2005 RISP International Workshop on Nonlinear
Circuit and Signal Processing (NCSP'05)
Hawaii, USA, Mar. 4-6, 2005.

Analysis of Chaotic Circuits using Wien Bridge Oscillator and a Resonator
Yasuteru Hosokawa† and Yoshifumi Nishio‡
Shikoku University†
Furukawa, Ohjin, Tokushima, Japan
Phone:+81-88-665-1300
Email: hosokawa@keiei.shikoku-u.ac.jp

Tokushima University‡
2-1 Minami-Josanjima, Tokushima, Japan
Phone/FAX:+81-88-656-7501
Email: nishio@ee.tokushima-u.ac.jp

Abstract
In this study, we analyze chaotic circuits using a Wien bridge
oscillator and a resonator. A Wien bridge oscillator and a resonator are coupled by diodes. These circuits are designed byour designing method of a chaotic circuit. Until now, we proposed and analyzed some chaotic circuits using this method.
We consider that this study fortifies our designing method.
1. Introduction
Recently, designing principles of chaotic circuits are studied in actively. These studies aim to understand chaos and toapply chaos. Because electric circuit is superior in its simplicity, repeatability and response in the experiment, a electric circuit is a very useful tool for understanding chaos. On
the other hand, many researchers developing chaotic circuit
applications use a few popular chaotic circuits. In order todevelop the excellent application, a chaotic circuit which hassuitable characteristics should be used for the application. In
our previous studies, a designing method of chaotic circuits
was proposed. This method is to couple an oscillator and aresonator with diodes. We investigated this method applied
to some oscillators and LC resonators. However, we haveconfirmed only one node per one oscillator.
In this study, we investigated combinations of a Wien
bridge oscillator and two type LC resonators. In particular,
we paid attention to a node that is coupled with diodes.
2. Designing method

Oscillator

Coupling DiodeResonator

Figure 2:
method.

Chaotic circuits developed by our designing

Figure 1 shows a designing method proposed by us. Oneoscillator and one resonator are coupled with a coupling diodewhich is coupled two diodes each other. We consider thatthese diodes play a role of nonlinearity which is essential forthe generation of chaos. Thus, circuits models of oscillatorsand resonators are modeled as linear model in our studies.
Figures 2 show chaotic circuit developed by our designing
method. van del Pol oscillators (parallel and series type), andWien bridge oscillator are used as a oscillator. LC resonators(parallel and series type) are used as a resonator.
3. Circuit Model
3.1. Wien bridge oscillator

Figure 1: Designing method.

Figure 3 shows the circuit schematic of a Wien bridge oscillator. The oscillating frequency is decided by C1 , R1 , C2- 73 -and R2 . The amplitude is decided by R3 and R4 . We investigate combinations of coupling nodes shown as nodes 1, 2, 3
and 4 in Fig. 3.

Table 1: Combinations of a node and a resonator
no.
1
2
3
45
6

node 2C2

R2

node 1

R1

node 3C1
R3

node
node 1
node 1
node 2
node 2
node 2 ( exchanged C2 and R2 )node 2 ( exchanged C2 and R2 )resonator
parallelseriesparallelseriesparallelseriesR4

node 4C2 and R2 means constructing another circuit. Therefore, six
combinations as shown in Table 1 are investigated.

Figure 3: Wien bridge oscillator.

4. Circuit experiments and computer calculations
4.1. Circuit equations

3.2. LC resonator and diodes.

v

did(a) Coupling diode model.

idVthgd-Vth1Vthvv

drd
1idd-Vth(b) Voltage controlled type.

(c) Current controlled type.

