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Bouncing droplets on a billiard table
1. A. A. Samarskii and A. N. Tikhonov, Equations of Mathematical Physics (Dover, 1963).
10. A. Eddi, E. Sultan, J. Moukhtar, E. Fort, M. Rossi, and Y. Couder, “ Information stored in faraday waves: The origin of a path memory,” J. Fluid Mech. 674, 433–463 (2011).
12. E. Fort, A. Eddi, A. Boudaoud, J. Moukhtar, and Y. Couder, “ Path-memory induced quantization of classical orbits,” Proc. Natl. Acad. Sci. U.S.A. 107, 17515–17520 (2010).
15. L. D. Landau and E. M. Lifshitz, Fluid Mechanics (Butterworth-Heinemann, 1959).
17.We remark that another natural (Faraday) length scale is set using the linear dispersion relation in combination with the period T. Such a choice, however, leads to a more complicated form for the coefficients in the dimensionless equation.
18. A. Oza, R. R. Rosales, and J. Bush, “ A trajectory equation for walking droplets: Pilot-wave hydrodynamics” (unpublished).
19.We omit the case of an oscillating instability when the derivative of the iterative map is less than −1.
20. D. Harris, J. Moukhtar, E. Fort, Y. Couder, and J. Bush, “ Pilot-wave dynamics in confined geometries” (unpublished).
21. J. Molacek and J. Bush, “ Droplets bouncing on a vibrating fluid bath” (unpublished).
22. J. Molacek and J. Bush, “ Drops walking on a vibrating fluid bath: Towards a hydrodynamic pilot-wave theory” (unpublished).
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In a set of experiments, Couder et al. demonstrate that an oscillating fluid bed may propagate a bouncing droplet through the guidance of the surface waves. I present a dynamical systems model, in the form of an iterative map, for a droplet on an oscillating bath. I examine the droplet bifurcation from bouncing to walking, and prescribe general requirements for the surface wave to support stable walking states. I show that in addition to walking, there is a region of large forcing that may support the chaotic motion of the droplet. Using the map, I then investigate the droplet trajectories in a square (billiard ball) domain. I show that in large domains, the long time trajectories are either non-periodic dense curves or approach a quasiperiodic orbit. In contrast, in small domains, at low forcing, trajectories tend to approach an array of circular attracting sets. As the forcing increases, the attracting sets break down and the droplet travels throughout space.
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