On unbounded motions in a real analytic bouncing ball problem
type: Item
instance of: scholarly article
title: On unbounded motions in a real analytic bouncing ball problem
zbMATH Open document ID: 1500.37047
author: Stefano Marò
published in: Qualitative Theory of Dynamical Systems
publication date: 2022-09-02T00:00:00Z
full work available at URL: https://arxiv.org/abs/2207.12737
review text: The paper is concerned with the model of a ball elastically bouncing on a racket moving in the vertical direction according to a given periodic real function \(f(t)\), and the ball is reflected according to the law of elastic bouncing when hitting the racket. The only force acting on the ball is the gravity, with acceleration \(g\). Moreover, the mass of the racket is assumed to be large with respect to the mass of the ball so that the impacts do not affect the motion of the racket. The main focus concerns the existence of unbounded motion, that is, the velocity of the ball tends to infinity. The first result in this direction is due to \textit{L. D. Pustyl'nikov} [Russ. Math. Surv. 50, No. 1, 145--189 (1995; Zbl 0846.70005); translation from Usp. Mat. Nauk 50, No. 1, 143--186 (1995)], who assumed that \(2\dot{f}(t_0) = g\), for some time \(t_0\). The author claims that with the same proof, it is possible to weaken the condition to \[ \max \dot{f}\geq \frac{g}{4}.\tag{1} \] This paper focuses on improving the upper bound given by (1), more precisely, the author gives an explicit example of a trigonometric polynomial \(p(t)\) with \[ \max \dot{p} < \frac{g}{4} \] admitting unbounded bouncing motions. Indeed, \(p(t)\) admits a continuum of unbounded orbits and moreover belongs to a family of trigonometric polynomial \(p_s (t)\), parametrized by \(s\) in a real open interval \(I\). More precisely, the author proves that if the function \(f(t)\) belongs to a class of trigonometric polynomials of degree 2 then there exists a one-dimensional continuum of initial conditions for which the velocity of the ball tends to infinity. This result improves the previous one by \textit{L. D. Pustyl'nikov} and gives a new upper bound to the applicability of KAM theory to this model. Moreover, the author discusses the optimality of condition (1) and construct the family \(p_s\) proving that the corresponding map admits one unbounded orbit. Thus, he shows how to extend the unbounded orb…
reviewed by: Martha Alvarez-Ramírez
Mathematics Subject Classification ID: 37N05
Mathematics Subject Classification ID: 37J60
Mathematics Subject Classification ID: 37J40
Mathematics Subject Classification ID: 70E50
Mathematics Subject Classification ID: 70H08
zbMATH DE Number: 7581585
zbMATH Keywords: KAM theory
zbMATH Keywords: trigonometric polynomials
zbMATH Keywords: mechanical problem
MaRDI profile type: Publication
OpenAlex ID: W4292316639
Wikidata QID: Q114220807
arXiv ID: 2207.12737
cites work: Q5848416
cites work: Bouncing balls in nonlinear potentials
cites work: Q3314610
cites work: Q4720438
cites work: The dynamics of repeated impacts with a sinusoidally vibrating table
cites work: Complete orbits for twist maps on the plane: extensions and applications
cites work: Regular and chaotic dynamics.
cites work: Converse KAM: theory and practice
cites work: A \(C^{1+\alpha}\) mechanical counterexample to Moser's twist theorem
cites work: Coexistence of bounded and unbounded motions in a bouncing ball model
cites work: A mechanical counterexample to KAM theory with low regularity
cites work: Chaotic dynamics in an impact problem
cites work: Diffusion and chaos in a bouncing ball model
cites work: Poincaré models, rigorous justification of the second element of thermodynamics on the basis of mechanics, and the Fermi acceleration mechanism
cites work: Periodic Motions of Linear Impact Oscillators via the Successor Map
cites work: Periodic solutions and chaotic dynamics in forced impact oscillators
cites work: Q4070368
cites work: A piecewise smooth Fermi–Ulam pingpong with potential
DOI: 10.1007/S12346-022-00644-4
description: scientific article