^{1,a)}

### Abstract

Deterministic pendula exhibit a spectrum of behavior ranging from periodic to chaotic and provide an opportunity for an introductory discussion on the application of probability techniques to a deterministic system. Analytic and simulation techniques are used to determine probability distributions for a range of dynamical possibilities. In particular, we obtain probability distributions of the pendulum’s angular displacement and distributions of first return times for regular and chaotic motion. For chaotic motion, the latter distribution is modeled by a simple two-state Bernoulli process. Further considerations suggest that not all distributions are probability distributions.

I would like to thank Professor James Blackburn and the anonymous reviewers for their comments on the manuscript.

I. INTRODUCTION

II. LINEARIZED PENDULUM

III. THE NONLINEAR PENDULUM

IV. THE DAMPED DRIVEN PENDULUM

A. Regular motion

B. Chaotic motion

V. DISCUSSION

VI. SUMMARY

### Key Topics

- Probability theory
- 13.0
- Chaos
- 12.0
- Attractors
- 9.0
- Chaotic systems
- 9.0
- Equations of motion
- 7.0

## Figures

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

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

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.

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.

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.

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.

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

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

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 .

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 .

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

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

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, .

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, .

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, .

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, .

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.

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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