RC Chaotic Oscillator Using Pulse Generator and Differ Amplifier

Yoshinori Doike†, Yoko Uwate†and Yoshifumi Nishio†

† Dept. of Electrical and Electronic Engineering, Tokushima University2-1 Minamijosanjima, Tokushima, 770-8506 JapanEmail: doike, uwate, nishio@ee.tokushima-u.ac.jp

Abstract—In this study, a simple chaotic oscillator com-posed of RC phase shift oscillators using operational am-plifier and pulse generator is proposed and investigated. Byinputing pulse wave to operational amplifier of RC phaseshift oscillator, chaotic phenomena are observed easily. Wecarry out computer calculation and circuit experiment in or-der to check the validity of the proposed model. And, wederive the one parameter bifurcation diagram.

1. Introduction

Recently, many researchers have taken an their interestin chaos [1][2]. Because, chaotic phenomena can be ob-served in various fields and chaotic systems are good mod-els to explain and describe the higher dimensional nonlin-ear phenomena. In the field of electrical and electronicengineering, various applications based on deterministicchaos have been proposed [3][4]. For realizing such ap-plications, we need chaotic oscillators.

There are various chaotic oscillators [5][6]. These oscil-lators are composed of RC circuits, pulse generators andoperational amplifiers. By not using inductor, it can beeasily integrated chaotic oscillators. When the parame-ters are varied, chaotic attractor are observed in this sim-ple oscillator. However, it is not difficult to analyze thesechaotic oscillators. These oscillators are pulse-driven nonautonomous and discontinuous systems. When the outputof 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. Theproposed model is composed of RC phase shift oscillatorsusing operational amplifier and pulse generators. By in-puting pulse wave to operational amplifier of RC phaseshift oscillator, chaotic phenomena are observed easily. Wecarry out computer calculation and circuit experiment in or-der to check the validity of the proposed model. And, wederive the one parameter bifurcation diagram.

2. Circuit Model

Figure 1 shows the circuit model in this study. The pro-posed model is composed of RC phase shift oscillator withan operational amplifier and a pulse generator. The pulse

wave is inputted to non-inverting input terminal of the oper-ational amplifier. The differ amplifier produces the outputvoltagevo which is their power supply voltage, accordingto the input signals.

v2v1

R

v3

vo

Vs

Ra

Rb

Ra

R R

Rb

C C C

Figure 1: Circuit model.

Figures 2 shows the approximatedvi − vo characteristicof the deffer amplifier.

Vs

vo

v

-E

E

Ra/Rb

-3

(a) (b)

Figure 2: Approximatedvi − vo characteristic of the defferamplifier.

vi corresponds toVS − 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 approximatedvi − vo characteristic of the deffer am-plifier is described as follows:

vo =12

|Ra

Rb(VS − v3) + E| − |Ra

Rb(VS − v3) − E|

.

(1)

Figure 3(a) shows the input voltage waveformVS(t). Vis the amplitude of the pulse voltage andT is the period of

2015 International Symposium on Nonlinear Theory and its ApplicationsNOLTA2015, Kowloon, Hong Kong, China, December 1-4, 2015

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E

－E

Vs(t)

t

T

β

－

(τ)

τ

γ

β

β

(a) (b)

Figure 3: Input voltage waveforms.

the waveform. Figure 3(b) shows the normalized voltagewaveformVβ(τ). Vβ corresponds toVS, γ corresponds toTandτ corresponds tot.

The proposed model consists of only simple RC phaseshift oscillators and a pulse generator. Namely, neither in-ductors nor negative resistors are included. Therefore, it isconsidered to be realized on the IC chip without complica-tion.

The circuit equations are described as follows:

RCdv1

dt= −2v1 + v2 + vo

RCdv2

dt= v1 − 2v2 + v3

RCdv3

dt= v2 − v3. (2)

By using the following variables and the parameters,

v1 = Ex, v2 = Ey, v3 = Ez,ddt= ˙

t = RCτ, α =Ra

Rb, VS = Eβ, T = RCγ, (3)

the normalized circuit equation is described as follows:

x = −2x+ y+ f (z)

y = x− 2y+ z

z= y− z. (4)

where f (z) is a piecewise-linear function corresponding tothe vi − vo characteristic of the deffer amplifier and is de-scribed as

f (z) =12|α(β − z) + 1| − |α(β − z) − 1| . (5)

3. Exact Solutions

Since the circuit equations (4) are piecewise-linear, exactsolutions in each linear region can be derived. At first, we

define three piecewise-linear regions as follows:

R1 : α(β − z) ≤ −1

R2 : −1 < α(β − z) < 1

R3 : α(β − z) ≥ 1. (6)

Namely,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 ampli-fier is maximum values. And,R3 corresponds to the regionwhere 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,

R1 and R3 : λa, λb, λc

R2 : λ2, σ2 ± jω2. (7)

The eigenvalues ofR1 andR3 can be derived from∣∣∣∣∣∣∣∣λ + 2 −1 0−1 λ + 2 −10 −1 λ + 1

∣∣∣∣∣∣∣∣ = 0. (8)

On the other hand, the eigenvalues ofR2 can be derivedfrom ∣∣∣∣∣∣∣∣

λ + 2 −1 α−1 λ + 2 −10 −1 λ + 1

∣∣∣∣∣∣∣∣ = 0. (9)