In order to carry out computer calculations, circuit equations of the circuits as shown in Table 1 are derived using
models of section 3. For example, the case of circuit no. 1 isshown as follows.
dv1
1
R4
1
11

=(−)v1 −v2 −id ,
dtCRRRRCC12 312 11
R41dv2 dt = C2 R2 R3 v1 − R2 C2 v2 ,
(1)
dv31


=(−i1 + id ),

dtC3





di11


 dt = L1 v3 ,Figure 4: Coupling diode model and its characteristics
where,
Series type LC resonator and parallel type LC resonator
id =are used as the resonator. We assume that an inductor and agd {v1 − v3 + (|v1 − v3 − Vth | − |v1 − v3 + Vth |) /2}
capacitor are ideal elements.
(2)Figure 4 shows coupling diode model. In the case of couChanging variable and parameter as follows.
pled with parallel type LC resonator, voltage controlled typemodel shown in Fig. 4 (b) is used. In the case of coupled
v1 = Vth x1 , √v2 = Vth x2 , v3 = V√th x3 ,with series type LC resonator, current controlled type modeli1 = px4 Vth / R1 R2 id = yd Vth / R1 R2
shown in Fig. 4 (c) is used.
(3)/R3 , γ = C1 /C3 ,α = R1 /R2 , β = R4√
δ = C1 R1 R2 /L1 , ε = R1 R2√
gd ,d
3.3. Circuit models= “ ˙ ”, t = C1 R1 R2 τ,
C1 = C2 , dτ
We investigate the circuits combining a node 1, 2, 3 or 4 Normalized circuit equation is shown as follows.
with a series/parallel type LC resonator via a coupling diode.
1
At first, circuit experiments were carried out. In the case ofx˙1 = (αβ − )x1 − αx2 − εyd ,using node 3 or 4, an oscillation on the saturation region ofα x˙2 = αβx1 − αx2 ,an operational amplifier and the convergence to an equilibrium point are observed. Chaotic phenomena could not bex˙3 = −γx4 + γεyd ,observed. Therefore, the case of nodes 1 and 2 are investi x˙4 = δx3 ,gated in this study. In the case of using node 2, exchanging- 74 -(4)

where,yd = x1 − x3 + (|x1 − x3 − 1| − |x1 − x3 + 1|)/2.

(5)
(1)

4.2. Circuit experiments and Computer calculations
Using normalized circuit equations, computer calculationsof each circuits are carried out. Figures 5 show the circuit experimental results and computer calculated results of circuit
no. 1. Fig. 5 (a) show the attractors of circuit experimentalresults, Fig. 5 (b) show the attractors of computer calculated
results, and Fig. 5 (c) show the Poincaré maps obtained bycomputer calculation. We choose β as control parameter. A
periodic orbit (1), quasi-periodic orbit (2), chaotic attractors(3) (4) (6) and a large periodic window (5) are observed. Figures 6, 7, 8, 9 and 10 show the circuit experimental results
and computer calculated results of circuit no. 2, 3, 4, 5 and
6, respectively. The Figs. (a) show the attractors of circuit
experimental results, and the Figs. (b) show the attractors ofcomputer calculated results.
We can observe almost same attractors and the same bifurcation scenario in circuit experimental results and computer calculated results. These result mean that our linearized
model of an oscillator does not lose features needed for chaos
generation. Namely, this mean that Wien bridge oscillator behave as simple divergently oscillating part and that only thenonlinearity of the coupling diode controls the amplitude. In
other words, the oscillator plays a role of expanding, while
the coupling diode plays folding. These are known as theessence of generating chaos.

R4 = 4.31[kΩ]β = 4.00

β = 4.00

R4 = 5.45[kΩ]β = 5.00

β = 5.00

R4 = 5.88[kΩ]β = 5.40

β = 5.40

R4 = 6.29[kΩ]β = 5.90

β = 5.90

R4 = 7.51[kΩ]β = 7.30

β = 7.30

R4 = 8.07[kΩ]β = 8.00

β = 8.00

(a)

(b)

(c)

(2)

(3)

(4)

5. Conclusions
In this study, we investigated circuits combining a oscillator with a resonator about chaotic circuits proposed by us.
The circuit consist of one Wien bridge oscillator, one resonator and one coupling diode. In particular, we paid attention to a node that is coupled with a coupling diode. As aresult, we confirmed six chaotic oscillators.

