Papers

International Journal of Bifurcation and Chaos, Vol. 10, No. 5 (2000) 959–970c© World Scientific Publishing Company

EXPERIMENTAL DEFINITION OF THE BASIN OFATTRACTION FOR CHUA’S CIRCUIT

GUIDO PEGNA, RITA MARROCU, ROBERTO TONELLI and FRANCO MELONIINFM – Dipartimento di Scienze Fisiche, Universita di Cagliari,

Cittadella Universitaria di Monserrato, 09042 Monserrato (CA), Italy

GIOVANNI SANTOBONICentre for Nonlinear Dynamics and its Applications,

University College London, Gower Street, London WC1E 6BT, UK

Received July 15, 1998; Revised September 2, 1999

In this paper we study the experimental determination of the basin of attraction for the Chua’scircuit by means of an electronic device that is able to select initial voltages and to showthe early stages of the subsequent trajectory on the oscilloscope. The results are shown anddiscussed in cases of multistability of periodic and chaotic solutions.

1. Introduction

Chaos is a dynamic phenomenon that can arisein physical nonlinear systems of very different na-tures, but nowhere else is it so ubiquitous and eas-ily observed as in electronic circuits [Thompson &Chua, 1995]. We can identify two main reasons forthis. First, the simplicity and inexpensive natureof the devices used, and second, the physics of elec-tronic devices is well understood, so that the mod-eling equations of motion are the best expressionof the particular physical phenomenon. These cir-cuits could be integrated into LSI chips if massivepractical applications should emerge.

The earliest observations of chaos in electroniccircuits were in forced nonlinear oscillators, like thesinusoidally excited neon bulb relaxation oscillator[van der Pol & van der Mark, 1927] and the forcednegative resistance oscillator [Ueda & Akamatsu,1980]. But among all electronic circuits, Chua’scircuit [Madan, 1993] deserves a special place, forits simplicity and universality. Simple because itconsists of all linear elements but one, a nonlin-ear resistor (Chua’s diode) with a piecewise-linearv–i characteristic. Universal because all charac-teristic features of chaotic motion (like period-

doubling, intermittency, torus breakdown, etc.)have all been observed in this circuit [Madan, 1993].

The presence of noise and inherent experimen-tal uncertainty in the definition of physical parame-ters make the task of an exact comparison betweenexperimental and numerical results unfeasible. Al-though the overall scenario may be qualitativelycomparable, particular thresholds, basin bound-aries, bifurcations, may be quantitatively hard tolocate. On the other hand, it is quite common forthe behavior of a chaotic circuit to be affected by theperturbation introduced by the switching mecha-nism at instant t = 0, when a rough switch-on takesplace. This peculiarity forbids the reproducibility ofa particular experimental situation, especially whena specific initial condition is required to be set up.

In this paper, we try to experimentally repro-duce the basin boundaries for different multistablesolutions of Chua’s circuit. Our task will be to setthe initial voltages in the circuit as we like, withgood precision. In order to carry out this investiga-tion, the whole chaotic circuit has been redesigned,as we will explain later. A similar approach, al-though with different components have been carriedout by [Heagy et al., 1994].

959

960 G. Pegna et al.

2. Chua’s Circuit

In Fig. 1 we show a schematic representation of asingle Chua’s circuit. The equations of motion rep-resenting the circuit are

C1dv1

dt=

1

R(v2 − v1)− f(v1)

C2dv2

dt=

1

R(v1 − v2) + iL

LdiLdt

= −v2 −R0iL

(1)

where v1, v2, and iL are, respectively, the voltagesacross capacitors C1 and C2, and the current flow-ing through inductor L. R and R0 are the resistanceat the top of the circuit and the internal resistanceof the inductor, respectively. In Eq. (1), f(v1) isthe piecewise-linear function

f(v1) = Gbv1 +1

2(Ga−Gb)(|v1 +Bp| − |v1−Bp|)

(2)

where Bp is the breakpoint voltage and Ga andGb are the slopes of the linear parts beyond thebreakpoints.

