Related papers: Birkhoff attractors of dissipative billiards

Let $M$ be a manifold or (more generally) a locally compact, metrizable ANR. If $K$ is an attractor for a flow in $M$, with basin of attraction $\mathcal{A}(K)$, it is well known that the inclusion $i : K \subseteq \mathcal{A}(K)$ is always…

We consider classical dynamical properties of a particle in a constant gravitational force and making specular reflections with circular, elliptic or oval boundaries. The model and collision map are described and a detailed study of the…

We consider a convex curve $\gamma$ lying on the Sphere or Hyperbolic plane. We study the problem of existence of polynomial in velocities integrals for Birkhoff billiard inside the domain bounded by $\gamma$. We extend the result by S.…

In this paper we study the ergodic properties of mathematical billiards describing the uniform motion of a point in a flat torus from which finitely many, pairwise disjoint, tubular neighborhoods of translated subtori (the so called…

We consider billiards with a single cusp where the walls meeting at the vertex of the cusp have zero one-sided curvature, thus forming a flat point at the vertex. For H\"older continuous observables, we show that properly normalized Birkoff…

We investigate a non-isothermal viscous relaxation of some nonlocal Cahn-Hilliard equations. This perturbation problem generates a family of solution operators, exhibiting dissipation and conservation. The solution operators admit a family…

We consider a linear implicit-explicit (IMEX) time discretization of the Cahn-Hilliard equation with a source term, endowed with Dirichlet boundary conditions. For every time step small enough, we build an exponential attractor of the…

This work continues the study of tiling billiards, a class of dynamical system introduced by Davis et al. in 2018. We develop the study of generalized tiling billiards in a cyclic polygon. This work shows that the behavior of generalized…

In this paper we study the viscous simplified Bardina equation on the two-dimensional closed manifold $M$ which is embedded in $\mathbb{R}^3$. First, we prove the existence and the uniqueness of the weak solutions and also the existence of…

We consider the integrable dynamics of a Kepler billiard in the plane bounded by a branch of a conic section focused at the Kepler center. We show that in this case, for non-zero-energy orbits, the lines of consecutive second orbital foci…

We extend the Birkhoff's theorem in Lovelock gravity for arbitrary base manifolds using an elementary method. In particular, it is shown that any solution of the form of a warped product of a two-dimensional transverse space and an…

The purpose of this paper is to study the dynamics of a square billiard with a non-standard reflection law such that the angle of reflection of the particle is a linear contraction of the angle of incidence. We present numerical and…

For billiards with $N$ obstacles on a torus, we study the behavior of specific kind of its trajectories, \emph{the so called admissible trajectories}. Using the methods developed in \cite{1}, we prove that the \emph{admissible rotation set}…

We study the classical motion in bidimensional polygonal billiards on the sphere. In particular we investigate the dynamics in tiling and generic rational and irrational equilateral triangles. Unlike the plane or the negative curvature…

The aim of this paper is to study dynamical and topological properties of a flow in the region of influence of an isolated non-saddle set. We see, in particular, that some topological conditions are sufficient to guarantee that these sets…

This paper addresses the long-time behavior of gradient flows of non convex functionals in Hilbert spaces. Exploiting the notion of generalized semiflows by J. M. Ball, we provide some sufficient conditions for the existence of a global…

The bending of bilayer plates is a mechanism which allows for large deformations via small externally induced lattice mismatches of the underlying materials. Its mathematical modeling, discussed herein, consists of a nonlinear fourth order…

In this paper I will investigate the bifurcation and asymptotic behavior of solutions of the Swift-Hohenberg equation and the generalized Swift-Hohenberg equation with the Dirichlet boundary condition on a one- dimensional domain $(0,L)$. I…

This paper investigates the behaviour of open billiard systems in high-dimensional spaces. Specifically, we estimate the largest Lyapunov exponent, which quantifies the rate of divergence between nearby trajectories in a dynamical system.…

An abstract framework for studying the asymptotic behavior of a dissipative evolutionary system $\mathcal{E}$ with respect to weak and strong topologies was introduced in [8] primarily to study the long-time behavior of the 3D Navier-Stokes…