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Probability, pendulums, and pedagogy
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 Binro, and H. J. T. Smith, “Driven pendulum for studying chaos,” Rev. Sci. Instrum.0034-6748 71, 422–426 (1989).
3.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.0031-899X 31, 891–894 (1928).
4.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.0002-9505 70, 525–531 (2002).
5.H. Poincare, The Foundation of Science: Science and Method, 1913, English translation (The Science Press, Lancaster, PA, 1946).
6.See, for example, R. C. Hilborn, Chaos and Nonlinear Dynamics (Oxford University Press, Oxford, 1994), p. 14.
7.E. Ott, Chaos in Dynamical Systems (Cambridge U.P., Cambridge, 1993), p. 261.
8.A. I. Khinchin, Mathematical Foundations of Statistical Mechanics (Dover, New York, 1949), 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,” CHAOS1054-1500 14, 975–981 (2004), and references therein.
11.R. W. Hamming, 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.0218-1274 9, 1907–1916 (1999).
14.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. A0375-9601 252, 191–197 (1999).
15.See Michael C. Mackey, Times Arrow: The Origins of Thermodynamic Behavior (Springer, New York, 1991), and
15.J. R. Dorfman, An Introduction to Chaos in Nonequilibrium Statistical Mechanics (Cambridge U. P., Cambridge, 1999).
16.John Maynard Keynes, A Treatise on Probability (Macmillan, London, 1921; Harper and Row, New York, 1962), p. 8.
17.R. Weatherford, Philosophical Foundations of Probability Theory (Routledge and Kegan Paul, London, 1982), p. 5.
19.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.0015-9018 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
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