We characterized the circuit by measuring allits parameters, and we found the values reported inTable 1. The experimentally determined v–i char-acteristic of Chua’s Diode is shown in Fig. 2, withthe measured points represented by open circles.The solid line represents the numerical fit used toderive those values present in Table 1; the fittingoperation shows us that the v–i characteristic is

Table 1. Experimental values used inthis paper. The additional resistancesRa and Rb described in Sec. 2 havevalues such that R/Ra = 0.09 and1/Rb = 0.05 mS.

Parameters Experimental

C1 (10.03± 0.02) nF

C2 (91.32± 0.02) nF

L (19.50 ± 0.05) mH

RL (2.6000 ± 0.0003) Ω

Ga (−0.725± 0.005) mS

G+b (−0.370± 0.005) mS

G−b (−0.371± 0.005) mS

B+p (1.045 ± 0.002) V

B−p (0.957 ± 0.002) V

slightly asymmetric (although in some parts withinthe uncertainty of the measurement), due to thepossibility of slight differences between the positiveand negative saturation voltages of the amplifiersused in our circuit. So, for parameters in the for-mula (2) we use G+

b , B+p , or G−b and B−p , whether

v1(t) is positive or negative. A better match be-tween positive and negative slopes of the v–i charac-teristic could likely be obtained by the use of betterquality operational amplifiers, such as, for instance,model AD712 by Analog Devices.

With a fourth-order Runge–Kutta integrationalgorithm with adaptive stepsize [Press et al., 1992]we simulated the flow generated by Eqs. (1). For

Fig. 1. Schematic representation of the Chua’s oscillator. The Chua’s oscillator consists of a linear inductor L with a seriesinternal resistance R0, a linear resistor R, two capacitors C1 and C2, and a nonlinear resistor NR.

Experimental Definition of the Basin 961

-10.0 -5.0 0.0 5.0 10.0vR (V)

-4.0

-2.0

0.0

2.0

4.0

i R (

A)

-Bp--B’p

B’p

Ga

Gb-

Gb+

Bp+

Fig. 2. Measured v–i characteristic of the nonlinear resistor NR. The circles represent the experimental points, while thesolid line represents the numerical fit used to extract the parameter values. In the negative slope regions, the inner part hasslope Ga, while for the outer part the slope is G±b . For higher |vR|-values the slope becomes positive, as for every physicalrealizable nonlinear resistor. The fitted values revealed that G+

b 6= G−b and B+p 6= B−p , so that the characteristic is asymmetric.

the values reported in Table 1 the numerical solu-tions are slightly different from experimental ones.Our opinion is that the difference is generated bythe presence of parasitic resistances in all the cir-cuit set-up, but mainly by the hysteresis cycle ofthe inductor L, that we call Ra, variable in vary-ing the resistance R at the top of the circuit, anda fixed contribution from the two capacitors, whoseeffect is to change slightly the form of the v–i char-acteristic (2). If the elaborated set-up carries withit an additional resistance Rb, the effect of connect-ing such a resistance in parallel with the negative-resistance diode is to change the slopes of the curve(2) as Ga → Ga + G, and Gb → Gb + G, whereG = 1/Rb. On the other hand, the presence of theresistance Ra modifies the Eqs. (1), as if there is aresistance in parallel with the capacitor C2, so thesecond equation of the circuit should be modified as

C2dv2

dt=

1

R[v1 − v2(1 + α)] + iL ,

where α = R/Ra. The modified version of the equa-tions of motion with the corrections α = 0.09, andG = 0.05 mS gives a much better agreement withthe numerical and the experimental behavior.

Figure 3 shows the bifurcation diagram ofChua’s circuit, Eqs. (1), in the interval R ∈[1.4, 2.2] kΩ. The different colors identify differentcoexisting solutions of Eqs. (1). The black pointsform the double-scroll chaotic attractor, ranging inthe interval R ∈ [R′, R′′]. The double-scroll at-tractor coexists, for a large interval of R-values,with a large-amplitude periodic solution, which welabel Cs, represented with blue points in Fig. 3.The double-scroll attractor loses its stability at R′

with trajectories converging towards Cs, and atR′′ through crises [Ott, 1993] giving birth to twonew chaotic solutions called single-scroll attractors.We label these attractors A±, whether the attrac-tor stays mostly in the positive (A+) or negative(A−) v1 axis. In Fig. 3, A+ and A− are repre-sented by red and yellow points, respectively. Atpoint R′′′ there is a bifurcation in which the large-amplitude periodic solution Cs loses its stability,giving way to one of the two periodic attractors C±.As resistance R increases, the periodic solutions ex-perience a reverse period-doubling bifurcation se-quence, until the two fixed points P± gain stabilityalong the way. In Fig. 3 two points are chosen onthe horizontal axis. The experimentally measured

