IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS II, VOL. X, NO. Y, DECEMBER 2015 1

Simple Two-Transistor Single-SupplyResistor-Capacitor Chaotic Oscillator

Lars Keuninckx, Guy Van der Sande and Jan Danckaert†

Abstract—We have modified an otherwise standard one-transistor self-biasing resistor-capacitor phase-shift oscillator toinduce chaotic oscillations. The circuit uses only two transistors,no inductors, and is powered by a single supply voltage. As suchit is an attractive and low-cost source of chaotic oscillations formany applications. We compare experimental results to Spicesimulations, showing good agreement. We qualitatively explainthe chaotic dynamics to stem from hysteretic jumps betweenunstable equilibria around which growing oscillations exist.

Index Terms—chaos, oscillator. nonlinear dynamics, RC-ladder.

Copyright (c) 2014 IEEE. Personal use of this material ispermitted. However, permission to use this material for anyother purposes must be obtained from the IEEE by sendingan email to pubs-permissions@ieee.org

I. INTRODUCTION

RECENTLY the interest in chaotic systems has beenrevived following the advent of novel and worthwhile

applications. Chaotic systems are now being used for so-called chaos encryption, in which data signals can be securedby hiding them within a chaotic signal. Relying on chaossynchronization the original messages can then be safelyrecovered [1]. Also, the long-term unpredictability of thechaotic signal makes these systems well suited for true randombit generation [2]. In robotics, chaotic signals are being usedin neural control networks or as chaotic path generators [3].Finally, from a more fundamental point of view, a shift ininterest towards networks of interconnected chaotic unit cellscan be observed [4]. In all these applications, a need existsfor flexible and preferably low cost chaotic circuits. Discretecomponents are preferable over operational amplifiers andspecialized multiplier chips, the latter only being produced bya few suppliers. Single supply and low voltage operation widenthe applicability, as does a wide easily tunable frequencyrange. Finally, using chaotic circuits as units in large networksadds the condition that the couplings between these units aredependable and stable. Requiring inductorless circuits willavoid such difficulties.

Since the development of Chua’s circuit [5], there has beena considerable interest in the construction of autonomouschaotic circuits. Well known circuits, such as the Wien bridgeoscillator, have been modified to produce chaos [6]. Often,these modifications involve the addition of an extra energystorage element such as an inductor at the right place, thereby

†The authors are with the Applied Physics Research Group (APHY), VrijeUniversiteit Brussel (VUB), Pleinlaan 2, 1050 Brussel, Belgium, correspon-dence e-mail: lars.keuninckx@vub.ac.be

adding a dimension to a predominantly two-dimensional limitcycle oscillator. In other cases, notably the Collpits oscilla-tor [7], a chaotic regime is already present for certain valuesof the design parameters. Others directly translate a knownchaotic system of differential equations to electronics. Thenecessary nonlinear terms are implemented by using dedicatedanalog multipliers as in reference [8] or operational amplifierbased piece wise continious functions [9]. Being able toavoid operational amplifiers and dedicated multipliers is aplus in terms of cost and circuit complexity, as there isno connection between complexity -in terms of used parts-of the circuit and complexity of the chaotic oscillations -interms of measurable quantitative properties such as entropy,attractor dimension, spectral content etc. Therefore chaoticoscillators using minimal and more basic components as inreferences [10] and [11] are desirable from an economic andlogistic viewpoint.

Elwakil and Kennedy conjecture that every autonomouschaotic oscillator contains a core sinusoidal or relaxationoscillator [12]. Accordingly, one can derive a chaotic oscillatorfrom any sinusoidal or relaxation oscillator. Motivated bythis, we set out to modify the well known resistor-capacitor(RC) phase shift oscillator, described as early as 1941 inreference [13] to produce chaos. It has been used previouslyas the basis for chaotic circuits. Hosokawa et al. reporton chaos in a system of two such oscillators, coupled bya diode [14]. They analyze the system as two nonlinearlycoupled linear oscillators. Ogorzalek describes a chaotic RCphase shift oscillator based on a piecewise-linear amplifierbuilt using operational amplifiers, however the RC-ladder isunmodified [15]. In our approach we use a transistor based RCoscillator to which a small subcircuit is added, directly inter-acting with the RC-ladder itself. The purpose of the subcircuitis to add a nonlinearity in the RC-ladder. The attractivenessof the resulting circuit lies in its simplicity and low cost andparts count. No specialized parts such as dedicated multipliersor rare inductor values are needed. Neither the componentvalues nor the supply voltage is critical. The operation does notdepend on the dynamic properties of the active components,as is often the case in even simpler chaotic circuits employingonly one transistor. The main oscillating frequency has a rangeof over five decades, from below 1 Hz up to several hundredkilohertz. which can be reached by scaling the capacitorsand/or resistors.

