Stability and instability results in a model of Fermi acceleration

Abstract

We consider the static wall approximation to the dynamics of a particle bouncing on a periodically oscillating infinitely heavy plate while subject to a potential force. We assume the case of a potential given by a power of the particle's height and sinusoidal motions of the plate. We find that for powers smaller than 1 the set of escaping orbits has full Hausdorff dimension for all motions and obtain existence of elliptic island of period 2 for arbitrarily high energies for a full-measure set of motions. Moreover we obtain conditions on the potential to ensure that the total (Lebesgue) measure of elliptic islands of period 2 is either finite or infinite. * jacopods@math.umd.edu Ij 1 j2 Figure 1: Choosing branches of F k for k = 2.Proposition III.1. If ∀ n ∈ N m n < Cm n , then the Hausdorff dimension of J is 1.

Dynamics of some piecewise smooth Fermi-Ulam models

“…The next result extends the work (Ref. 15) where a similar statement is proven for a smooth model of Fermi acceleration.…”

Bouncing balls in non-linear potentials

“…For α < 1 E is non-empty (in fact, it has Hausdorff dimension 2 [3]). However Conjecture 1 holds at least for small α.…”

“…However Conjecture 2 seems much more difficult than Conjecture 1. Indeed according to [3] there is a large set of parameters where elliptic islands appear at arbitrary large heights.…”

“…To see why we expect velocity to decrease eventually recall that Lemma 5.3 states that the dynamics of the fast variable is chaotic. Therefore we expect (due to (3)) that the change of velocity obeys the Central Limit Theorem, that is, it can be approximated by the Brownian Motion. Since the Brownian Motion is recurrent we expect the same for our system.…”

Chaotic Dynamics in an Impact Problem

“…Actually, in [11] we proved that if f is regular then for every real number ω sufficiently large, there exists a solution with rotation number ω. Among the huge amount of results concerning such model we cite [4,7,10,13,17]. It is worth mentioning also the paper by Dolgopyat [5] dealing with non-gravitational potentials and the paper by Kunze and Ortega [9] dealing with non-periodic functions f .…”