2014 RISP International Workshop on Nonlinear Circuits,Communications and Signal Processing
NCSP'14, Honolulu, Hawaii, USA. February 28 - March 3, 2014

Chaotic Circuit Using a Colpitts Oscillator and a Resonator
Hayato Amo1 , Yasuteru Hosokawa1 and Yoshifumi Nishio2
1

Shikoku University
Furukawa, Ohjin, Tokushima, Japan
Phone:+81-88-665-1300
FAX:+81-88-637-1318
E-mail: s002337004@edu.shikoku-u.ac.jp
E-mail: hosokawa@keiei.shikoku-u.ac.jp

2

Tokushima University
2-1 Minami-Josanjima, Tokushima, Japan
Phone:+81-88-656-7470
FAX:+81-88-656-7471
E-mail: nishio@ee.tokushima-u.ac.jp

Abstract
In this study, a novel chaotic circuit is proposed. The circuit
consists of a Colpitts oscillator, a resonator and a diode. By
using a idealized diode model, exact solutions are derived.
Chaotic attractors are shown in circuit experiments and computer calculations of exact solutions.

Oscillator

1. Introduction

Resonator

Figure 1: Designing method for a chaotic circuit.

Chaos can be observed various field of the natural world.
Therefore, many researchers pay attention to this phenomenon and related phenomena. In a field of electric engineering, chaotic phenomena have been investigated using achaotic circuit. Elements of an electric circuit can be obtainedin very low cost and the specific is very high. Additionally,a chaotic circuit has a simple structure. Thus, it is easy togenerate chaotic phenomena on an electric circuit.
However, designing a novel chaotic circuit is difficult except experts like [1]–[3] . A designing method is proposedin [4]. This method is that coupling an oscillator and a resonator with diodes as shown in Fig. 1. The advantage is thatthe method is simple. Therefore, this method is very useful method for many researchers. On the other hand, thisadvantage is that there is no rule for setting the parameters.
Breaking the disadvantage is essential to make this method
perfected.
In this study, a novel chaotic circuit applied this method
is proposed. The circuit consists of a Colpitts oscillator, aresonator and a diode. Developing many sample circuits is an
important task for making this method perfected.

R2

R1f(v - v )
2v

1

C1i

1

L11i L2v

2

R3
2C2 v 3C3

Figure 2: Proposed system.

the analysis becomes easily. In order to investigate validity ofthis model, a circuit equation is derived.

dv1


(C1 + C2 )

dt




dv
2. Circuit Model3


C3


dt
Figure 2 is a proposed circuit in this study. A Colpitts oscildi1

lator connects to a resonator via a diode. In this model, all el- L1dt
ements except a diode are modeled as linear elements. Diode di2model is shown in Fig. 3. Its v-i characteristics is shown in L2
dtFig. 4. This model is an idealized model. Using this model,- 177 -1(v1 + Vth ) − i1 − i2 ,R1{}
1R2= −v3 +(v1 + Vth ) + i2 ,R3
R1= −= v1 ,= v1 − v3 + Vth ,

(1)

ONi

Equations (1), (2) and (3) are normalized asεẋ1 =(−αx1 − x4 − x5 − α),ε+1 ẋ = ε(−αβx31 − βx3 + x5 − αβ),ẋ4 = x1 , ẋ = δ(x − x + 1),
513OFFv

VthFigure 3: Idealized diode model.

{i ẋ2ẋ3 ẋ5ONVthOFFv

ẋ1ẋ4(5)= x4 ,= x1 ,(6)= ε(−αx2 − x5 ),= ε(−γx3 − αβx2 + x5 ),= δ(x2 − x3 ),(7)

We define two piecewise-linear regions as follows.
R1 ≡ {(x1 , x3 , x4 , x5 )|ẋ1 + i1 > 0}R2 ≡ {(x1 , x2 , x3 , x4 , x5 )|ẋ2 − x1 − 1 < 0}(8)

Figure 4: v-i characteristic of the idealized diode model as These regions are corresponding to the two state of the diode.
Note that the circuit equations in R2 are completely decoushown in Fig. 3.
pled into two and three dimensional equations. In order toderive exact solutions, eigenvalues in two regions are derived.
dv1
The characteristic equation of the circuit equations in each C1 dt = −i1 ,linear region is given as follows.
(2)
In R1 :di1
 L1 dt = v1 ,(ε + 1)m4 + ε(α + β + βε)m3 + ε(αβε + 2δ
dv2C2dtdv3C3dtdi2 L2 dt)
1=− v2 − i2 ,R1()
1
R2= −v3 +v2 + i2 ,R3
R1C1C2+δε + 1)m2 + ε2 (αβδ + αδ + βδ + β)m + δε2= 0.

(In R2 : For the resonator,
m2 + 1 = 0.
(3)

= v2 − v3 .

(10)

For the Colpitts Oscillator,
m3 + ε(α + γ)m2 + ε(αγε + 2δ)m + δε2 (α + αβ+γ) = 0.

