2008 International Symposium on Nonlinear Theory and its Applications
NOLTA'08, Budapest, Hungary, September 7-10, 2008

Chaotic Circuit Using a Ring Oscillator and a van der Pol Oscillator
Yasuteru Hosokawa† and Yoshifumi Nishio‡†Faculty of Management and Information Science, Shikoku University
123-1 Furukawa, Ohjin-cho, Tokushima, Japan‡Department of Electrical and Electronics Engineering, Tokushima University
2-1 Minami-Josanjima, Tokushima, Japan
Email: hosokawa@keiei.shikoku-u.ac.jp, nishio@ee.tokushima-u.ac.jp
Abstract—In this study, we propose a chaotic circuit using a ring oscillator and a van der Pol oscillator. The structure of this circuit is very simple. Therefore, IC implementation of this circuit is not so diﬃcult. The exact solution
and its Poincaré map are derived and the largest Lyapunov
exponent are calculated. As a result, chaotic phenomena
are conﬁrmed. This chaotic circuit is suitable for using investigations of coupled chaotic systems, because this circuit is implemented as a electric circuit and IC implementation is not so diﬃcult.

1. Introduction
Electric circuits are one of the natural physical system and it can be represented by a diﬀerential equations.
Namely, it is considered that electric circuits are corresponding to many kinds of natural phenomena and it isexpected that these phenomena be analyzed by using differential equations. Especially, electric circuits are suitablefor investigating large scale chaotic coupled systems. Many
investigation results can be obtained by circuit experiments
immediately. Generally, large scale coupled systems canbe described by high dimensional equations. Therefore, its
analysis is not so easy and its computer simulation needs
a long time. Additionally, the value of electric circuit elements is comparatively stable. If many same circuits aremade, almost circuits will behave the same. These are very
important advantages of using electric circuits for investigations of coupled systems.
However, implementing large scale coupled systems is
not so easy. The one solution of this problem is IC implementation. Though circuit experiments of the large scale
coupled chaotic system become easy by IC implementation, many money and time are needed. Therefore, thechaotic circuit which can be implemented as IC and canbe also implemented as normal electric circuit is needed.
In this study, we propose a chaotic circuit using a ringoscillator and a van der Pol oscillator. The ring oscillator
included the circuit consists of Op-Amps and resistors. The
structures of the ring oscillator and the van der Pol oscillator are very simple. Therefore, IC implementation of thiscircuit is not so diﬃcult.

2. Circuit Model
2.1. Circuit Schematic
Figure 1 shows the schematic of the proposed circuit.
The left side is a ring oscillator which consists of three inverters. Resistor Rvr1 is for adjusting the amplitude. The
right side is a van der Pol oscillator. The Op-Amp partworks as a negative resistor. These two oscillators are coupled via diodes which is placed the center.
In order to propose the new chaotic circuit, we used ourdesigning technique. This technique is that two oscillators
are coupled via diodes. It is very simple. However, thistechnique can not determine the parameters. In this study,parameters are determined by circuit experiments.
Ring Osc.

VDP Osc.
DiodesRR

Rn
Rvr1+R+

RCR+-L

R
Rvr2

Rn+

Figure 1: Circuit schematic.

2.2. Circuit Model
In order to make clear the mechanism of generating
chaos, the proposed circuit is modeled as follows. Figure 2
shows diodes model of the proposed circuit. Diodes aremodeled as a piece-wise linear function shown in Fig 2 (b).
Figure 3 shows the circuit model of the proposed circuit
shown in Fig. 1. Inverters of the ring oscillator are modeled as a inverter including a time delay. All elements except diodes are modeled as linear elements. It means that
diodes play a role as non-linearity which is needed for generating chaos. Additionally, analysis becomes easy because
the elements except diodes are linear elements. This is theuseful advantage for investigating large scale coupled systems. Using this model, circuit equations are described as- 285 -

id

vd

Equations (1) and (2) are normalized as

0id(a)

ẋ1







ẋ2




 ẋ3






ẋ4





 ẋ5rd-VthVthvdrd(b)

==
=
=
=

−x1 − αx3 ,−x2 − αx1 ,−βx3 − αx2 − εxd ,δ(γx4 − x5 + εxd ),ζ x4 ,(4)whereFigure 2: Diodes model.
Gm v 3C1R1

1xd = x3 − x4 + (|x3 − x4 − 1| − |x3 − x4 + 1|.
2(5)

v1

3. Exact solutions and Poincaré Map
Gm v 1C1

R1

Since the circuit equations (4) are piecewise-linear, solutions in each linear region can be derived. At ﬁrst, wedeﬁne three piecewise-linear region as follows.

v2 idi1
C2

Gm v 2

C1

R2

L1-R3v4v3

R1 :x3 − x4 > 1.
R2 : −1 < x3 − x4 < 1.
R3 :x3 − x4 < −1.