(5)

(6)
6. Acknowledgment
This work was supported by the Ministry of Education,
Culture, Sports, Science and Technology.
(KAKENHI15760284)

Figure 5: (a)circuit experimental results, (b)computer calculated results, (c) Poincar/’e maps. (a)horizontal axis:[1] Y. Hosokawa, Y. Nishio and A. Ushida, “A Design v1 [0.5v/div], vertical axis:v3 [0.5v/div], R1 = 2.0[kΩ], R2 =Method of Chaotic Circuits Using an Oscillator and a 5.6[kΩ], R3 = 1.0[kΩ], C1 = C2 = C3 = 33[nF],Resonator,” Proc. of ISCAS2001, Vol. 3, pp. 373–376, L1 = 10[mH] R4 . (b)horizontal axis: x1 , vertical axis: x3 ,α = 0.6, γ = 1.1, δ = 43, ε = 30.
2001.
References- 75 -

(a)(b)Figure 6: circuit experimental result (a) and computer calculated result (b) of circuit no. 2. (a)horizontal axis:
v1 [0.5v/div], vertical axis:v3 [0.5v/div], R1 = 3.3[kΩ], R2 =(a)(b)2.4[kΩ], R3 = 1.0[kΩ], R4 = 2.7[kΩ], C1 = C2 = C3 =33[nF], L1 = 50[mH]. (b)horizontal axis: x1 , vertical axis:
Figure 9: circuit experimental result (a) and computer calx3 , α = 1.2, β = 2.7, γ = 1.0, δ = 7, ε = 20.
culated result (b) of circuit no. 5. (a)horizontal axis:
v1 [0.2v/div], vertical axis:v3 [0.2v/div], R1 = 5.6[kΩ], R2 =5.6[kΩ], R3 = 1.0[kΩ], R4 = 3.1[kΩ], C1 = C2 = C3 =33[nF], L1 = 50[mH]. (b)horizontal axis: x1 , vertical axis:
x3 , α = 1.0, β = 3.0, γ = 1.0, δ = 20, ε = 40.

(a)(b)

Figure 7: circuit experimental result (a) and computer calculated result (b) of circuit no. 3. (a)horizontal axis:
v1 [0.5v/div], vertical axis:v3 [0.5v/div], R1 = 5.6[kΩ], R2 =5.6[kΩ], R3 = 1.0[kΩ], R4 = 5.0[kΩ], C1 = C2 = C3 =33[nF], L1 = 50[mH]. (b)horizontal axis: x1 , vertical axis:
x3 , α = 1.0, β = 4.5, γ = 1.0, δ = 20, ε = 40.

(a)

(b)

Figure 10: circuit experimental result (a) and computer
calculated result (b) of circuit no. 6. (a)horizontal axis:
v1 [0.5v/div], vertical axis:v3 [0.5v/div], R1 = 3.3[kΩ], R2 =(a)(b)3.3[kΩ], R3 = 1.0[kΩ], R4 = 4.5[kΩ], C1 = C2 = C3 =33[nF], L1 = 50[mH]. (b)horizontal axis: x1 , vertical axis:
Figure 8: circuit experimental result (a) and computer calx3 , α = 1.1, β = 3.4, γ = 1.0, δ = 15, ε = 20.
culated result (b) of circuit no. 4. (a)horizontal axis:
v1 [0.5v/div], vertical axis:v3 [0.5v/div], R1 = 3.3[kΩ], R2 =2.7[kΩ], R3 = 1.0[kΩ], R4 = 3.1[kΩ], C1 = C2 = C3 =33[nF], L1 = 50[mH]. (b)horizontal axis: x1 , vertical axis:
x3 , α = 1.6, β = 2.1, γ = 1.0, δ = 6, ε = 20.

- 76 -