By substituting the normalized variables and the parameters,
vnfor n = 1, 2, 3,Vth√
L1 in−3xn =for n = 4, 5,C1 Vthxn =√1L1d1= ” · ”, τ = √,t, α =dt
RCL1 C111√
R21L1L1β=, γ=, δ=andR3R2 C1L2C1C1ε==.
C2C3(9)(11)

The eigenvalues in each region can be calculated from thecharacteristic equations ( 9 ) - ( 11)
From circuit experiment and computer simulation using
Eq. (5) - (7), The eigenvalues are calculated as follows:
R1 : λ1 , λ2 , σ ± jωR2 (Resonator) : ± j(4)(12)R2 (Colpitts Osc.) : λ, σ1 ± jω1
Further, the equilibrium points in R1 is described as follows:
B=[- 178 -− α − β − αβ000]T .

(13)

S

These values are calculated by making the right-hand sideof Eq. (5) to be equal to zero. The equilibrium point in R2 isthe origin. Then, we can describe the exact solutions in eachlinear region as follows.
In R1 :x(τ ) − B = F(τ ) · F−1 (0) · (x(0) − B).

R1T1R2T2(14)In R2 :For the resonator,xa (τ ) = Ga (τ ) · Ga −1 (0) · (xa (0)).

Figure 5: Route map.

(15)

For the Colpitts osc.,(4). These results shows this model does not lose important(16) features of the original circuit related with the generation ofxb (τ ) = Gb (τ ) · Gb (0) · (xb (0)).
chaos. Additionally, the nonlinear element is only the diode.
In order to derive the Poincaré map, Let us define the follow- It is mean that the Colpitts oscillator plays a role of expanding subspace
ing and the diode play a role of folding. These are known as
the essence of generating chaos.
S = S1 ∩ S2 .
(17)
−14. Conclusion

whereε(−αx1 − x4 − x5 − α).
(ε + 1)In this study, the novel chaotic circuit has been proposed.
(18) By applying a idealized diode model, exact solutions of each
S2 : x2 − x1 = 1.
regions are derived. Additionally, the Poincaré map is defined. In circuit experiments and computer calculations,
The subspace S1 corresponds to the boundary condition be- chaotic phenomena are observed.
tween R1 and R2 , while the subspace S2 corresponds to theIn our future works, calculating the largest Lyapunov exON state of the diode. Namely, S corresponds to the transi- ponents, the circuit analysis using exact solutions and so on
tional condition from R1 to R2 . Let us consider the solution will be carried out.
starting from an initial point on S. The solution returns backto S again after two subspaces as shown in Fig. 5. Hence, theReferences
Poincaré map can be derived as follows:
S1 : ẋ1 = −x4 =(19) [1] L. O. Chua, M. Komuro and T. Matsumoto, “The double
scroll family,” IEEE Trans. Circuits Syst., vol. 33, no. 11,
pp. 1073–1118, 1986.
where x0 is an initial pont on S, while T(x0 ) is the pint
at which the solution starting from x0 hits S again. The[2] N. Inaba, S. Mori and T. Saito, “Chaotic Phenomena inPoincaré map can be represented as a composit map of the3rd Order Autonomous Circuits with a Linear Negative
submaps T1 and T2 . Namely, the Poincaré map can be obResistance and an Ideal Diode II,” Technical Report oftained as
IEICE, Vol. 87, No. 271, pp.9-15, Nov. 1987.
T : S → S,

x 7→ T(x0 ).

T = T1 ◦ T2 .

(20) [3] M. Shinriki, M. Yamamoto and S. Mori, “Multimode oscillations in a modified van der Pol Oscillator Containing
a Positive Nonlinear Conductance,” Proc. IEEE, vol. 69,
3. Circuit Experiments and Computer Calculations
pp. 394–395, 1981.
Figure 6 shows the circuit experimental results and com- [4] Y. Hosokawa, Y. Nishio and A. Ushida, “A Designputer calculated results. In the circuit experiments, parameMethod of Chaotic Circuits Using an Oscillator and aters are set as R1 = 1 [kΩ], R3 = 200 [Ω], C1 = 0.10 [µF],
Resonator,” Proceedings of IEEE International SympoC2 = C3 = 0.022 [µF], L1 = 200[mH] and L1 = 30[mH].
sium on Circuits and Systems (ISCAS’01), vol. 3, pp. 373In the computer calculations, parameters are set as α = 1.41,376, 2001.
γ = 4.71, δ = 6.67 and ε = 4.5. R2 and β are set as control parameters. A periodic orbit is observed in Fig. 6 (1)and (5). Chaotic attractors are observed in Fig. 6 (2), (3) and- 179 -(1)R2 = 8.60[kΩ]β = 24.5

R2 = 8.80[kΩ]β = 29.5

R2 = 10.1[kΩ]β = 33.0

R2 = 11.3[kΩ]β = 39.5

(2)

(3)

(4)

(5)

R2 = 12.0[kΩ](a)β = 41.0(b)(c)

Figure 6: Circuit Experimental Results and Computer Calculation Results. (a) Circuit Experiments. Horizontal axis: v1 ( 1.0[V/div.] ). Vertical axis: v2 ( 2.0 [V/div.] ). (b) Attractors. Horizontal axis: x1 . Vertical axis: x2 . Gridlines show 2.5. (c)Poincaré maps of (b). Horizontal axis: x1 . Vertical axis: x2 .

- 180 -