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

We 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.

On the Qualitative Behavior of a Class of Generalized Liénard Planar Systems

We study the problem of existence/nonexistence of limit cycles for a class of Liénard generalized differential systems in which, differently from the most investigated case, the function F depends not only on x but also on the y-variable. In this framework, some new results are presented, starting from a case study which, actually, already exhibits the most significant properties. In particular, the so-called “superlinear case” presents some new phenomena of escaping orbits which will be discussed in detail.

A piecewise smooth Fermi–Ulam pingpong with potential

In this paper we study a Fermi–Ulam model where a pingpong ball bounces elastically against a periodically oscillating platform in a gravity field. We assume that the platform motion $f(t)$ is 1-periodic and piecewise $C^3$ with a singularity, $\dot {f}(0+)\ne \dot {f}(1-)$ . If the second derivative $\ddot {f}(t)$ of the platform motion is either always positive or always less than $-g$ , where g is the gravitational constant, then the escaping orbits constitute a null set and the system is recurrent. However, under these assumptions, escaping orbits co-exist with bounded orbits at arbitrarily high energy levels.

Machine learning and serving of discrete field theories

A method for machine learning and serving of discrete field theories in physics is developed. The learning algorithm trains a discrete field theory from a set of observational data on a spacetime lattice, and the serving algorithm uses the learned discrete field theory to predict new observations of the field for new boundary and initial conditions. The approach of learning discrete field theories overcomes the difficulties associated with learning continuous theories by artificial intelligence. The serving algorithm of discrete field theories belongs to the family of structure-preserving geometric algorithms, which have been proven to be superior to the conventional algorithms based on discretization of differential equations. The effectiveness of the method and algorithms developed is demonstrated using the examples of nonlinear oscillations and the Kepler problem. In particular, the learning algorithm learns a discrete field theory from a set of data of planetary orbits similar to what Kepler inherited from Tycho Brahe in 1601, and the serving algorithm correctly predicts other planetary orbits, including parabolic and hyperbolic escaping orbits, of the solar system without learning or knowing Newton’s laws of motion and universal gravitation. The proposed algorithms are expected to be applicable when the effects of special relativity and general relativity are important.

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

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.

The escaping set of transcendental self-maps of the punctured plane

We study the different rates of escape of points under iteration by holomorphic self-maps of $\mathbb{C}^{\ast }=\mathbb{C}\setminus \{0\}$ for which both zero and infinity are essential singularities. Using annular covering lemmas we construct different types of orbits, including fast escaping and arbitrarily slowly escaping orbits to either zero, infinity or both. We also prove several properties about the set of fast escaping points for this class of functions. In particular, we show that there is an uncountable collection of disjoint sets of fast escaping points, each of which has $J(f)$ as its boundary.

Trapped and Escaping Orbits in an Axially Symmetric Galactic-Type Potential

In the present article, we investigate the behavior of orbits in a time-independent axially symmetric galactic-type potential. This dynamical model can be considered to describe the motion in the central parts of a galaxy, for values of energies larger than the energy of escape. We use the classical surface-of-section method in order to visualize and interpret the structure of the phase space of the dynamical system. Moreover, the Lyapunov characteristic exponent is used in order to make an estimation of the degree of chaoticity of the orbits in our galactic model. Our numerical calculations suggest that in this galactic-type potential there are two kinds of orbits: (i) escaping orbits and (ii) trapped orbits, which do not escape at all. Furthermore, a large number of orbits of the dynamical system display chaotic motion. Among the chaotic orbits, there are orbits that escape quickly and also orbits that remain trapped for vast time intervals. When the value of a test particle's energy slightly exceeds the energy of escape, the number of trapped regular orbits increases as the value of the angular momentum increases. Therefore, the extent of the chaotic regions observed in the phase plane decreases as the energy value increases. Moreover, we calculate the average value of the escape period of chaotic orbits and try to correlate it with the value of the energy and also with the maximum value of the z component of the orbits. In addition, we find that the value of the Lyapunov characteristic exponent corresponding to each chaotic region for different values of energy increases exponentially as the energy increases. Some theoretical arguments are presented in order to support the numerically obtained outcomes.