962 G. Pegna et al.

1.4 1.6 1.8 2.0 2.2R (kΩ)

−10.0

−5.0

0.0

5.0

10.0

Ext

rem

a v 1

(t)

(V)

double scroll

Cs

R2 R1

R’ R’’ R’’’

C+

C-

A+P+

P-

A-

Fig. 3. Schematic bifurcation diagram of the Chua’s circuit, Eqs. (1), obtained by sampling, for several values of R ∈[1.4, 2.2] Ω the relative maxima and minima of the time history of v1(t). The different colors show all qualitatively differentsolutions. The black points belong to the double-scroll chaotic attractor, ranging, in the figure, in the interval R ∈ [R′, R′′]. Inthis range of R-values, the double-scroll attractor and the large periodic solution Cs (blue points) coexist. The red and yellowpoints belong, respectively, to A+ and A− just after the crises in R′′, so in the small interval R ∈ [R′′, R′′′] the three solutionsA± and Cs coexist. Finally, the red and yellow branches give birth, reversing the period-doubling sequence in increasing R,to C± and P±.

value of R2 = 1.870 kΩ, corresponds to a chaoticsingle-scroll solution just before (in decreasing R)the crises leading to the double-scroll solution, andR1 = 2.020 kΩ, is located in the interval of stabilityof the C± periodic solution. We will experimen-tally define and numerically compute the basins ofattraction for these two values of the resistance R.For R = R1 we have the multistablilty of C±, whilefor R = R2 the multistability of A± and Cs.

3. Experimental Apparatusand Operations

We now describe all the technical details of the sys-tem built to set the initial conditions to a preciselyknown value, and to show the first orbits in theirfree-running evolutions. In order to set initial con-ditions v10 = v1(0) and v20 = v2(0), the circuit must

be put in a state, which we call SET, in which thefollowing conditions are fulfilled:

(a) The two capacitors C1 and C2 are disconnectedfrom the circuit.

(b) Chua’s diode is set to a high impedence state(see Appendix A), to avoid any flow of currentto the rest of the circuit.

(c) Generators of initial voltages v10 across C1 andv20 across C2, are connected to the respectivecapacitors.

(d) Inductor L is disconnected from the circuit, toassure that the third initial condition is set toiL(0) = 0, as is usually done. Moreover, dis-connecting L when applying at point H the ini-tial voltage v20 is a must in this circuit, owingto the high current which would otherwise flowinto the low series resistance of the inductor,which would require an impractical high powervoltage generator for v20.

Experimental Definition of the Basin 963

Having accomplished the setting of initial condi-tions, steps (a)–(d) in reverse order must be ex-ecuted in the shortest possible time to start thefree-running dynamics of the chaotic circuit: dis-connect C1 and C2 from the voltage generators andreconnect them to the circuit together with L; re-set Chua’s diode to its normal state. We call thissecond state RUN. For the observation of the firstorbits with an analog oscilloscope the sequence ofthe alternate SET and RUN states is cyclically re-peated at least fifty times per second to have a vis-ible enough display. Moreover, the least possibletime must be spent in the SET state, while the timein the RUN state must be variable to let observa-tion time vary in order to display as much of theorbit as we like.

The complete diagram of the circuit is shownin Fig. 4 with related subfigures in boxes. At thetop of Fig. 4 is shown the skeleton of the circuit,with relevant points H, G, F , and D. Figure 4(top left) shows the circuits used to switch C1 andC2 from the SET to the RUN state, as explainedbelow. Figure 4 (top-right box) reports our imple-mentation of the Chua’s diode (op-amps D1 andD2) [Kennedy, 1992] with its switching mechanismto a high-impedance state. In the bottom-left aresketched the two options available for inductor L:The real and the synthetic one, together with theirswitching circuits. Finally, in the bottom-right box,there is a scheme of the command signals to all theanalog switches.