Additionally, the circuit has the attractive feature that theunderlying core oscillator is clearly visible in the structure.This makes it a great introductory pedagogical tool for studentsinterested in chaos.

IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS II, VOL. X, NO. Y, DECEMBER 2015 2

vBE2 B2i

B1i

R R R

C C C

Vp

v v21

v

R

R C Q

Q1

24 2

1

1

CE1

R23

vCE2

v

R

BE1

Figure 1. A two-transistor chaotic RC phase shift oscillator. The componentsin the dashed line box cause chaotic dynamics in an otherwise standard self bi-asing RC oscillator. The components have the following values: R = 10 kΩ,R1 = 5 kΩ, R2 = 15 kΩ, R3 = 30 kΩ, C = 1 nF, C2 = 360 pF, andVP = 5 V.

0

0.5

1

1.5

2

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

v CE

1 v

CE

2 [

V]

t [ms]

vCE1 vCE2

Figure 2. Timetrace of vCE1 (upper trace, red in color) and vCE2 (lowertrace, blue in color) for R4 = 44 kΩ, Vp = 5 V, other component valuesas stated in the text.

II. CIRCUIT AND EXPERIMENTAL RESULTS

Figure 1 introduces the circuit that we have designed. Itconsists of a standard single-supply self-biasing RC phaseshiftoscillator with an added subcircuit (inside the dashed line box)interacting with the RC-ladder. Unless stated otherwise, thefollowing component values are used: R = 10 kΩ, R1 =5 kΩ, R2 = 15 kΩ, R3 = 30 kΩ, C = 1 nF, C2 = 360 pF.The transistors Q1 and Q2 are of the type BC547C althoughthis is not critical. Both the first resistor of the RC-ladder andthe collector resistor of transistor Q1, have been chosen equalto 1/2R. Since the output resistance of a common emitteramplifier roughly equals the collector resistor, the combinedresistance equals R, thus forming the first resistor of the ladder.The frequency at which the RC-ladder has 180 phase lag nec-essary for oscillation, is given by f =

√6/2πRC ≈ 39 kHz.

Due to the loading of the last RC stage by transistor Q1’s base,the free running frequency is shifted to 54 kHz. The subcircuit

consisting of Q2, R2, R3, R4 and C2 is responsible for thechaotic behaviour. This subcircuit adds a new equilibriumin which transistor Q1 is also biased as an active amplifier,enabling oscillations. It is clear that for low R4 or low Vp,transistor Q2 does not conduct and the circuit reduces tothe unmodified phase shift oscillator. The base oscillationfrequency in the chaotic operating regime for R4 = 44 kΩ,Vp = 5 V is approximately 44 kHz. The circuit consumes3.1 mW at 5 V. In Figure 2, we show the time trace of vCE1.The dynamical behavior consists of jumps between two statesof high and low voltage of the collector of Q2. In betweenthese jumps growing oscillations are observed on the collectorof Q1. Note that the collector voltage of Q2 has an almostbinary distribution. In Figure 3(a), we show an oscilloscopepicture of vCE1 vs. vCE2 for R4 = 44 kΩ and Vp = 5 V, fromwhich the bistable oscillations around two unstable equilibriaare clearly visible. Figures 3(b) and 3(c), showing vCE1 vs.v2, and v1 vs. v2 respectively, also attest to this, wherethe openings or ’eyes’ of the attractor indicate a possibleunstable equilibrium at their center. To estimate the largestLyapunov exponent, we use the Time Series Analysis ’Tisean’package[16], from a timetrace consisting of 50000 points ofvCE1, representing 10 ms, as λ ≈ (0.045± 0.005) /µs. Apositive Lyapunov exponent is a strong indication of chaoticdynamics. In Figure 4, we show oscilloscope pictures of v1vs. vCE1 for increasing values of R4 at Vp = 5 V, indicatesthe evolution of the dynamics to chaos. For R4 = 0 kΩ thesubcircuit is inactive such that a simple limit cycle exists (notshown). Close to R4 = 30 kΩ [Figure 4(a)] we see a perioddoubling bifurcation. At R4 = 31.6 kΩ [Figure 4(b)] a criticalslowing down occurs as shown by the point of high intensityon the analog oscilloscope picture. This value marks the onsetof a bistable operation. Subsequent period doublings lead tochaos as seen in Figure 4(d) and Figure 4(e). The attractorconsists of oscillations around two unstable equilibria withlower and higher average vCE1. The asymmetry between theupper and lower opening of the attractors is caused by differentaverage collector current of transistor Q1, resulting in differentgain. This asymmetry is seen regardless of which projection ischosen. Further increase of R4 [Figure 4(f)] leads to a periodhalving sequence.