Next, we define the equilibrium points in each regions as

E1 = [E11 E12 E13]T

E2 = [E21 E22 E23]T

E3 = [E31 E32 E33]T, (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 andR3:

xa(τ) − En = F(τ) · F−1(0) · (xa(0)− En),

where

xa(τ) = [xn(τ) yn(τ) zn(τ)]T,

F(τ) = [fa(τ) fb(τ) fc(τ)]T,

fc(τ) = [λa λb λc],

fb(τ) =dfc(τ)

dτ+ fc(τ),

fa(τ) =dfb(τ)

dτ+ fb(τ). (11)

(n = 1,3)

In R2:

xb(τ) − E2 = G(τ) ·G−1(0) · (xb(0)− E2),

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where

xb(τ) = [x2(τ) y2(τ) z2(τ)]T,

G(τ) = [ga(τ) gb(τ) gc(τ)]T,

gc(τ) = [eλ2τ eσ2τ cosω2τ eσ2τ sinω2τ]T,

gb(τ) =dgc(τ)

dτ+ gc(τ),

ga(τ) =dgb(τ)

dτ+ gb(τ). (12)

4. Results

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

x− y-0.5

0.5

0.25

-0.25

y− z-0.25

0.25

0.125

-0.125

z− x -0.125

0.125

0.5

-0.5

(1) (2)

Figure 4: The chaotic attractors obtained from the proposedmodel. (1) Computer calculation. (α = 50, β = 0.03,andγ = 7.3) (2) Circuit Experiment. (R = 10[kΩ],Ra =

750[kΩ],Rb = 15[kΩ],C = 103[nF],VS = 0.3[V] andf = 140[Hz])

We can observed almost same attractors. The attractors ofRC chaotic oscillator which consists in a pulse generatorare angular[5][6]. On the other hand, the attractors of theproposed model are round.

In order to confirm the generation of chaos and to inves-tigate bifurcation scenario, we derive the Poincare map.

Let us define the following subspace

(d)(c)(b)(a) (e) (f) (g) (h) (i)

Figure 5: One parameter bifurcation diagram whenα isvaried.

S= S1 ∩ S2, (13)

where

S1 : β > 0

S2 : τ = nγ (n = 1,2, 3, ...). (14)

The subspaceS1 shows that the output of pulse generatoris maximum, while the subspaceS2 shows that the outputof pulse generator varies. Namely,S corresponds to thetransitional condition from−β to β.

We choseα as the control parameter and other parame-ters are fixed asβ = 0.03, γ = 7.3.

Figure 5 shows the one parameter bifurcation diagramwhenα is varied from 1 to 50. The vertical axis showsx1,the horizontal axis showsα, 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 val-ues. From these results, we can observe the periodic or-bits, torus and chaotic phenomena. In particular, we focuson the chaotic phenomena whenα is 23.83 and 50. In or-der to compare two types chaotic phenomena, we show thePoincare maps and the largest Lyapunpv exponents whenαis 23.83 and 50. The largest Lyapunov exponents are givenby

µ =1n

limn → ∞

n∑k=1

log2x1(τ + γ)

x1(τ). (15)

Table 1: Comparative results.α Lyapunov exponent Transitional condition

22.83 0.000594 Continuity50 0.000818 Discontinuity

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-0.5

0.5

0.25

-0.25

-0.5

0.5

0.25

-0.25

-0.5

0.5

0.25

-0.25

(a)α = 10 (b) α = 18.1 (c) α = 22

-0.5

0.5

0.25

-0.25

-0.5

0.5

0.25

-0.25

-0.5

0.5

0.25

-0.25

(d) α = 23.83 (e)α = 27.3 (f) α = 30

-0.5

0.5

0.25

-0.25

-0.5

0.5

0.25

-0.25

-0.5

0.5

0.25

-0.25

(g) α = 40 (h) α = 47.3 (i) α = 50

Figure 6: Computer calculated results.

-0.33

0.33

0.17

-0.17

-0.33

0.33

0.17

-0.17

α = 23.83 α = 50

Figure 7: The Poincare maps.

From Fig. 7, it is seen that two types chaotic attractorsare different. In case thatα is 50, piecewise-linear regionsis changed fromR1 to R3 or R3 to R1. Namely, discontinu-ity is observed on the proposed model. Almost RC chaoticoscillators containing pulse generator have discontinuity.On the other hand, in case thatα is 23.83, discontinuity isnot observed. By using differ amplifier, piecewise-linearregions is changed smoothly. Therefore, continuity is ob-served on the proposed model.

5. Conclusion

In this study, we have investigated a RC chaotic oscil-lator. The proposed model is composed of RC phase shiftoscillator with an operational amplifier and a pulse gener-ator. By inputing pulse wave to operational amplifier ofRC phase shift oscillator, chaotic phenomena are observedeasily. We have carried out computer calculation and cir-cuit experiment to investigate. From computer calculationsand circuit experiments, we have checked the validity ofthe proposed model. And, we have observed the periodic

orbits, torus and chaotic phenomena. Moreover, we haveobserved two types chaotic phenomena. The one is chaoticphenomena with discontinuity. The other is chaotic phe-nomena with continuity in spite of using pulse generator.

In our future works, we will derive the Lyapunov expo-nents by using Jacobian in order to investigate the proposedmodel in detail. And, we will investigate the correlationsof between chaotic phenomena and other parameters.

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