Figure 3: Circuit model.

The eigenvalues in each region are calculated from Eq. (4).
The eigenvalues in each region are described as follows.

follows:

dv1



C1


dt






dv

2


C1


dt





 C1 dv3

dt





dv4



C2



dt




di1


L1


dt

R1 and R3 : λ11 , λ12 , λ13 , σ11 ± jω11 .
R2 :λ21 , σ21 ± jω21 , σ22 ± jω22 .
===
=
=

1
v1 − Gm v3 ,R1
1− v2 − Gm v1 ,R1
1− v3 − Gm v2 − id ,R2
1v 4 − i1 + id ,R3−(1)

v4 ,{}
11v3 − v4 + (|v3 − v4 − Vth | − |v3 − v4 + Vth )| .
rd2(7)The eigenvalues in R1 and R3 are obtained numericallyfrom the 5th order eigenequation of Eq. (4). On the otherhand, the eigenvalues in R2 can be derived from the 3rdand 2nd order eigenequations of Eq. (4). The equilibrium
points of each region are calculated by putting the right sideof Eq. (4) to be equal to zero. Then, the solutions in eachlinear region can be described. We omit the description,
because it is complicated enough to write here.
Now, we are deriving the Poincaré map and the Jacobianmatrix.
Let us deﬁne the following subspacewhere
id =(6)

where,(2)

S = S1 ∩ S2 .

(8)

S1 : x3 − x4 = 1.
S2 : −αx2 − βx3 − γδx4 + δx5 < 0.

(9)

By substituting the variables and the parameters,vn(n = 1, 2, 3, 4),Vth1
τ=t,α = G m R1 ,C1 R1xn =δ=C1,
C2ε=R1R2andR1R1i1 ,xd =id ,VthVthR1R1β=,γ=,R2R3x5 =ζ=C1 R21.
L1(3)The subspace S1 corresponds to the boundary condition between R1 and R2 , while the subspace S2 corresponding tothe condition ẋ3 − ẋ4 > 0. Namely, S corresponds to thetransitional condition from R1 to R2 .
Let us consider the solution starting from an initial pointon S. The solution returns back to S again after wanderingR1 , R2 and R3 . Hence, we can derive the Poincaré map asfollows.
T : S → S, x0 7→ T(x0 ).
(10)- 286 -

where, x0 is an initial point on S, while T(x0 ) is the point
at which the solution starting from x0 hits S again. T(x0 )can be derived by using the exact solutions. The Jacobianmatrix DT of the Poincaré Map T can be also derived.
We can calculate the largest Lyapunov exponent by1 ∑log |DT j · e j |.
N→∞ N
j=0N

µ = lim(11)(a)

here e j is a normalized base.

Rvr1 = 55[Ω]β = 12

Rvr1 = 62[Ω]β = 10

Rvr1 = 98[Ω]β = 6.7

Rvr1 = 106[Ω]β = 5.9

Rvr1 = 126[Ω]β = 5.2

4. Circuit Experiments And Computer Calculations
Figure 4 shows the results of circuit experiments andcomputer calculated results. Circuit experiment results areshown in Fig. 4 (1). Projections of attractors onto x3 andx4 and their Poincaré maps are shown in Figs 4 (2) and (3),respectively. Rows of Fig. 4 are corresponding to each others. In circuit experiments, circuit parameters are ﬁxed as
R = 74.0[kΩ], C = 0.015[µF], L = 10[mH], Rn = 2.7[kΩ]
and Rvr2 = 380[kΩ]. The Op-Amp is type TL082. Corresponding parameters of computer calculations are ﬁxed as
α = 4, γ = 3, δ = 0.2, ε = 100 and ζ = 0.5. Control
parameters are set as Rvr1 and β. By changing control parameters, periodic orbits are observed in Figs. 4 (a), (b) and(f). Chaotic phenomena are observed in Figs. 4 (c) - (e).
One-parameter bifurcation diagram and the calculated
largest Lyapunov exponent are shown in Fig. 5 and Fig. 6,respectively. Bifurcation phenomena of periodic orbits,chaos, window and so on are observed. For 4.20 <β < 6.80, the largest Lyapunov exponent becomes positive though two large windows are observed. By using thisresult, We can say the generation of chaos is conﬁrmed numerically. By changing other parameters, period doublingbifurcation phenomena, tours and so on are also observedin this circuit.