III. MODEL AND SPICE SIMULATION

The circuit of Figure 1 is described by:

RCdv1dt

= − v1(

1 +R

R1− RR3

R1(2R3 +R1)

)(1)

+ v2 +RR3

2R3 +R1

(VpR1− iC1 +

vBE2

R3

)RC

dv2dt

= − 2v2 + v1 + vBE1 − iC2R, (2)

RCdvBE1

dt= − vBE1 + v2 − iB1R, (3)

(2R3 +R1)C2dvBE2

dt= − vBE2

(2 +

2R3 +R1

R4

)(4)

+ v1 + Vp − iC1R1

− iB2 (2R3 +R1) .

IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS II, VOL. X, NO. Y, DECEMBER 2015 3

(a) (b) (c)

Figure 3. Attractor projections from the circuit for R4 = 44 kΩ, Vp = 5 V. (a) horizontal: vCE1, 0.2 V/div, vertical: vCE2, 0.1 V/div, (b) horizontal:vCE1, 0.2 V/div, vertical: v2, 20 mV/div, (c) horizontal: v1, 0.1 V/div, vertical: v2, 20 mV/div.

(a) R4 = 30 kΩ. (b) R4 = 31.6 kΩ. (c) R4 = 37.5 kΩ.

(d) R4 = 39.5 kΩ. (e) R4 = 44 kΩ. (f) R4 = 75 kΩ.

Figure 4. The route to chaos. Horizontal: vCE1, vertical: v1 for increasing values of R4 at Vp = 5 V. The asymmetry between the upper and lower ’eye’of the attractors is caused by different average collector current of Q1, resulting in different gain.

The currents iB1, iB2, iC1 and iC2 are determined by thetransistor model and are a function of the base-emitter andcollector voltages vBE1, vBE2, vCE1 and vCE2. These formthe nonlinearities in the circuit. Note R2 is absent from (1)-(4), however since vCE2 = v2 − iCE2R2, current iC2 isinfluenced by R2. In Figure 5, we show timetraces of vCE1

and vCE2 as generated from Spice. These again show bistableoscillations for vCE1 and jumping between a high and a lowstate for vCE2 and have a good qualitative agreement with themeasurements in Figure 2. In Figure 6, a three dimensionalplot of this simulation adding v1 as third variable, is shownwith color intensity coding used to indicate time from orange(beginning of the simulation) to light yellow (towards the end

of the simulation). The bistable nature of the circuit in thisoperating regime is clear. A projection on the vCE1-v1 plane isshown for comparison with Figure 4(e). Figure 7(a) shows theone dimensional bifurcation diagram of vCE1 for parameterR4, with Vp = 5 V, where the dynamics start for low valuesof R4 with a limit cycle, evolving to chaos at R4 ≈ 45 kΩthrough period doubling bifurcations. The chaotic dynamicsare interspersed with small periodic windows indicating thepossible co-existance of a chaotic attractor with a limit cycle.Similar results can be obtained by varying R3 while keepingother parameters fixed. In general it must be noted that thelocation of the features in these plots depends much on thetransistor model, even for the same type of transistor, as

IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS II, VOL. X, NO. Y, DECEMBER 2015 4

0

0.5

1

1.5

2

10 10.1 10.2 10.3 10.4 10.5 10.6 10.7

v CE

1 v

CE

2 [

V]

t [ms]

vCE1 vCE2

Figure 5. vCE1 (black) and vCE2 (grey) via Spice simulation for R4 =44 kΩ and Vp = 5 V.

0 0.4 0.8 1.2 1.6 2

0.4

0.8

1.2

1.6

-0.4

0

0.4

0.8

vCE2 [V]

vCE1 [V]

v1 [V]

vCE2 [V]

Figure 6. A three dimensional view of the attractor at Vp = 5 V andR4 = 44 kΩ, generated from a 5 ms Spice generated timeseries in 100 nsincrements. Coloring intensity indicates transistion time during the simulation,from orange at the beginning, to light yellow, near the end of the simulation.

these models differ somewhat per manufacturer. Figure 7(b)indicates that the circuit can produce chaos over a wide rangeof supply voltages as confirmed in our experiments.

