On unbounded motions in a real analytic bouncing ball problem
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 orbit previously found to a continuum. It is done a detailed discussion of the result on the persistence of the stable manifold under nonautonomous perturbations.
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