(b)

(c)

(d)

(e)

(f)

5. Conclusions
In this study, we proposed the chaotic circuit using a ringoscillator and a van der pol oscillator. By circuit experiments, chaotic attractors have been observed and by using
a linearized model, the generation of chaos have been conﬁrmed numerically.
The characteristic of this circuit is a suitable for IC implementation and it is easy to implement as a normal electric circuit. This characteristic is useful for investigationsof large scale coupled chaotic circuits. Therefore, it is expected that this circuit contributes to studies of large scale
coupled chaotic circuits.

Rvr1 = 148[Ω](1)

β = 4.0(2)(3)

Figure 4: Circuit experiments and Computer calculation
results. (1) Circuit experiments. (2) Projections of attractors. (3) Poincaré maps. Circuit experiments: Horizontal axis is v3 [1.0V/div]. Vertical axis is v4 [1.0V/div].
R = 74.0[kΩ], C = 0.015[µF], L = 10[mH], Rn = 2.7[kΩ]
and Rvr2 = 380[kΩ]. Computer calculations: α = 4, γ = 3,
δ = 0.2, ε = 100 and ζ = 0.5.

- 287 -Figure 5: One-parameter bifurcation diagram. Horizontal: β. Vertical: x1

Figure 6: Largest Lyapunov exponents. Horizontal: β. vertical: µ.
References

a Class of Hysteresis Chaotic Oscillators,” Proc. NOLTA’99,Vol. 1, pp.111–114, 1999.

[1] M. Itoh, “Chaotic Attractors in Nonlinear Circuits,” Tech. Report of IEICE, Vol.85, No.261, pp.19–24, 1986.
[2] N. Inaba, S. Mori and T. Saito, “Chaotic Phenomena in 3rdOrder Autonomous Circuits with a Linear Negative Resistance and an Ideal Diode II,” Tech. Report of IEICE, Vol.87,No.271, pp.9–15, 1987.
[3] T. Saito, “An Approach Toward Higher Dimensional Hysteresis Chaos Generators,” IEEE Trans. Circuits Syst., Vol.37,
No.3, pp.399–409, 1990.
[4] K. Mitsubori and T. Saito, “A Four-Dimensional Plus Hysteresis Chaos Generator,” IEEE Trans. Circuits Syst., Vol.41,
No.12, pp.782–789, 1994.
[5] T. Suzuki and T. Saito, “On Fundamental Bifurcations froma Hysteresis Hyperchaos Generator,” IEEE Trans. Circuits
Syst., Vol.41, No.12, pp.876–884, 1994.
[6] A. S. Elwakil and M. P. Kennedy, “Systematic Realization of[7] Y. Hosokawa, Y. Nishio and A. Ushida, “Analysis of Chaotic
Phenomena in Two Second Order Oscillators Coupled byDiodes,” Proc. NDES2000, pp.263–267, 2000.
[8] Y. Hosokawa and Y. Nishio, “A Design Method for Chaotic
Circuits Using Two Oscillators,” Chaos in Circuits and Systems, Guanrong Chen and Tetsushi Ueta (Ed.), Chapter 3, pp.
51–69, World Scientiﬁc, 2002.
[9] Y. Hosokawa and Y. Nishio, “Simple Chaotic Circuit Using
CMOS Ring Oscillators,” Int. J. of Bifurcation and Chaos,Vol.14, No.7, pp.2513–2524, 2004.
[10] Yasuteru Hosokawa, Yoshifumi Nishio and Akio Ushida, “A
Design Method of Chaotic Circuits Using Two Oscillators,”
Proc. NOLTA’00, Vol.1, pp.17–21, 2000.
[11] Yasuteru Hosokawa, Yoshifumi Nishio and Akio Ushida, “A
Design Method of Chaotic Circuits Using an Oscillator and aResonator,” Proc. ISCAS01, Vol.3, pp.373-376, 2001.

- 288 -