A first tentative version of the circuit was basedon reed relay as commutation devices. Unfor-tunately, in spite of its great simplicity, contactbouncing had a detrimental effect on the repeata-bility of trajectories. This fact led us to an en-tirely new design based on electronic commuta-tors in order to have fast, clean and reproduciblecommutations.

The four analog switches contained in the 4066CMOS integrated circuit have an ON resistancewhich is too high (80–350 Ω) and too dispersed,to be used as commutators in the insertion mecha-nism of initial voltages. For this reason, we intro-duced “ideal electronic switches” whose principle issketched in Fig. 5.

In such a structure, the operational amplifiertransforms the real analog switch inserted in itsfeedback loop into an ideal switch. When the switchis closed, the operational amplifier forces at its

inverting input, which is the output of the circuit,a voltage identical to the voltage applied to its non-inverting input, and it behaves as an ideal voltage

generator. On the contrary, when the electronicanalog switch is in the open state, if a FET input

op-amp is used, the circuit is seen at its terminal asan infinite resistance.

As already reported in [Zhong & Ayrom, 1985],

we found that small changes in R0 cause apprecia-ble shifts in the period-doubling bifurcations, so, in

order to give a more suitable definition of the valueof R0, we set this resistance to be the sum of tworesistances in series, R0 = RL+RS , where RL is the

internal resistance of inductor L in Fig. 4 (bottom-left), and RS is variable.

With reference to the complete circuit diagramsin Fig. 4 (and to command signals shown in thebottom-right box), during the SET cycle opera-

tional amplifier A1 and A2 apply the inital volt-ages across C1 and C2 respectively, while A3 and

A4 transform the values of resistances R3 and R6

in the Chua diode at infinity, causing point D tobe seen by the rest of the circuit as having infi-

nite impedance. During the same cycle the idealswitch around op-amp A5 is open, disconnecting

from ground the lower terminal of inductor L. Insuch a configuration, during the SET state the ini-tial voltages across the two capacitors are iden-

tical to v10 and v20, both within the [−6, 6] Vinterval.

In our experimental configuration, v10 and v20

are displayed using high precision digital volt-meters. As shown in the bottom left box of Fig. 4,

a synthetic inductor can be alternatively inserted inthe circuit in place of the real one. It is built up by

the circuit of op-amp A6 (see Appendix B). Syn-thetic inductors are interesting because of specificadvantages they have over real ones: Their series

resistance is adjustable down to very low values, al-lowing very high quality factors Q to be obtained

thus introducing another control parameter on thebehavior of the chaotic circuit, which might be ofrelevance in possible future studies on coupled cir-

cuits. Moreover, synthetic inductors do not sufferthe effects of the hystheresis of the core, and are eas-

ily integrated in LSI. In our circuit the switching ofthe synthetic inductor is simply obtained by insert-

ing one of the analog switches of the 4066 i.c. intoits feedback loop.

Fig. 4. The complete schematic diagram of the circuit. At the top of the figure is shown the skeleton of the circuit then, inthe relative boxes are shown, the switching mechanism from the SET to the RUN state (top left), the structure of op-ampsD1 and D2 (top right), the structure of inductor L (bottom left), and the scheme of the command signals to all the analogswitches (bottom right), respectively. The practical implementation uses two op-amps and six resistors to realize the Chua’sdiode. R varies between 1.483 kΩ and 2.483 kΩ, R1 = R2 = 218.6 Ω, R3 = 2.195 kΩ, R4 = 21.79 kΩ, R5 = 21.86 kΩ,R6 = 3.28 kΩ, in addition to the values previously reported in Table 1. The relative uncertainty on the resistances R1 ÷ R6

is 10−4.

964

Experimental Definition of the Basin 965

Fig. 5. Ideal switch mechanism.

The logic of the driving signals supplied to theintegrated analog switches is as follows:

a b c d

SET 0 0 1 1

RUN 1 1 0 0

where a and b are all the switches indicated in theschematic diagrams of Fig. 4 with indices 1 and 2.A suitable command circuit (not reported here) isbuilt up on another board which supplies drivingsignals in true and complementary form and whichallows the timing of the RUN state to be variedwithin large limits.

