2015 International Symposium on Nonlinear Theory and its Applications
NOLTA2015, Kowloon, Hong Kong, China, December 1-4, 2015

RC Chaotic Oscillator Using Pulse Generator and Diﬀer Amplifier
Yoshinori Doike† , Yoko Uwate† and Yoshifumi Nishio†† Dept. of Electrical and Electronic Engineering, Tokushima University
2-1 Minamijosanjima, Tokushima, 770-8506 Japan
Email: {doike, uwate, nishio}@ee.tokushima-u.ac.jp
Abstract—In this study, a simple chaotic oscillator composed of RC phase shift oscillators using operational amplifier and pulse generator is proposed and investigated. By
inputing pulse wave to operational amplifier of RC phaseshift oscillator, chaotic phenomena are observed easily. We
carry out computer calculation and circuit experiment in order to check the validity of the proposed model. And, we
derive the one parameter bifurcation diagram.

wave is inputted to non-inverting input terminal of the operational amplifier. The diﬀer amplifier produces the output
voltage vo which is their power supply voltage, according
to the input signals.
RbRaR

Rb1. Introduction

Vs

Recently, many researchers have taken an their interestin chaos [1][2]. Because, chaotic phenomena can be observed in various fields and chaotic systems are good models to explain and describe the higher dimensional nonlinear phenomena. In the field of electrical and electronicengineering, various applications based on deterministicchaos have been proposed [3][4]. For realizing such applications, we need chaotic oscillators.
There are various chaotic oscillators [5][6]. These oscillators are composed of RC circuits, pulse generators andoperational amplifiers. By not using inductor, it can beeasily integrated chaotic oscillators. When the parameters are varied, chaotic attractor are observed in this simple oscillator. However, it is not diﬃcult to analyze thesechaotic oscillators. These oscillators are pulse-driven nonautonomous and discontinuous systems. When the output
of pulse generators varies, operational amplifier’s outputvaries. To analyze discontinuous system, it is necessary tofollow complex procedures [7].
In this study, we propose new RC chaotic oscillator. The
proposed model is composed of RC phase shift oscillators
using operational amplifier and pulse generators. By inputing pulse wave to operational amplifier of RC phaseshift oscillator, chaotic phenomena are observed easily. We
carry out computer calculation and circuit experiment in order to check the validity of the proposed model. And, we
derive the one parameter bifurcation diagram.

2. Circuit Model
Figure 1 shows the circuit model in this study. The proposed model is composed of RC phase shift oscillator with
an operational amplifier and a pulse generator. The pulse

RRvoCRav1

Cv2

Cv3

Figure 1: Circuit model.
Figures 2 shows the approximated vi − vo characteristicof the deﬀer amplifier.

voERa/RbVs - v3-E(a)(b)

Figure 2: Approximated vi − vo characteristic of the deﬀer
amplifier.
vi corresponds to VS − v3 . By using diﬀer amplifiers, theproposed model has a continuity unlike using comparators.
The diﬀer amplifier’s output varies smooth even though theoutput of pulse generators varies. The following equationof the approximated vi − vo characteristic of the deﬀer amplifier is described as follows:{}
Ra1 Ra| (VS − v3 ) + E| − | (VS − v3 ) − E| .
vo =2 RbRb
(1)
Figure 3(a) shows the input voltage waveform VS (t). V
is the amplitude of the pulse voltage and T is the period of- 463 -Vs (t)β(τ)

Tdefine three piecewise-linear regions as follows:γ

R1 : α(β − z) ≤ −1
R2 : −1 < α(β − z) < 1
R3 : α(β − z) ≥ 1.

βEtτNamely, R1 corresponds to the region where the output ofdiﬀer amplifier is minimum values. On the other hand, R3
corresponds to the region where the output of diﬀer amplifier is maximum values. And, R3 corresponds to the region
where the output of diﬀer amplifier is varied.
We calculate the eigenvalues in each region from Eq. (4).
The eigenvalues in each region are described as follows,－β－E(a)(b)

Figure 3: Input voltage waveforms.
the waveform. Figure 3(b) shows the normalized voltage
waveform Vβ (τ). Vβ corresponds to VS , γ corresponds to Tand τ corresponds to t.
The proposed model consists of only simple RC phaseshift oscillators and a pulse generator. Namely, neither inductors nor negative resistors are included. Therefore, it isconsidered to be realized on the IC chip without complication.
The circuit equations are described as follows:dv1= −2v1 + v2 + vo
dt
dv2
RC
= v1 − 2v2 + v3
dt
dv3
RC
= v2 − v3 .
dt

R1 and R3 : λa , λb , λc
R2 : λ2 , σ2 ± jω2 .

