Chaotic pendulum: The complete attractorRobert DeSerioa)Department of Physics, University of Florida, Gainesville, Florida 32611!Received 16 April 2002; accepted 11 October 2002"A commercial chaotic pendulum is modified to study nonlinear dynamics, including thedetermination of Poincare´sections, fractal dimensions, and Lyapunov exponents. The apparatus isdriven by a simple oscillating mechanism powered by a 200 pulse per revolution stepper motorrunning at constant angular velocity. A computer interface generates the uniform pulse train neededto run the stepper motor and, with each pulse, reads a rotary encoder attached to the pendulum axle.Ten million readings from overnight runs of 50 000 drive cycles were smoothed and differentiatedto obtain the pendulum angle#and the angular velocity$at each pulse of the drive. A plot of the50 000 !#,$" phase points corresponding to one phase of the drive system produces a single Poincare´section. Thus, 200 Poincare´sections are experimentally available, one at each step of the drive.Viewed separately, any one of them strikingly illustrates the fractal geometry of the underlyingchaotic attractor. Viewed sequentially in a repeating loop, they demonstrate the stretching andfolding of phase point density typical of chaotic dynamics. Results for four pendulum dampingconditions are presented and compared. ©2003 American Association of Physics Teachers.%DOI: 10.1119/1.1526465&I. INTRODUCTIONThe tremendous interest in nonlinear dynamics hasbrought with it a need for suitable introductory experiments.Although the chaotic pendulum is an obvious candidate, onlya few implementations have been proposed for use in teach-ing laboratories.1–3A torsional pendulum1and a mechanicalDuffing oscillator2have been described, but experimentalPoincare´sections were not collected. Three commercial cha-otic pendulums were reviewed in 19983and components ofone of those systems were used here. Our apparatus is par-ticularly well suited for an introductory treatment because itprovides quality data, and its simple design means that theequations of motion can be readily derived and experimen-tally verified for both regular and chaotic behavior.Not including a desktop personal computer and an assort-ment of clamps and rods, the total system cost can be keptbelow $1000. The rotary encoder and pendulum componentsare available from Pasco for less than $300;4the counter/timer board, connector block, and cable are available fromNational Instruments for less than $500;5and the steppermotor system can be put together for less than $200.6A 500MHz Pentium III PC was available, as was the LabVIEWsoftware development system used for all data acquisitionand analysis.7A data acquisition program is available on theUniversity of Florida Physics Department web site8and doesnot require a LabVIEW license. The acquisition and analysisprograms !virtual instruments" are also available,8but areunsupported and require a LabVIEW software license. Othernonlinear analysis and graphing packages such as TISEAN9and gnuplot10could also be used.The stepper motor and data acquisition system are criticalreplacements for the corresponding components currentlysupplied by Pasco. A Poincare´section requires that the pen-dulum angle and angular velocity be determined each timethe drive system passes a single point in its oscillatory mo-tion, and an optical or magnetic pickup is often placed on thedrive mechanism for this purpose. For example, the Daeda-lon pendulum reviewed in Ref. 3 passes such a pickup signalto its data acquisition interface and is the only one of thethree reviewed systems capable of creating Poincare´sec-tions. Moreover, each hardware pickup normally permitsonly a single Poincare´section to be collected at one time. Ineffect, our implementation creates a software pickup at everyangular position of the motor, thus making possible the cre-ation of Poincare´sections for almost all drive phases andproviding accurate three-dimensional phase space coordi-nates of points along a trajectory.The mechanical components and the data acquisition hard-ware and software are described in Sec. II. Data fitting ex-amples and calculations of fractal dimensions and Lyapunovexponents are described in Sec. III.II. APPARATUSA schematic of the apparatus is shown in Fig. 1. The pen-dulum, rotary encoder, and stepper motor are mounted on a 6foot, 1/2 in. diameter steel rod. Two identical springs areattached on either side of the pendulum by a string wrappedtwice around the pulley to prevent slipping. The end of onespring is driven up and down by a stepper motor running atconstant angular velocity via a string passing through a guidehole in a cross rod. The drive amplitude A is adjusted bychanging the length of the shaft attached to the motor. Theend of the other spring is fixed with a short string to a tuningpeg also mounted on the cross rod.The pendulum axle is part of the rotary encoder whichtransmits logic pulses !1440/rev" on separate phonojacks forclockwise and counterclockwise rotations. One of the dataacquisition counters performs up/down counting of thesepulses without any need for preprocessing. The pulse count,directly proportional to the pendulum rotation angle, can besaved to a counter register at any time by strobing thecounter’s gate.The stepper motor requires 200 pulses per revolution andruns at a frequency near one revolution per second. The near200 Hz square wave sent to the controller is divided downfrom a 20 MHz clock by another data acquisition counter.The frequency resolution !around 10 ppm" and stability !lessthan 50 ppm drift in a 24 hour period" are more than ad-250 250Am. J. Phys. 71 !3", March 2003 http://ojps.aip.org/ajp/ © 2003 American Association of Physics Teachersequate for this experiment. This square wave also strobes thegate of the angular counter with the result that 200 readingsare collected per drive period. The raw readings are trans-ferred to main memory by direct memory access hardware.Thus, all data acquisition is hardware timed and the angularreadings stay synchronized to the drive phase even over longruns. After programming the counters for the correct operat-ing modes and drive frequency, the stepper motor pulses areinitiated and the direct memory access buffer fills using al-most no computer resources. The buffer is written to diskevery few seconds and with typical overnight runs of 50 000drive cycles, two bytes per reading, and 200 readings perperiod, file