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Probability, pendulums, and pedagogy
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View: Figures


Image of Fig. 1.
Fig. 1.

Angular distribution of the linearized pendulum. The solid line is the analytic expression and the dots are obtained from the Monte Carlo sampling.

Image of Fig. 2.
Fig. 2.

The distribution of first return times to the angular interval of centered at , for the linearized pendulum. Because the return angle is , there is only one peak observed at a time equal to one-half of the period. is the period of the pendulum.

Image of Fig. 3.
Fig. 3.

Power spectrum for the time series of the angular displacement of the nonlinear pendulum [from a numerical solution to Eq. (5)]. Note the presence of the odd harmonics of the fundamental frequency , the largest peak.

Image of Fig. 4.
Fig. 4.

Angular distribution of the nonlinear pendulum. The amplitude is . The bin counting technique is used. Note the slightly different shape from Fig. 1.

Image of Fig. 5.
Fig. 5.

A bifurcation diagram showing the regular and chaotic behavior of the pendulum. The angular velocity is sampled once every forcing cycle. Period doubling occurs at forcing amplitudes where two points occur in the region . Chaos is evident when many points occur. The other pendulum parameters are .

Image of Fig. 6.
Fig. 6.

The fundamental and one subharmonic generate the angular distribution given by Eq. (8).

Image of Fig. 7.
Fig. 7.

Period doubling. (a) Semilog power spectrum of a period doubled time series of the angle. The doubling is shown by the peak at about which is half of the fundamental forcing frequency, that is indicated by the largest peak at about . Other peaks occur at half-integral values of the forcing frequency. (b) Angular distribution for period doubling. Note the double singularities at the edges of the distribution. (c) The distribution of first return times. Note that the range of the data is about double the forcing period, .

Image of Fig. 8.
Fig. 8.

Period-15 motion. . (a) A bifurcation diagram near the regime of period-15 motion. . (b) Angular distribution in which the 15 singularities on either side of indicate the presence of period-15 motion. . (c) Semilog plot of the first return distribution, . The time is in units of the forcing period, .

Image of Fig. 9.
Fig. 9.

Chaotic motion, . (a) Angular distribution. The many peaks of the period- regular motion have now coalesced into a bumpy, but approximately rectangular distribution. The distribution is apparently symmetric around the . (b) First return distribution, to the interval . The exponential decay model approximates the data.


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752b84549af89a08dbdd7fdb8b9568b5 journal.articlezxybnytfddd
Scitation: Probability, pendulums, and pedagogy