λ + 2 −10−1 λ + 2 −10−1 λ + 1

λ + 2 −1α−1 λ + 2 −10−1 λ + 1

E3 = [E31 E32 E33 ]T ,
(3)

(10)

respectively. These values are calculated by putting rightside of Eq. (4) to be equal to zero.
Then, we can described the exact solutions as follows.
In R1 and R3 :xa (τ) − En = F(τ) · F−1 (0) · (xa (0) − En ),ẋ = −2x + y + f (z)ẏ = x − 2y + z

where
xa (τ) = [xn (τ) yn (τ) zn (τ)]T ,(4)

F(τ) = [fa (τ) fb (τ) fc (τ)]T ,fc (τ) = [λa λb λc ],dfc (τ)+ fc (τ),
fb (τ) =dτdfb (τ)+ fb (τ).
fa (τ) =dτ

where f (z) is a piecewise-linear function corresponding to
the vi − vo characteristic of the deﬀer amplifier and is described as1{|α(β − z) + 1| − |α(β − z) − 1|} .
2(9)E2 = [E21 E22 E23 ]T

the normalized circuit equation is described as follows:f (z) == 0.

E1 = [E11 E12 E13 ]Td=˙dtż = y − z.

(8)

Next, we define the equilibrium points in each regions as(2)Ra, VS = Eβ, T = RCγ,Rb= 0.

On the other hand, the eigenvalues of R2 can be derived
from

By using the following variables and the parameters,

t = RCτ, α =(7)The eigenvalues of R1 and R3 can be derived fromRC

v1 = Ex, v2 = Ey, v3 = Ez,(6)

(5)

(11)
(n = 1, 3)

3. Exact Solutions
In R2 :
Since the circuit equations (4) are piecewise-linear, exactsolutions in each linear region can be derived. At first, we- 464 -xb (τ) − E2 = G(τ) · G−1 (0) · (xb (0) − E2 ),( a)where( b)(c) (d) (e) (f)( g)(h) (i)xb (τ) = [x2 (τ) y2 (τ) z2 (τ)]T ,G(τ) = [ga (τ) gb (τ) gc (τ)]T ,gc (τ) = [eλ2 τ eσ2 τ cos ω2 τ eσ2 τ sin ω2 τ]T ,dgc (τ)gb (τ) =+ gc (τ),
dτdgb (τ)ga (τ) =+ gb (τ).
dτ(12)

4. Results
We show the chaotic attractors which are obtained bycomputer calculation and circuit experiment as shown
Fig. 4.

Figure 5: One parameter bifurcation diagram when α isvaried.

0.25

S = S1 ∩ S2 ,x−y-0.5

where0.5

S1 : β > 0-0.25

S2 : τ = nγ0.125y−z-0.25
0.25-0.1250.5

0.125

z−x-0.125

-0.5

(1)

(13)

(2)

Figure 4: The chaotic attractors obtained from the proposedmodel. (1) Computer calculation. (α = 50, β = 0.03,
and γ = 7.3) (2) Circuit Experiment. (R = 10[kΩ], Ra =750[kΩ], Rb = 15[kΩ], C = 103[nF], VS = 0.3[V] andf = 140[Hz])(n = 1, 2, 3, ...).

(14)

The subspace S1 shows that the output of pulse generator
is maximum, while the subspace S2 shows that the outputof pulse generator varies. Namely, S corresponds to thetransitional condition from −β to β.
We chose α as the control parameter and other parameters are fixed as β = 0.03, γ = 7.3.
Figure 5 shows the one parameter bifurcation diagram
when α is varied from 1 to 50. The vertical axis shows x1 ,the horizontal axis shows α, parameter step is 0.001 andplotted point is 1000[τ]. And, Figures 6 show the attractorsof computer calculation.
For 1 < α < 14.29, the oscillation is not observed fromFig. 6(a) and the values vary depending on the initial values. From these results, we can observe the periodic orbits, torus and chaotic phenomena. In particular, we focus
on the chaotic phenomena when α is 23.83 and 50. In order to compare two types chaotic phenomena, we show thePoincare maps and the largest Lyapunpv exponents when αis 23.83 and 50. The largest Lyapunov exponents are givenby

