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. 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: 10.1119/1.1621031$I. INTRODUCTIONThe study of nonlinear systems and chaos provides a fas-cinating gateway into the world of research for students.With 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 nature.Many articles have been published over the past 15 yearsregarding chaotic behavior in systems ranging from a bounc-ing ball to various electronic circuits.1–8In 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, returnmaps and Poincare´sections.Nonlinear 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 map.1Continuous-time flows are somewhateasier to model electronically. One of the best-known chaoticcircuits of this latter type is Chua’s circuit.9–11The 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 beendescribed.12–14Recent work has highlighted several newchaotic circuits that are very simple to construct andanalyze.15,16These 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", and resistors. Fur-thermore, with slight modifications, they hold the potentialfor very precise comparisons between theory andexperiment.17The differential equations corresponding tothese circuits are among the simplest third-order differentialequations that lead to chaotic behavior.18–22As 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 communication.24Furthermore, several such cir-cuits could in principle be linked together to investigatehigher-dimensional chaos.One class of simple circuits that leads to chaotic behavioris described by the following third-order differentialequation,17x!! " Ax¨ " x˙ #D!x""%, !1"where x represents the voltage at a particular node in thecorresponding circuit. In Eq. !1" A and%are constants, thedots denote derivatives with respect to a dimensionless time,and D(x) is a nonlinear function that characterizes the non-linearity in the circuit.In 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 Ref.15, but uses a more precise implementation of thenonlinearity.25The increase in precision allows for a detailedcomparison between theory and experiment. Such compari-sons yield agreement to within 1% for quantities such asbifurcation points. The data taken from the circuit also canbe used in a variety of ways to illustrate many aspects ofchaotic and periodic behavior.The 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 re-marks.II. 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 nonlinearfunction D(x).16,17The variable resistor Rvacts as a controlparameter, moving the system in and out of chaos, and theinput voltage V0may be either positive or negative.26Allunlabeled resistors !capacitors" have the same nominal resis-tance R !capacitance C). The box labeled D(x) in Fig. 1represents the nonlinearity in the circuit, which is necessary503 503Am. J. Phys. 72 !4", April 2004 http://aapt.org/ajp © 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 relation Vout! D(Vin).The circuit in Fig. 1 contains three successive invertingintegrators with outputs at the nodes labeled V2, V1, and x,as well as a summing amplifier with its output at V3. If weuse Kirchhoff’s rules at nodes a-d !along with the ‘‘goldenrules’’ for op amps27", we obtain the following relationsamong the voltages:28V1! " RCdxdt! " x˙ , !2"V2! " RCdV1dt! x¨ , !3"RCdV2dt! "!RRv"V2"!RR0"V0" V3, !4"V3! " V1" D!x", !5"where the dots denote derivatives with respect to the dimen-sionless variable t˜! t/(RC). The substitution of Eqs. !2",!3", and !5" into Eq. !4" yieldsx!! "!RRv"x¨ " x˙ # D!x""!RR0"V0. !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 values.In Ref. 17 the nonlinearity in the circuit was taken to