Output signals v1 and v2 (taken from the pointsX and Y ) are bufferized by op-amps A7 and A8,while points X and Y are directly accessible. Fi-nally, for monitoring purposes, an additional signalproportional to iL is made available at the outputof amplifier A5.

4. Experimental and NumericalBasin Boundaries

We experimentally mapped two different basins ofattraction for two different values of resistance R.Both cases will be compared with their numericalcounterparts. The experimental basins were ob-tained by first selecting the edge point (v10, v20) =(−6, −6) V, still with iL(0) = 0. We then variedthe value of v20 continuously, observing the con-vergence of each trajectory, until we reached pointv20 = +6 V.

After this, we repeat the procedure with an ini-tial increment in the v10 axis of 0.5 V, but choos-ing smaller increments whenever the boundariesseem to be more irregular, until we arrive at thepoint where (v10, v20) = (+6, +6) V. The crucialpoint is that the whole basin is determined “onthe flight”, i.e. observing the evolution of the or-bits never switching off the circuit.

Following this procedure, we determined thebasins of attraction for the two different valuesR1 = 2.020 kΩ (Fig. 6) and R2 = 1.870 kΩ (Fig. 8).In the periodic case, there are slight differences be-tween experimental and numerical values. The pe-culiar characteristic of this kind of basin is thatit is arranged by alternating stripes, but lookingat Figs. 6 and 7, the number and width of thesestripes are slightly different. In both figures thecolors red/yellow define initial conditions converg-ing towards C+/C− respectively. Now we turn tothe chaotic case. Looking at the basin reproducedin Fig. 9, the outer blue zone marks all initial con-ditions whose convergence is towards the large peri-odic orbit, while the internal red/yellow zones withtheir peculiar boundaries represent convergence to-wards the single-scroll chaotic attractors A+/A−

pointing, respectively, in the positive or negative v1-direction. Figure 9 has been produced with numer-ical integration of Eqs. (1). After 100 000 integra-tion steps, if the trajectory has a positive/negativev1-value, the point is marked as belonging to thered/yellow basin.

For R2 = 1.870 kΩ (numerical) the resultingorbits A+ and A− are chaotic single-scroll solu-tions, and from Fig. 3, that value of R is closerto the crises giving rise to the chaotic double-scroll

966 G. Pegna et al.

-6.0 -4.0 -2.0 0.0 2.0 4.0 6.0-6.0

-4.0

-2.0

0.0

2.0

4.0

6.0

v (V)10

v(V

)20

Fig. 6. Experimentally defined basin of attraction for R = 2.020 kΩ. Initial conditions in the red zone converge towardsthe periodic solution C+, while initial conditions in the yellow zone converge to the solution C−. The comparison with thenumerical result in Fig. 7 shows that qualitatively the agreement is good, but few differences can be recognized; this plotshows an evident asymmetry, e.g. the width of the stripes is bigger for the red ones, a peculiarity not seen in the numericalresult.

−6.0 −4.0 −2.0 0.0 2.0 4.0 6.0v10 (V)

−6.0

−4.0

−2.0

0.0

2.0

4.0

6.0

v 20

(V)

Fig. 7. Numerical basin of attraction for the convergence of Eqs. (1) onto the period-1 orbit C+ or C− for R = 2.020 kΩ.The meaning of the red and yellow zones is the same as in the previous figure.

Experimental Definition of the Basin 967

-6.0 -4.0 -2.0 0.0 2.0 4.0 6.0-4.0

-2.0

0.0

2.0

4.0

v (V)10

v(V

)20

Fig. 8. Experimentally defined basin of attraction for R = 1.870 kΩ. This value of R is slightly bigger than R′′ so, according tothe bifurcation diagram in Fig. 3, the system is close to the transition from single-scroll to double-scroll solutions. Trajectoriesstarting from the internal red and yellow zones converge respectively towards A+ or A−, but two additional zones are present.The outermost blue zone indicates a part of the basin belonging to the large periodic solution Cs (as recognizable in Fig. 3)while the green zone in the middle indicates initial conditions which we have not been able to locate in either the red oryellow zone.