We can observed almost same attractors. The attractors ofRC chaotic oscillator which consists in a pulse generator
are angular[5][6]. On the other hand, the attractors of theproposed model are round.
In order to confirm the generation of chaos and to investigate bifurcation scenario, we derive the Poincare map.
Let us define the following subspace- 465 -µ=

α22.8350n∑
1x1 (τ + γ)
lim
log2.
n n → ∞ k=1x1 (τ)(15)

Table 1: Comparative results.
Lyapunov exponent Transitional condition0.000594
Continuity
0.000818
Discontinuity

0.25

0.25-0.5-0.5-0.5

0.5

0.5-0.25

0.5-0.25(a) α = 10-0.25(b) α = 18.1(c) α = 22
0.25

0.25

0.25

-0.5-0.5-0.5(e) α = 27.30.25(f) α = 30

-0.5-0.5
0.5

0.5

0.5

-0.25

-0.25-0.25(g) α = 40

[2] J. E. Skinner, M. Molnar, T. Vybiral and M. Mitra, Application of Chaos Theory to Biology andMedicine, Integrative Physiological and Behavioral
Science, vol. 27, Issue 1, pp. 39-53, January-March,
1992.

0.25

0.25

-0.5

[1] S. Vlad, P. Pascu and N. Moraiu, Chaos Models inEconomics, Journal of Computing, vol. 2, Issue 1,pp. 79-83, January, 2010.

-0.25

-0.25

-0.25

References

0.5

0.5

0.5(d) α = 23.83

orbits, torus and chaotic phenomena. Moreover, we haveobserved two types chaotic phenomena. The one is chaotic
phenomena with discontinuity. The other is chaotic phenomena with continuity in spite of using pulse generator.
In our future works, we will derive the Lyapunov exponents by using Jacobian in order to investigate the proposedmodel in detail. And, we will investigate the correlationsof between chaotic phenomena and other parameters.

0.25(h) α = 47.3(i) α = 50

[3] K. Aihara, Chaos Engineering and Its Applications
to Parallel Distributed Processing with Chaotic Neural Networks,” Proc. of IEEE 90.5, pp. 919-930, May,
2002.

Figure 6: Computer calculated results.
0.17

0.17

[4] K. M. Cuomo, A. V. Oppenheim and S. H. Strogatz,
Synchronization of Lorenz-based Chaotic Circuitswith Applications to Communications,” IEEE Trans.
Circuits and Systems, vol.40, no.10, pp, pp. 626-633,
October, 1993.

-0.33

-0.33

0.33

0.33

-0.17

α = 23.83-0.17

α = 50[5] A. S. Elwakil and S. Ozoguz, “Pulse-excited RC
Nonautonomous Chaotic Oscillator Structures,” Journal of Bifurcation and Chaos, vol. 15, no. 7, pp. 22572261, August, 2005.

Figure 7: The Poincare maps.

From Fig. 7, it is seen that two types chaotic attractors
are diﬀerent. In case that α is 50, piecewise-linear regions
is changed from R1 to R3 or R3 to R1 . Namely, discontinuity is observed on the proposed model. Almost RC chaoticoscillators containing pulse generator have discontinuity.
On the other hand, in case that α is 23.83, discontinuity is
not observed. By using diﬀer amplifier, piecewise-linear
regions is changed smoothly. Therefore, continuity is observed on the proposed model.

[6] S. Masuda, Y. Uchitani, and Y. Nishio, “Simple
Chaotic Oscillator Using Two RC Circuits,” Proc. ofNCSP’09, pp. 89-92, March, 2009.
[7] R. I. Leine, D. H. Van Campen, and B. L. Van
De Vrande, “Bifurcations in Nonlinear Discontinuous Systems,” Nonlinear dynamics, vol. 23, Issue. 2,pp. 105-164, October, 2000.

5. Conclusion
In this study, we have investigated a RC chaotic oscillator. The proposed model is composed of RC phase shiftoscillator with an operational amplifier and a pulse generator. By inputing pulse wave to operational amplifier ofRC phase shift oscillator, chaotic phenomena are observed
easily. We have carried out computer calculation and circuit experiment to investigate. From computer calculations
and circuit experiments, we have checked the validity ofthe proposed model. And, we have observed the periodic- 466 -