Existence of Invariant Curves for Maps Close to Degenerate Maps, and a Solution of the Fermi-Ulam Problem

Abstract

Characteristics of the National Semiconductor Corporation LX1600A pressure transducer are examined with the object of using the device as a short-range altimeter for probing the lowest kilometre or two of the atmosphere. In this application the device is subject to large temperature changes and stringent power restrictions.It is shown that pressure determination to i 1.5 % over ranges of 2 and 20 kPa is possible provided that temperature effects are calibrated out. Operation of the device in a pulsed mode results in a considerable saving in power consumption.

A symmetry break in energy distribution and a biased random walk behavior causing unlimited diffusion in a two dimensional mapping

“…Indeed the returning mechanism for a further collision of the particle with the moving wall is the gravitational field only while it was a rebound collision with a fixed wall in the FUM. For such a system and for specific combinations of both control parameters and initial conditions the phenomenon of unlimited energy growth can be observed [9]. However, the introduction of dissipation also has been shown to be a sufficient condition to suppress the unlimited energy growth [31].…”

“…He argued that charged particles could be accelerated by interactions with time-dependent magnetic structures. Since then many authors from both physics as well as mathematics expertise have addressed the topic trying to contribute towards a better understanding and explanation of Fermi's idea [2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17]. One of the most studied versions of the problem is the one-dimensional Fermi-Ulam model (FUM) [18][19][20][21].…”

Fermi acceleration and adiabatic invariants for non-autonomous billiards

“…Then, provided the motion is sufficiently smooth, Kolmogorov-Arnold-Moser (KAM) theory may be applied. [46][47][48] The resulting KAM curves present permanent barriers for the Fermi acceleration and therefore the particle energy stays bounded for all times. Thus, one proves that Fermi's intuition fails in the smooth time-periodic onedimensional case.…”

Thermodynamics of Dispersing Billiards With Time-Dependent Boundaries

“…Later various models (the Fermi-Ulam model and others, see, e.g. [Brahic, 1971;Lichtenberg & Lieberman, 1990;Lichtenberg et al, 1980;Pustyl'nikov, 1987Pustyl'nikov, , 1994Pustyl'nikov, , 1995Pustyl'nikov et al, 1995;Ulam, 1961;Zaslavsky, 1970]) have been developed which explained this phenomenon to a certain extent.…”