scholarly journals Escaping Orbits Are also Rare in the Almost Periodic Fermi–Ulam Ping-Pong

2021 ◽  
Vol 21 (1) ◽  
Author(s):  
Henrik Schließauf
Keyword(s):  
Lebesgue Measure ◽  
Measure Zero ◽  
Initial Condition ◽  
Almost Periodic ◽  
Periodic Forcing ◽  
One Dimensional ◽  
Forcing Function ◽  
Ping Pong ◽  
The One ◽  
Escaping Orbits

AbstractWe study the one-dimensional Fermi–Ulam ping-pong problem with a Bohr almost periodic forcing function and show that the set of initial condition leading to escaping orbits typically has Lebesgue measure zero.

Escaping orbits are rare in the quasi-periodic Fermi–Ulam ping-pong

2018 ◽  
Vol 40 (4) ◽  
pp. 975-991 ◽  
Author(s):  
MARKUS KUNZE ◽  
RAFAEL ORTEGA
Keyword(s):  
Initial Data ◽  
Lebesgue Measure ◽  
Measure Zero ◽  
Lebesgue Measure Zero ◽  
Diophantine Condition ◽  
Ping Pong ◽  
Escaping Orbits

We consider the quasi-periodic Fermi–Ulam ping-pong model with no diophantine condition on the frequencies and show that typically the set of initial data which leads to escaping orbits has Lebesgue measure zero.

scholarly journals Сlassical solution of the mixed problem for the one-dimensional wave equation with the nonsmooth second initial condition

2020 ◽  
Vol 64 (6) ◽  
pp. 657-662
Author(s):  
V. I. Korzyuk ◽  
J. V. Rudzko
Keyword(s):  
Boundary Condition ◽  
Wave Equation ◽  
Mixed Problem ◽  
Method Of Characteristics ◽  
Smooth Boundary ◽  
Analytical Form ◽  
Initial Condition ◽  
One Dimensional ◽  
The One ◽  
Dimensional Wave

In this article, we study the classical solution of the mixed problem in a quarter of a plane and a half-plane for a one-dimensional wave equation. On the bottom of the boundary, Cauchy conditions are specified, and the second of them has a discontinuity of the first kind at one point. Smooth boundary condition is set at the side boundary. The solution is built using the method of characteristics in an explicit analytical form. Uniqueness is proved and conditions are established under which a piecewise-smooth solution exists. The problem with linking conditions is considered.

scholarly journals Exact statistical properties of the Burgers equation

2000 ◽  
Vol 417 ◽  
pp. 323-349 ◽  
Author(s):  
L. FRACHEBOURG ◽  
Ph. A. MARTIN
Keyword(s):  
White Noise ◽  
Large Distance ◽  
Burgers Equation ◽  
Higher Order ◽  
Statistical Properties ◽  
Inviscid Limit ◽  
Initial Condition ◽  
Spatial Correlations ◽  
One Dimensional ◽  
The One

The one-dimensional Burgers equation in the inviscid limit with white noise initial condition is revisited. The one- and two-point distributions of the Burgers field as well as the related distributions of shocks are obtained in closed analytical forms. In particular, the large distance behaviour of spatial correlations of the field is determined. Since higher-order distributions factorize in terms of the one- and two- point functions, our analysis provides an explicit and complete statistical description of this problem.

scholarly journals Forced quasi-periodic oscillations in strongly dissipative systems of any finite dimension

2019 ◽  
Vol 21 (07) ◽  
pp. 1850064 ◽  
Author(s):  
Guido Gentile ◽  
Alessandro Mazzoccoli ◽  
Faenia Vaia
Keyword(s):  
Potential Energy ◽  
Finite Dimension ◽  
Dissipative Systems ◽  
Diophantine Condition ◽  
Frequency Vector ◽  
Periodic Forcing ◽  
One Dimensional ◽  
Forcing Term ◽  
Degeneracy Condition ◽  
The One

We consider a class of singular ordinary differential equations describing analytic systems of arbitrary finite dimension, subject to a quasi-periodic forcing term and in the presence of dissipation. We study the existence of response solutions, i.e. quasi-periodic solutions with the same frequency vector as the forcing term, in the case of large dissipation. We assume the system to be conservative in the absence of dissipation, so that the forcing term is — up to the sign — the gradient of a potential energy, and both the mass and damping matrices to be symmetric and positive definite. Further, we assume a non-degeneracy condition on the forcing term, essentially that the time-average of the potential energy has a strict local minimum. On the contrary, no condition is assumed on the forcing frequency; in particular, we do not require any Diophantine condition. We prove that, under the assumptions above, a response solution always exists provided the dissipation is strong enough. This extends results previously available in the literature in the one-dimensional case.

scholarly journals Revisiting the 1D and 2D Laplace Transforms

2020 ◽  
Author(s):  
Manuel Duarte Ortigueira ◽  
José Tenreiro Machado
Keyword(s):  
Transfer Function ◽  
Laplace Transform ◽  
Linear Systems ◽  
Laplace Transforms ◽  
Dimensional Case ◽  
Initial Condition ◽  
Two Dimensional ◽  
One Dimensional ◽  
The One

This paper reviews the unilateral and bilateral, one- and two-dimensional Laplace transforms. The unilateral and bilateral Laplace transforms are compared in the one-dimensional case, leading to the formulation of the initial-condition theorem. This problem is solved with all generality in the one- and two-dimensional cases with the bilateral Laplace transform. General two-dimensional linear systems are introduced and the corresponding transfer function defined.

On the stability of a class of self-similar solutions to the filtration-absorption equation

2002 ◽  
Vol 13 (2) ◽  
pp. 179-194 ◽  
Author(s):  
ALINA CHERTOCK
Keyword(s):  
The Self ◽  
Initial Condition ◽  
Cauchy Problems ◽  
One Dimensional ◽  
Self Similar ◽  
Analytical Result ◽  
Axisymmetric Solutions ◽  
The Stability ◽  
The One ◽  
Similar Solutions

We consider the one-dimensional and two-dimensional filtration-absorption equation ut = uΔu−(c−1)(∇u)2. The one-dimensional case was considered previously by Barenblatt et al. [4], where a special class of self-similar solutions was introduced. By the analogy with the 1D case we construct a family of axisymmetric solutions in 2D. We demonstrate numerically that the self-similar solutions obtained attract the solutions of non-self-similar Cauchy problems having the initial condition of compact support. The main analytical result we provide is the linear stability of the above self-similar solutions both in the 1D case and in the 2D case.

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