References and Notes

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[1]
For some recent examples of engineering applications in which the bouncing ball problem arises, see M.-O. Hongler, P. Cartier, and P. Flury, Numerical study of a model of vibro-transporter, Phys. Lett. A 135 (2), 106-112 (1989); M.-O. Hongler and J. Figour, Periodic versus chaotic dynamics in vibratory feeders, Helv. Phys. Acta 62, 68-81 (1989); T. O. Dalrymple, Numerical solutions to vibroimpact via an initial value problem formulation, J. Sound Vib. 132 (1), 19-32 (1989).

[2]
The bouncing ball problem has proved to be a useful system for experimentally exploring several new nonlinear effects. Examples of some of these experiments include S. Celaschi and R. L. Zimmerman, Evolution of a two-parameter chaotic dynamics from universal attractors, Phys. Lett. A 120 (9), 447-451 (1987); N. B. Tufillaro, T. M. Mello, Y. M. Choi, and A. M. Albano, Period doubling boundaries of a bouncing ball, J. Phys. (Paris) 47, 1477-1482 (1986); K. Wiesenfeld and N. B. Tufillaro, Suppression of period doubling in the dynamics of the bouncing ball, Physica D 26, 321-335 (1987); P. Pieranski, Jumping particle model. Period doubling cascade in an experimental system, J. Phys. (Paris) 44, 573-578 (1983); P. Pieranski, Z. Kowalik, and M. Franaszek, Jumping particle model. A study of the phase of a nonlinear dynamical system below its transition to chaos, J. Phys. (Paris) 46, 681-686 (1985); M. Franaszek and P. Pieranski, Jumping particle model. Critical slowing down near the bifurcation points, Can. J. Phys. 63, 488-493 (1985); P. Pieranski and R. Bartolino, Jumping particle model. Modulation modes and resonant response to a periodic perturbation, J. Phys. (Paris) 46, 687-690 (1985); M. Franaszek and Z. J. Kowalik, Measurements of the dimension of the strange attractor for the Fermi-Ulam problem, Phys. Rev. A 33 (5), 3508-3510 (1986); P. Pieranski and J. Ma ecki, Noisy precursors and resonant properties of the period doubling modes in a nonlinear dynamical system, Phys. Rev. A 34 (1), 582-590 (1986); P. Pieranski and J. Ma ecki, Noise-sensitive hysteresis loops and period doubling bifurcation points, Nuovo Cimento D 9 (7), 757-780 (1987); P. Pieranski, Direct evidence for the suppression of period doubling in the bouncing ball model, Phys. Rev. A 37 (5), 1782-1785 (1988); Z. J. Kowalik, M. Franaszek, and P. Pieranski, Self-reanimating chaos in the bouncing ball system, Phys. Rev. A 37 (10), 4016-4022 (1988).

[3]
A description of some simple experimental bouncing ball systems, suitable for undergraduate labs, can be found in A. B. Pippard, The physics of vibration, Vol. 1 (New York: Cambridge University Press, 1978), p. 253; N. B. Tufillaro and A. M. Albano, Chaotic dynamics of a bouncing ball, Am. J. Phys. 54 (10), 939-944 (1986); R. L. Zimmerman and S. Celaschi, Comment on ``Chaotic dynamics of a bouncing ball,'' Am. J. Phys. 56 (12), 1147-1148 (1988); T. M. Mello and N. B. Tufillaro, Strange attractors of a bouncing ball, Am. J. Phys. 55 (4), 316-320 (1987); R. Minnix and D. Carpenter, Piezoelectric film reveals f versus t of ball bounce, The Physics Teacher, 280-281 (March 1985). The KYNAR piezoelectric film used for the impact detector is available from Pennwalt Corporation, 900 First Avenue, P.O. Box C, King of Prussia, PA, 19046-0018. The experimenter's kit containing an assortment of piezoelectric films  costs about $50.

[4]
Some approximate models of the bouncing ball system are presented by G. M. Zaslavsky, The simplest case of a strange attractor, Phys. Lett. A 69 (3), 145-147 (1978); P. J. Holmes, The dynamics of repeated impacts with a sinusoidally vibrating table, J. Sound Vib. 84 (2), 173-189 (1982); M. Franaszek, Effect of random noise on the deterministic chaos in a dissipative system, Phys. Lett. A 105 (8), 383-386 (1984); R. M. Everson, Chaotic dynamics of a bouncing ball, Physica D 19, 355-383 (1986); A. Mehta and J. Luck, Novel temporal behavior of a nonlinear dynamical system: The completely inelastic bouncing ball, Phys. Rev. Lett.\ 65 (4), 393-396 (1990).

[5]
A detailed comparison between the exact model and the high bounce model is given by C. N. Bapat, S. Sankar, and N. Popplewell, Repeated impacts on a sinusoidally vibrating table reappraised, J. Sound Vib. 108 (1), 99-115 (1986).

[6]
A. D. Bernstein, Listening to the coefficient of restitution, Am. J. Phys. 45 (1), 41-44 (1977).

[7]
H. M. Isomäki, Fractal properties of the bouncing ball dynamics, in Nonlinear dynamics in engineering systems, edited by W. Schiehlen (Springer-Verlag: New York, 1990), pp. 125-131.