2015 BFY II Abstract Detail Page
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||W08 - Chaotic pendulum: the complete attractor
||With this apparatus your students can study one of the simplest systems capable of displaying chaotic behavior. It is constructed from a physical pendulum mounted on a Pasco rotary encoder and coupled to a stepper motor drive via strings, a pair of springs and a pulley. Raw data consists of pendulum angles measured synchronously with the uniform pulse train that drives the stepper motor. The motion is governed by a nonlinear differential equation readily modeled according to Newton's second law with basic forces due to gravity, springs and friction. Many behaviors of nonlinear systems such as multiple steady state solutions, limit cycles and chaos are observable. Chaos is a condition where the dynamical variables describing the evolution of the system are predictable over short time intervals, but unpredictable over longer intervals. In the steady state, the dynamical variables for chaotic motion do not repeat and evolve on a fractal subset of phase space called a strange attractor. Data sets collected over 75,000 drive cycles (24 hours) produce beautiful cross sections of the attractor called Poincare sections that can be further analyzed to determine the system parameters, the Lyapunov exponents characterizing the rate at which predictability is lost, and the fractal dimension of the attractor characterizing the self-similar structures common to chaotic systems. The same analyses can also be performed on data from numerical simulations of the differential equations. The workshop will cover the construction details of the apparatus including the data acquisition hardware and software and the design of an inexpensive stepper motor drive. The algorithms and LabVIEW programs for data acquisition, for calculating Lyapunov exponents and fractal dimensions, for fitting the data to the differential equations, and for numerical simulations will be described and the programs will be available for participants to use in their implementations.
University of Florida