IV. THE ’HIDDEN’ BISTABLE CIRCUIT

Here we offer a qualitative explanation for the appearance ofchaos in this circuit. Intuitively the circuit can be understoodas a Schmitt-trigger combined with an oscillator as follows.If one removes capacitors C from Figure 1 as in Figure 8,one ends up with a circuit in which there is only one energystoring element, C2. The node voltages in equilibrium ofthis one dimensional circuit equal those of the original fourdimensional circuit. Often when a circuit shows bistability,in at least one of the states the active components are eithersaturated or non-conducting such that there is no gain availableto support oscillation. We now show that transistor Q1 hasgain in both states. Consider the circuit with capacitors Cremoved as described above. Intuitively, if vBE2 < 0.6 V,Q2 is non conducting, then vCE1 ≈ vBE1 ≈ 0.6 V. If bysome external disturbance or a variation of R4, vBE2 risesabove ≈ 0.6 V, Q2 will begin to conduct. Consequently vBE1

decreases, resulting in an increase of vCE1, increasing vBE2,etc. until Q2 is saturated. At this point, with the choice ofcomponents where R1 +R = R2, we have vCE1 ≈ 2×0.6 V.

0

0.5

1

1.5

2

20 40 60 80 100

vC

E1

loca

l e

xtr

em

a

[V]

R4 [kΩ]

(a)

0

0.5

1

1.5

2

3 4 5 6 7 8

vC

E1

loca

l e

xtr

em

a

[V]

Vp [V]

(b)

Figure 7. One dimensional bifurcation diagrams generated from Spice data.(a) over R4 with V p = 5 V , (b) over Vp for R4 = 44 kΩ. Both plots showperiod doubling routes to chaos, periodic windows and crises.

R C Q24 2

vBE2

R2R3

vCE2

BE1v

R R R

Vp

v v21

v

R

Q1

1

1

CE1

Figure 8. By removing the capacitors C associated with the oscillator inFigure 1, the embedded bistable circuit becomes visible.

IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS II, VOL. X, NO. Y, DECEMBER 2015 5

Thus intuitively there two states exist, distinguished by Q2

being either conducting or not, while Q1 is biased as an activeamplifier in both states. These states are stable for the circuitof Figure 8. In the case of the full four dimensional circuithowever, these states are unstable and form the centers of theobserved oscillations. Indeed both experimentally and numer-ically we find that transistor Q2 is most of the time eithernon conducting or saturated, as shown in Figure 2. Hystereticjumping as a basis of chaotic dynamics has been found inother chaotic oscillators, as described in references [17] and[18].

V. DISCUSSION

A chaotic oscillator derived from a self biasing discreteresistor-capacitor ladder oscillator has been introduced. Thecircuit does not need specialty components and operates from asingle supply voltage. It has a wide frequency range which canbe chosen by scaling the capacitors and/or resistors. Thereforeit is suited for many applications such as robotics, randomnumber generators and chaos encryption. Being inductorlessand thus avoiding hard-to-manage magnetic couplings betweennearby units, it also lends itself well for research into thedynamics of chaotic networks. Spice simulations confirmthe chaotic dynamics found experimentally. Intuitively thecircuit can be explained as an oscillator modified such thatgrowing oscillations exist around two unstable fixed points,with hysteretic jumping between them.

ACKNOWLEDGMENT

This research was supported by the Interuniversity Attrac-tion Poles program of the Belgian Science Policy Office, undergrant IAP P7-35 ”Photonics@be”. The authors acknowledgethe Research Foundation-Flanders (FWO) for project support,the Research Council of the VUB and the Hercules Founda-tion. We thank S. T. Kingni for the stimulating discussions.

REFERENCES

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[12] A. S. Elwakil and M. P. Kennedy, “Chua’s circuit decomposition: asystematic design approach for chaotic oscillators,” J FRANKL INST,vol. 337, February 2000.

[13] E. L. Ginzton and L. M. Hollingsworth, “The phase shift oscillator,” PIRE, vol. 29, 1941.

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[18] S. Nakagawa and T. Saito, “Design and control of rc vccs 3-d hysteresischaos generators,” IEEE TCAS I, vol. 45(2), February 1998.