−6.0 −4.0 −2.0 0.0 2.0 4.0 6.0v10 (V)

−4.0

−2.0

0.0

2.0

4.0

v 20

(V)

Fig. 9. Numerical basin of attraction for R = 1.870 kΩ. The meaning of the red, yellow and blue zones is the same as in theexperimental counterpart, Fig. 8.

968 G. Pegna et al.

solution, so some intermittent behavior can bepresent, in which the trajectories wander across A+

and A−, before suddenly converging towards oneof them.

Figure 8 represents the experimentally definedbasin for the same value of the parameter. The out-ermost blue part represents the part of the basinconverging towards the large external periodic so-lution Cs, but between this part and the internalbistable part of the basin we have indicated in greenan annular zone of the basin whose convergencewe have not been able to assign to either A+ orA− because of the intermittent behavior previouslydiscussed.

4.1. Role of the parameters

The experimental and numerical results for thisdynamical system show reasonable agreement, al-though we have to keep in mind the slight differ-ences in the arrangment of the red/yellow strips inboth basins. In this part of the paper we try toexplain how differences in the circuit’s parameters,even within the limits of experimental uncertainty,may change the shape of the basins.

First of all, we check the behavior of the basinof attraction on changing the value of resistance R,as in Fig. 1. As the value of R is lowered, the period-doubling sequence is passed through in the directionof the chaotic region, as can be seen in Fig. 3, andit also leads to a mixing of the two zones as seen inFigs. 6–9. So the red and yellow zones of the basinstill equally share the available phase space (in thesection iL(0) = 0), but they seem to be much moreintertwined.

Regarding any deviation from the almost sym-metric v–i characteristic (2), its effect is to breakthe symmetry of the basins of attraction. For ex-ample, setting |G+

b | > |G−b | the result is a significant

enlargement of the basin for the C+, in correspon-dence to the larger |Gb|-value. Conversely, if theasymmetry is reversed, the visible effect is to en-large the width of the yellow stripes. In conclusion,the steeper the slope of G±b , the greater the portionof phase space occupied by C±.

5. Conclusions

In this paper we described Chua’s circuit with anelectronic device capable of selecting initial volt-ages for it. We have detected a small disagreementbetween the experimental and the numerical behav-

ior and we have proposed an improvement of themodel which gives reasonable agreement. This de-vice allowed us to map basins of attraction for mul-tistable solutions, and the comparison between nu-merical and experimental results showed reasonableagreement.

We wish now to direct our next experimenttowards a mapping of the initial conditions whenthe situation is far more critical and the systemsare of higher dimensionality, i.e. when we dealwith problems of synchronization of coupled circuits[Pecora & Carroll, 1990]. In this case, the possibil-ity of very complex basins of attraction must betaken into account [Ott & Sommerer, 1994; Heagyet al., 1994], and some experimental evidencesof riddled/intermingled behavior have appearedin recent scientific literature, using Chua’s circuit[Kapitaniak, 1995; Kapitaniak et al., 1997].

Acknowledgments

The authors wish to thank P. Sirigu, A. Chessa andA. Bosin for their precious help in the practical andnumerical realization of this paper. During the sub-mission of the first version of the manuscript, thefollowing paper, dealing with the same problem,appeared in the scientific literature [Virgin et al.,1998]. We acknowledge S. R. Bishop for pointingit out to our attention, and also for a critical read-ing of the manuscript. In particular, G. Sartoboniwishes to thank G. M. H. van der Heijden for manyenjoyable discussions.

ReferencesBruton, L. T. [1969] “Network transfer functions us-

ing the concept of frequency-dependent negativeresistance,” IEEE Trans. Circuit Th. 16, 406–408.

Heagy, J. F., Carroll, T. L. & Pecora, L. M. [1994] “Ex-perimental and numerical evidence of riddled basinsin coupled chaotic systems,” Phys. Rev. Lett. 73,3528–3531.

Kapitaniak, T. [1995] “Experimental observation of rid-dled behaviour,” J. Phys. A28, 63–66.

Kapitaniak, T., Chua, L. O. & Zhong, G.-Q. [1997] “Ex-perimental evidence of intermingled basins of attrac-tion in coupled Chua’s circuit,” Chaos Solit. Fract. 8,1517–1522.

