Precision measurements of a simple chaotic circuit

1 Precision measurements of a simple chaotic circuitKen Kiersa)and Dory Schmidtb)Department of Physics, Taylor University, Upland, Indiana 46989J. C. Sprottc)Department of Physics, University of Wisconsin, Madison, Wisconsin 53706~Received 10 June 2003; accepted 27 August 2003!We describe a simple nonlinear electrical circuit that can be used to study chaotic phenomena. Thecircuit employs simple electronic elements such as diodes, resistors, and operational amplifiers, andis easy to construct. A novel feature of the circuit is its use of an almost ideal nonlinear element,which is straightforward to model theoretically and leads to excellent agreement betweenexperiment and theory.

2 For example, comparisons of bifurcation points and power spectra giveagreement to within 1%. The circuit yields a broad range of behavior and is well suited forqualitative demonstrations and as a serious research tool. 2004 American Association of Physics Teachers.@DOI: #I. INTRODUCTIONThe study of nonlinear systems and chaos provides a fas-cinating gateway into the world of research for the growing use of nonlinear analysis techniques inmany areas of science, it also is becoming increasingly im-portant to provide undergraduate students with a good intro-duction to nonlinear systems. Undergraduate chaos experi-ments that are available commercially tend either to berelatively expensive or to be somewhat qualitative in articles have been published over the past 15 yearsregarding chaotic behavior in systems ranging from a bounc-ing ball to various electronic 8In many of thesearticles the authors have made clever use of low cost orreadily available equipment to illustrate well-known aspectsand analytical techniques associated with chaos, such as bi-furcation diagrams, periodic and chaotic attractors.

3 Returnmaps and Poincare electronic circuits provide an excellent tool forthe study of chaotic behavior. Some of these circuits treattime as a discrete variable, employing sample-and-hold sub-circuits and analog multipliers to model iterated maps suchas the logistic flows are somewhateasier to model electronically. One of the best-known chaoticcircuits of this latter type is Chua s 11 The originalversion of this circuit contains an inductor~making it diffi-cult to model and to scale to different frequencies!, but in-ductorless versions of Chua s circuit have also 14 Recent work has highlighted several newchaotic circuits that are very simple to construct ,16 These circuits correspond to simple third-orderdifferential equations, are easy to scale to different frequen-cies, and contain only simple electronic elements such asdiodes, operational amplifiers~op amps!

4 , and resistors. Fur-thermore, with slight modifications, they hold the differential equations corresponding tothese circuits are among the simplest third-order differentialequations that lead to chaotic 22As noted inRefs. 16 and 17, several of these circuits may be groupedtogether and regarded as an analog computer for the preciseexperimental study of chaotic phenomena. Some possibleuses of these circuits involve studies of synchronization23and secure , several such cir-cuits could in principle be linked together to investigatehigher-dimensional class of simple circuits that leads to chaotic behavioris described by the following third-order differentialequation,17x^52Ax 2x 1D~x!

5 2a,~1!wherexrepresents the voltage at a particular node in thecorresponding circuit . In Eq.~1!Aandaare constants, thedots denote derivatives with respect to a dimensionless time,andD(x) is a nonlinear function that characterizes the non-linearity in the this paper we describe an investigation of a new circuitbelonging to the class of circuits described by Eq.~1!. Thenonlinearity in the circuit models a function proportional tomin(x,0) . The circuit is similar to the one described in , but uses a more precise implementation of increase in Precision allows for a detailedcomparison between theory and experiment. Such compari-sons yield agreement to within 1% for quantities such asbifurcation points.

6 The data taken from the circuit also canbe used in a variety of ways to illustrate many aspects ofchaotic and periodic paper is structured as follows. In Sec. II we describethe circuit and provide several technical details. Section IIIcontains the experimental results and compares these to the-oretical expectations. Section IV offers some concluding CIRCUITA. General remarksFigure 1 shows a schematic diagram of the circuit used tomodel Eq.~1!. The circuit has a modular design and may,with small changes, be used to study any of several differentchaotic systems, each corresponding to a different nonlinearfunctionD(x).16,17 The variable resistorRvacts as a controlparameter, moving the system in and out of chaos, and theinput voltageV0may be either positive or resistors~capacitors!

7 Have the same nominal resis-tanceR~capacitanceC) . The box labeledD(x) in Fig. 1represents the nonlinearity in the circuit , which is necessary503503Am. J. ~4!, April 2004 2004 American Association of Physics Teachersfor the circuit to exhibit chaotic behavior. The voltage at theoutput of the box~on the left!is related to that at its input bythe functional relationVout5D(Vin).The circuit in Fig. 1 contains three successive invertingintegrators with outputs at the nodes labeledV2,V1, andx,as well as a summing amplifier with its output Kirchhoff s rules at nodesa-d~along with the goldenrules for op amps27!, we obtain the following relationsamong the voltages:28V152 RCdxdt52x ,~2!

8 V252 RCdV1dt5x ,~3!RCdV2dt52 SRRvDV22 SRR0DV02V3,~4!V352V12D~x!,~5!where the dots denote derivatives with respect to the dimen-sionless variablet 5t/(RC) . The substitution of Eqs.~2!,~3!, and~5!into Eq.~4!yieldsx^52 SRRvDx 2x 1D~x!2 SRR0DV0.~6!Equation~6!may be compared with Eq.~1!. It is straightfor-ward to generalize Eq.~6!to the case where the resistors andcapacitors differ slightly from their nominal Ref. 17 the nonlinearity in the circuit was taken to havethe form of an absolute value ,D(x)5uxu. The solutions ofthe differential equation corresponding to this form becomeunbounded whenRvexceeds a certain threshold. In the cir-cuit itself, such unbounded solutions manifest themselvesthrough saturated op amps, making the circuit somewhat dif-ficult to work with.

9 In particular, it was found that certainpower supplies to the circuit had to be turned on in a specificorder and in quick succession or the circuit would instability manifested itself for all values ofRv, not justthose beyond the the present work we employ a different nonlinear sub- circuit than in Ref. 17. The nonlinearity used here models thefunctionD(x)526 min(x,0) and does not lead to un-bounded solutions. The resulting circuit is generally muchmore stable to work with, making it ideal for use with un-dergraduate students and for other 2 shows the nonlinear subcircuit used in this workto model the functionD(x) noted above.

10 Slight variations ofthis circuit27are used widely in various electronic applica-tions such as AC voltmeters. To show that the circuit yieldsthe desired functional form, we use the Shockley equation tomodel theI Vcurves for the diodes,ID5IS~eaVD21!,~7!whereIDandVDrepr esent the current through and voltageacross each diode, respectively. For the BAV20 silicon di-odes that we use, the reverse bias currentISis of order a fewnA andais of order 20 V21. If we employ Kirchhoff s rulesat nodesaandbin Fig. 2, we obtain the following transcen-dental equation relating the input and output voltages in :29 Vout11a2lnF11 VoutIS2R2G521a1lnF111IS1 SVinR11 VoutR2DG.