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.
REFERENCES
1.
See, for example,
G. L.
Baker
and J. A.
Blackburn
, The Pendulum: A Case Study in Physics
(Oxford University Press
, Oxford, 2005
).2.
J. A.
Blackburn
, S.
Vik
, Wu
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, and H. J. T.
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,” Rev. Sci. Instrum.
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G. L.
Baker
and J. P.
Gollub
, Chaotic Dynamics: An Introduction
(Cambridge U. P.
, Cambridge, UK, 1996
), 2nd ed.4.
The quantum pendulum, which is not considered in this discussion, brings its own version of probability through its wave function. The original treatment of the quantum pendulum is found in
E. U.
Condon
, “The physical pendulum in quantum mechanics
,” Phys. Rev.
31
, 891
–894
(1928
).For a recent discussion, see
G. L.
Baker
, J. A.
Blackburn
, and H. J. T.
Smith
, “The quantum pendulum: Small and large
,” Am. J. Phys.
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(2002
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H.
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See, for example,
R. C.
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(Oxford University Press
, Oxford, 1994
), p. 14
.7.
8.
A. I.
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, Mathematical Foundations of Statistical Mechanics
(Dover
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), p. 52
.9.
In dynamical systems, a Bernoulli system refers to a type of map, the Bernoulli shift map which is tied to coin flipping. In turn, coin flipping is the elementary Bernoulli process that is modeled by the binomial probability distribution.
10.
Return time statistics have been applied to a variety of chaotic systems. See,
E. G.
Altmann
, E. C.
da Silva
, and I. L.
Caldas
, “Recurrence time statistics for finite size intervals
,” CHAOS
14
, 975
–981
(2004
), and references therein.11.
R. W.
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, Numerical Methods for Scientists and Engineers
, 2nd ed. (Dover
, New York, 1986
), p. 132
.12.
In the region between and the pendulum’s motion is still periodic with the forcing period, but an asymmetry develops. That is, the motion is not symmetric about the vertical but depends on the initial conditions. Both even and odd harmonics of the forcing frequency appear. There are said to be two basins of attraction. If many initial states are used to create the bifurcation diagram, then in the stated region, two branches would occur. However, Fig. 5 was generated using only one initial condition and therefore the pendulum “chose” only one of the asymmetric orbits. Period doubling of this single orbit occurs for the interval as indicated in Fig. 5.
13.
See, for example, Ref. 6, p. 191.
14.
It is perhaps coincidental but interesting to note that the distribution of long synchronization times for chaotic coupled pendulums may also be modeled by a two-state Bernoulli model. Supporting physical data may be found in
H. J. T.
Smith
, J. A.
Blackburn
, and G. L.
Baker
, “Experimental observations of intermittency in coupled chaotic pendulums
,” Int. J. Bifurcation Chaos Appl. Sci. Eng.
9
, 1907
–1916
(1999
).The two-state model is presented in
G. L.
Baker
, J. A.
Blackburn
, and H. J. T.
Smith
, “A stochastic model of synchronization for chaotic pendulums
,” Phys. Lett. A
252
, 191
–197
(1999
).15.
See
Michael C.
Mackey
, Times Arrow: The Origins of Thermodynamic Behavior
(Springer
, New York, 1991
), andJ. R. Dorfman, An Introduction to Chaos in Nonequilibrium Statistical Mechanics (Cambridge U. P., Cambridge,
1999
).16.
John Maynard
Keynes
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(Macmillan
, London, 1921
; Harper and Row, New York, 1962), p. 8
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R.
Weatherford
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(Routledge and Kegan Paul
, London, 1982
), p. 5
.18.
See, for example,
R. T.
Cox
, “Probability, frequency, and reasonable expectation
,” Am. J. Phys.
14
, 1
–l3
(1946
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Information entropy contains the idea of missing information. The greater the information entropy, the greater the information that is missing for complete specification of the system’s state. See
L.
Brillouin
, Science and Information Theory
(Academic
, London, 1962
).20.
H.
Atmanspacher
and H.
Scheingraber
, “A fundamental link between system theory and statistical mechanics
,” Found. Phys.
17
, 939
–963
(1987
).21.
Equation (7) can be written as three first-order differential equations in three dynamical variables , and . Thus, the phase space is three-dimensional and requires three Lyapunov exponents to quantify the stretching and shrinking of an initial ball of phase points.
22.
Technically, the pendulum is not hyperbolic because not every point in the phase space possesses distinct directions for stable and unstable manifolds. There are tangencies between the stable and unstable manifolds, and therefore the degree of randomness is limited to that of a -system; Edward Ott, private communication (2005).
23.
Deadelon, Pasco, and TelAtomic each sell a version of the chaotic pendulum. For a review of these products and further information, see
J. A.
Blackburn
and G. L.
Baker
, “A comparison of commercial chaotic pendulums
,” Am. J. Phys.
66
, 821
–830
(1998
).© 2006 American Association of Physics Teachers.
2006
American Association of Physics Teachers
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