Proportional feedback control of chaos in a simple electronic oscillatorRichard J. Wiener,a!Kristine E. Callan, and Stephen C. HallDepartment of Physics, Pacific University, Forest Grove, Oregon 97116Thomas OlsenDepartment of Physics, Lewis & Clark College, Portland, Oregon 97219!Received 10 July 2005; accepted 16 December 2005"We demonstrate the control of chaos in a nonlinear circuit constructed from readily availableelectronic components. Control is achieved using recursive proportional feedback, which isapplicable to chaotic dynamics in highly dissipative systems and can be implemented usingexperimental data in the absence of model equations. The application of recursive proportionalfeedback to a simple electronic oscillator provides an undergraduate laboratory problem forexploring proportional feedback algorithms used to control chaos. ©2006 American Association of PhysicsTeachers.#DOI: 10.1119/1.2166367$I. INTRODUCTIONChaos theory is the study of deterministic systems whosedynamics are aperiodic and depend sensitively on initialconditions.1Chaotic systems have long term behavior that isunpredictable. Nonlinear dynamical systems often behavechaotically for certain ranges of system parameters; for otherparameter ranges they behave periodically and thus predict-ably. Proportional feedback control of chaos involves per-turbing a system parameter, while maintaining its valuewithin its normally chaotic range, to achieve stabilization ofa selected trajectory on the system’s chaotic attractor. In thispaper we focus on a proportional feedback control strategythat stabilizes periodic dynamics.In 1990, Ott, Grebogi, and Yorke !OGY"2introduced aproportional feedback algorithm suitable for a large class ofnonlinear oscillators. Their approach employs a feedbackloop that applies small perturbations to a system at the end ofeach oscillation, with each perturbation proportional to thedifference between the current state and a desired state. Thisstrategy is an extension of engineering control theory.3,4TheOGY algorithm precipitated an outpouring of experimentaland theoretical work on controlling chaotic dynamics.5Pro-portional feedback control has been demonstrated for a widevariety of dynamical systems including mechanical,6,7fluid,8electronic,9optical,10chemical,11,12and biological13systems.Control via proportional feedback is now a central topic ofresearch in nonlinear dynamics and has been extended ex-perimentally to chaotic spatial patterns.14,15Given its cur-rency and prominence, it is desirable to introduce this topicin undergraduate courses on chaos.Baker16has provided a clear presentation of the OGY al-gorithm. However, the algorithm is challenging to implementexperimentally because it requires sampling more than onedynamical variable in real time. Dressler and Nitsche17showed that the OGY algorithm can be modified to allow formeasurements of a single dynamical variable. But there aremuch simpler alternatives to the OGY algorithm for highlydissipative systems, that is, systems whose dynamics can bereduced to one-dimensional !1D" return maps. For this spe-cial but not uncommon case, Peng, Petrov, and Showalter18introduced simple proportional feedback, and Rollins,Parmananda, and Sherard19derived recursive proportionalfeedback. These less complicated proportional-feedback al-gorithms illustrate key ideas of control and are considerablyless mathematically demanding than the OGY algorithm,which is applicable to a wider range of systems.In this paper we present a derivation of recursive propor-tional feedback at a level suitable for an introductory courseon chaos, and we explain why recursive proportional feed-back is more generally applicable than simple proportionalfeedback. Flynn and Wilson20and Corron, Pethel, andHopper21have presented other simple methods of controllingchaos that are also suitable for introducing undergraduates tothis topic. However, a discussion of simple proportionalfeedback and recursive proportional feedback allows us toaddress the issues of the stability and the range of applica-bility of individual control algorithms.Recently, Kiers, Schmidt, and Sprott !KSS"22introduced asimple nonlinear electronic circuit that can be used to studychaotic phenomena. This circuit employs readily availableelectronic components and is well-suited for advanced un-dergraduate instructional laboratories. A novel feature of theKSS circuit is the presence of an almost ideal nonlinear ele-ment, which results in excellent agreement between the ex-perimental circuit and numerical solutions of the differentialequation that models the dynamics of the circuit. The circuitallows for precise measurements of bifurcation diagrams,phase portraits, return maps, power spectra, Lyapunov expo-nents, and the fractal dimension of chaotic attractors. A fur-ther advantage of using the KSS circuit for undergraduateexperiments is that the time scale can be adjusted so that theperiods of oscillation are on the order of a second, makingthe circuit an ultra-low-frequency electronic oscillator. Stu-dents can observe the dynamics in real time, and there issufficient time during the oscillations for a digital processorto compute the requisite perturbations for chaos control. Un-dergraduates can readily wire the circuit, interface it to acomputer-based data acquisition board, and write a programto acquire data and apply a proportional feedback loop.There are many data acquisition, output, and analysis sys-tems in use in undergraduate laboratories that could be em-ployed.In this paper we show that the KSS circuit can be used toillustrate proportional feedback control of chaos by applyingrecursive proportional feedback to its dynamics. We showhow chaos can be controlled experimentally, even in the ab-sence of model equations, by determining the values of thecoefficients in the recursive proportional feedback algorithmonly from experimental data, without reference to the differ-200 200Am. J. Phys. 74 !3", March 2006 http://aapt.org/ajp © 2006 American Association of Physics Teachersential equation that models the dynamics of the circuit. How-ever, because the model equation is known, an importantfeature of this circuit is that one can perform analytic andnumerical investigations in parallel with the experiment. Ourdemonstration of control of a chaotic electronic oscillator fitswell in an undergraduate course on nonlinear dynamics orcomputer interfaced experimentation.II. CHAOS IN A SIMPLE