Kennedy, M. P. [1992] “Robust op amp realization ofChua’s circuit,” Frequenz 46, 66–80.

Madan, R. N. [1993] Chua’s Circuit: A Paradigmfor Chaos, ed. Madan, R. N. (World Scientific,Singapore).

Experimental Definition of the Basin 969

Ott, E. [1993] Chaos in Dynamical Systems (CambridgeUniv. Press).

Ott, E. & Sommerer, J. C. [1994] “Blowout bifurcations:The occurrence of riddled basins and on-off intermit-tency,” Phys. Lett. A188, 39–47.

Pecora, L. M. & Carroll, T. L. [1990] “Synchronizationin chaotic systems,” Phys. Rev. Lett. 64, 821–824.

Press, W. H., Teukolsky, S. A., Vetterling, W. T. &Flannery, B. P. [1992] Numerical Recipes in C(Cambridge Univ. Press).

Thompson, J. M. T. & Chua, L. O. [1995] “Chaotic be-haviour in electronic circuits,” R. Soc. Phil. Trans.353.

Ueda, Y. & Akamatsu, N. [1980] “Chaotically transi-tional phenomena in the forced negative-resistanceoscillators,” IEEE Trans. Circuits Syst. 28, 217–224.

van der Pol, B. & van der Mark, J. [1927] “Frequencydemultiplication,” Nature 120, 363–364.

Virgin, L. N., Todd, M. D., Begley, C. J., Trickey, S. T.& Dowell, E. H. [1998] “Basins of attraction in exper-imental nonlinear oscillators,” Int. J. Bifurcation andChaos 8, 521–533.

Zhong, G.-Q. & Ayrom, F. [1985] “Experimental confir-mation of chaos from Chua’s circuit,” Int. J. CircuitTh. Appl. 13, 93–98.

Appendix A

The Chua’s diode actually constitutes of theparallel of two negative conductance devices with

suitable v–i slopes and breakpoints. One such de-vice is sketched in Fig. 10, together with its v–icharacteristic. Its conductance, as seen at pointD, is given by G = −(R2/R1R3), so in our caseG = −1/R3, since R1 = R2. In order to set thedevice to a high-impedance state, one of the possi-bilities is to let R3 →∞, which is accomplished bymeans of perfect switches given by the op-amps A3and A4 and real switches b1 and b2, respectively (seeFig. 4). Owing to the fact that at the instant whenR3 is disconnected, the initial voltge v10 is simulta-neously applied across C1 (and hence at point D,Fig. 4), both inputs of the op-amp together with itsoutput will be driven at the voltage v10, no satura-tion of the op-amp occurring. The amplifier simplyreacts forcing voltage v10 at its inverting input. Be-ing the noninverting input and the output of the op-amp at the same voltage v10, no current flows intoR1, as the impedance at point D is actually seenas infinite.

Appendix B

The circuit shown for our synthetic inductor is likelyto be a derivation of the so-called FDNR (FrequencyDependent Negative Resistor), a well assessed de-vice introduced by Bruton [1969], widely used inthe design of active filters. The version of the

Fig. 10. The parallel of two negative conductance device constituting the Chua’s diode and its v–i characteristic.

970 G. Pegna et al.

Fig. 11. Structure and working principle for the synthetic inductor.

circuit proposed here is exactly as reported in:Linear applications, Vol. 1, National SemiconductorCo., Santa Clara, Ca. 1973, p. AN31-15. A semi-qualitative and somewhat approximate explanationof its working principle is as in Fig. 11.

At time t = 0 imagine the start of a currentvariation dI into point H. Point B is at this timea virtual ground; therefore point H will experiencethe voltage variation dV = R1dI. This voltage vari-ation will produce the current I = C(dV/dt) in thecapacitor and a finite voltage Vk = R2C(dV/dt)across R2. The amplifier will react to this non-

inverting input voltage by forcing and if the valueof R1 is small, approximately the same voltage willdevelop at point H. In conclusion, we have

V = R1R2CdI

dt.

This functional relation between V and dI/dt showsthat the circuit behaves as an inductor of valueL = R1R2C. Actually, it behaves as a “nonper-fect” inductor, its equivalent series resistance beingR1, while a parallel resistance R2 appears connectedacross it.