin ω2 τ
f1 (τ ) =f3 (τ ) = · · · ,f4 (τ ) = · · · ,Region R2x(τ ) − E2 = G(τ ) · G−1 (0) · (x(0) − E2 ),
x1 (τ )
g1 (τ )
 x2 (τ )  g2 (τ ) x(τ ) =  x3 (τ )  , G(τ ) =  g3 (τ )  ,x4 (τ )g4 (τ )
e λ1 τ e λ1 τg2 (τ ) =  eστ cos ωτ  ,eστ sin ωτ(10)dg2, g3 (τ ) = · · · , g4 (τ ) = · · · ,dτWe omit the descriptions of fk and gk (k = 3, 4).
g1 (τ ) =(5)IV. P OINCAR É M AP

Since the circuit equations are piecewise-linear, exact solutions in each linear region can be derived. We deﬁne twopiecewise-linear regions as follows.
R1 : x1 − x3 + 1 ≥ 0,
R2 : x1 − x3 + 1 < 0,df2,dτ(9)(3)

where,0.5(ε + 1) + 0.5(1 − ε)(x1 − x3 )−0.5(ε + 1)|x1 − x3 + 1|.

x(τ ) = F(τ ) · F−1 (0) · x(0),x1 (τ )
f1 (τ )
 f2 (τ )  x2 (τ ) x(τ ) = 
 x3 (τ )  , F(τ ) =  f3 (τ )  ,f4 (τ )x4 (τ )

(2)

The normalized circuit equations are described by the following four-dimensional piecewise-linear differential equations.
ẋ1 = −α1 x1 − x2 − βy,
 ẋ2 = x1 ,(4)ẋ3 = −γ(α2 x3 + x4 − βy),




 ẋ4 = δx3 ,y=respectively. These values are calculated by putting the rightside of Eq. (4) to be equal to zero.
Then, we can describe the exact solutions in each linear
region as follows.
Region R1R1R2S(6)The eigenvalues in each region can be calculated numericallyfrom Eq. (4). The eigenvalues in each region are described asfollows.
R1 : σ1 ± jω1 , σ2 ± jω2(7)R2 : λ1 , λ2 , σ ± jωNext, we deﬁne the equilibrium points in E1 and E2 as0 −β(ε + 1) ,
E1 = 0 and E2 = (8)
0β(ε + 1)

Fig. 4.

Derivation of the Poincaré map.

In order to conﬁrm the generation of chaos, we derive thePoincaré map.

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Let us deﬁne the following subspace
S = S1 ∩ S2(11)

(a)

where
S1 : x1 − x3 + 1 = 0S2 : (−α1 + α2 γ)x1 − x2 + γx4 + α2 γ − βε − βγε < 0(12)The subspace S1 corresponds to the boundary condition between R1 and R2 , while the subspace S2 corresponds tothe condition ẋ1 − ẋ3 < 0. Namely, S corresponds to thetransitional condition from R1 to R2
Let us consider the solution starting from an initial point on
S. The solution returns back to S again after wandering R2
and R1 as shown in Fig. 4. Hence, we can derive the Poincaré
map as follows.
T : S → S,

x0 7→ T(x0 )g1 = 5.66[kΩ]ε = 0.0100

ε = 0.0100

g1 = 5.07[kΩ]ε = 0.0130

ε = 0l0130

g1 = 3.35[kΩ]ε = 0.0190

ε = 0.0190

g1 = 3.09[kΩ]ε = 0.0198

ε = 0.0198

g1 = 2.70[kΩ]ε = 0.0210

ε = 0.0210

g1 = 2.35[kΩ]ε = 0.0230

ε = 0.0230

g1 = 2.33[kΩ]ε = 0.0240

ε = 0.0240(1)(2)(3)(b)(13)(c)

where x0 is an initial point on S, while T(x0 ) is the point at
which the solution starting from x0 hits S again.
The concrete representation of T(x0 ) is given as followsusing the exact solutions in Eqs. (9) and (10).
Suppose that the solution starting from x0=(X 10 , X20 , X30 , X40 ) when τ = 0 hits S1 and enters
R1 at x1 = (X11 , X21 , X31 , X41 ) when τ = τ1 . In this case,(d)x1 is given by
X11X10
 X21 + β(ε + 1)  = G(τ1 ) · G−1 (0) ·  X20 + β(ε + 1)  X11 + 1
 X10 + 1

X41 − β(ε + 1)
X40 − β(ε + 1)
(14)(e)where τ1 is given by using the ﬁrst and third rows of Eq. (14).
The solution hits S again at x2 = (X12 , X22 , X32 , X42 ) when
τ = τ1 + τ2 . x2 is given by
X12X11
 X22 X21

−1



(15)
 X12 + 1  = F(τ1 ) · F (0) ·  X11 + 1 X42X41(f)where τ2 is given by using the ﬁrst and third rows of Eq. (15).
Finally, we get
T(x0 ) = x2(16)The jacobian matrix DT of the Poincaré map T can be also
derived rigorously from Eqs. (14)-(16). We can calculate thelargest Lyapunov exponent by1 ∑µ = limlog |DTj · ej |N →∞ Nj=1(g)

N(17)where ej is a normalized base.
Figures 5(1) show circuit experiment results. Horizontal
axes are x1 and vertical axes are x3 . We choose g1 as thecontrol parameter. Figures 5(2) show computer calculated
results of the exact solutions in Eqs. (9) and (10). We choose
ε as the control parameter. Figures 5(3) show Poincaré mapsFig. 5. System models of designing method of chaotic circuits. (1) CircuitExperiments. R1 = R2 = 20.0[kΩ], g2 = 0.0[Ω], C1 = 100.0[nF], C2 =10.0[nF], L1 = 50.0[mH] and L2 = 30.0[mH]. (2) Attractors. (3) Poincarémap. α1 = 0.05, α2 = 0.05, β = 9.0, γ = 10.0 and δ = 1.6.

- 77 -Fig. 6.

One-parameter bifurcaition diagram. Horizontal: ε. Vertical: x2 .

obtained by calculating Eqs. (14)-(16). One-parameter bifurcation diagram is shown in Fig. 6. Control parameter is ε andother parameters are ﬁxed as Fig. 5
We can observe almost same attractors and the same bifurcation scenario as follows. In Figs. 5(a), a periodic orbit isobserved. By increasing the control parameter, quasi periodic
orbit in Figs. 5(b), period-doubling bifurcation in Figs. 5(c)(d)chaos in Figs. 5(e)(g), and window in Figs. 5(f) are observed. We also calculated the largest Lyapunov exponents. InFigs. 5(e) and (g), the value of the largest Lyapunov exponents
are µ = 0.0958 and µ = 0.1395, respectively.
V. C ONCLUSIONSIn this study, we have analyzed a chaotic circuit using
one nonlinear resistor and two resonators. we have conﬁrmed
the generation of chaos by calculating the largest Lyapunov
exponent.
Our future research is the development of a designing
method of chaotic circuit based on this study.
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