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A piecewise smooth Fermi–Ulam pingpong with potential

Part of: Dynamical systems with hyperbolic behavior Ergodic theory Applications - Dynamical systems and ergodic theory

Published online by Cambridge University Press:  04 March 2021

JING ZHOU Open the ORCID record for JING ZHOU [Opens in a new window]
JING ZHOU*
Affiliation:
Department of Mathematics, University of Maryland, College Park, MD20742, USA
*
e-mail: jingzhou@umd.edu
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Abstract

In this paper we study a Fermi–Ulam model where a pingpong ball bounces elastically against a periodically oscillating platform in a gravity field. We assume that the platform motion $f(t)$ is 1-periodic and piecewise $C^3$ with a singularity, $\dot {f}(0+)\ne \dot {f}(1-)$ . If the second derivative $\ddot {f}(t)$ of the platform motion is either always positive or always less than $-g$ , where g is the gravitational constant, then the escaping orbits constitute a null set and the system is recurrent. However, under these assumptions, escaping orbits co-exist with bounded orbits at arbitrarily high energy levels.

Keywords

Fermi accelerationFermi–Ulamergodic theorybouncing ballescaping orbit

MSC classification

Primary: 37N05: Dynamical systems in classical and celestial mechanics 37D20: Uniformly hyperbolic systems (expanding, Anosov, Axiom A, etc.)
Secondary: 37A25: Ergodicity, mixing, rates of mixing 70F15: Celestial mechanics
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 42 , Issue 5 , May 2022 , pp. 1847 - 1870
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Copyright
© The Author(s), 2021. Published by Cambridge University Press

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