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Computer-Assisted Methods for Analyzing Periodic Orbits in Vibrating Gravitational Billiards

Kevin E. M. Church

Department of Mathematics and Statistics, McGill University, 805 Sherbrooke St W, Montreal, Quebec, H3A 0B9, Canada

E-mail Address: [email protected]

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and

Clément Fortin

Department of Physics, McGill University, 805 Sherbrooke St W, Montreal, Quebec, H3A 0B9, Canada

E-mail Address: [email protected]

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https://doi.org/10.1142/S0218127421300214Cited by:0 (Source: Crossref)

Abstract

Using rigorous numerical methods, we prove the existence of 608 isolated periodic orbits in a gravitational billiard in a vibrating unbounded parabolic domain. We then perform pseudo-arclength continuation in the amplitude of the parabolic surface’s oscillation to compute large, global branches of periodic orbits. These branches are themselves proven rigorously using computer-assisted methods. Our numerical investigations strongly suggest the existence of multiple pitchfork bifurcations in the billiard model. Based on the numerics, physical intuition and existing results for a simplified model, we conjecture that for any pair $(k, p)$ , there is a constant $ξ$ for which periodic orbits consisting of $k$ impacts per period $p$ cannot be sustained for amplitudes of oscillation below $ξ$ . We compute a verified upper bound for the conjectured critical amplitude for $(k, p) = (2, 2)$ using our rigorous pseudo-arclength continuation.

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