Volume 26, Number 1
Volume 26, Number 1, 2021
Cox C., Feres R., Zhao B.
Rolling Systems and Their Billiard Limits
Abstract
Billiard systems, broadly speaking, may be regarded as models of mechanical
systems in which rigid parts interact through elastic impulsive (collision) forces. When it is
desired or necessary to account for linear/angular momentum exchange in collisions involving
a spherical body, a type of billiard system often referred to as noslip has been used. In recent
work, it has become apparent that noslip billiards resemble nonholonomic mechanical systems
in a number of ways. Based on an idea by Borisov, Kilin and Mamaev, we show that noslip
billiards very generally arise as limits of nonholonomic (rolling) systems, in a way that is akin to
how ordinary billiards arise as limits of geodesic flows through a flattening of the Riemannian
manifold.

Kaur N., Goyal K.
Uncertainty Quantification of Stochastic Epidemic SIR Models Using Bspline Polynomial Chaos
Abstract
Reallife epidemic situations are modeled using systems of differential equations
(DEs) by considering deterministic parameters. However, in reality, the transmission parameters
involved in such models experience a lot of variations and it is not possible to compute them
exactly. In this paper, we apply Bspline waveletbased generalized polynomial chaos (gPC) to
analyze possible stochastic epidemic processes. A sensitivity analysis (SA) has been performed
to investigate the behavior of randomness in a simple epidemic model. It has been analyzed
that a linear Bspline wavelet basis shows accurate results by involving fewer polynomial chaos
expansions (PCE) in comparison to cubic Bspline wavelets. We have carried out our developed
method on two real outbreaks of diseases, firstly, influenza which affected the British boarding
school for boys in North England in 1978, and secondly, Ebola in Liberia in 2014. Real data
from the British Medical Journal (influenza) and World Health Organization (Ebola) has been
incorporated into the SusceptibleInfectedRecovered (SIR) model. It has been observed that
the numerical results obtained by the proposed method are quite satisfactory.

Araujo G. C., Cabral H. E.
Parametric Stability of a Charged Pendulum with an Oscillating Suspension Point
Abstract
We consider a planar pendulum with an oscillating suspension point and with the bob carrying an electric charge $q$. The pendulum oscillates above a fixed point with a charge $Q.$ The dynamics is studied as a system in the small parameter $\epsilon$ given by the amplitude of the suspension point. The system depends on two other parameters, $\alpha$ and $\beta,$ the first related to the frequency of the oscillation of the suspension point and the second being the ratio of charges. We study the parametric stability of the linearly stable equilibria and use the DepritHori method to construct the boundary surfaces of the stability/instability regions.

Chierchia L., Koudjinan C. E.
V. I. Arnold’s ''Global'' KAM Theorem and Geometric Measure Estimates
Abstract
This paper continues the discussion started in [10] concerning Arnold's legacy on classical KAM theory and (some of) its modern developments. We prove a
detailed and explicit ''global'' Arnold's KAM theorem, which yields, in particular, the Whitney conjugacy of a nondegenerate,
realanalytic, nearlyintegrable Hamiltonian system to an integrable system on a closed, nowhere dense, positive measure subset of the phase space. Detailed measure estimates on the Kolmogorov set are provided in case the phase space is: (A)
a uniform neighbourhood of an arbitrary (bounded) set times the $d$torus and
(B) a domain with $C^2$ boundary times the $d$torus. All constants are explicitly given.

Tzemos A. C., Contopoulos G.
Integrals of Motion in Timeperiodic Hamiltonian Systems: The Case of the Mathieu Equation
Abstract
We present an algorithm for constructing analytically approximate integrals of motion in
simple timeperiodic Hamiltonians of the form $H=H_0+
\varepsilon H_i$, where $\varepsilon$ is a perturbation parameter. We apply our algorithm in a Hamiltonian system whose dynamics is governed by the Mathieu equation and examine in detail the orbits and their stroboscopic invariant curves for different values of $\varepsilon$. We find the values of $\varepsilon_{crit}$ beyond which the orbits escape to infinity and construct integrals which are expressed as series in the perturbation parameter $\varepsilon$ and converge up to $\varepsilon_{crit}$. In the absence of resonances the invariant curves are concentric ellipses which are approximated very well by our integrals. Finally, we construct an integral of motion which describes the hyperbolic stroboscopic invariant curve of a resonant case.

Hosseini K., Matinfar M., Mirzazadeh M.
Soliton Solutions of Highorder Nonlinear Schrödinger Equations with Different Laws of Nonlinearities
Abstract
In the present paper, highorder nonlinear Schrödinger equations in nonKerr law media with different laws of nonlinearities are studied. In this respect, after considering a complex envelope and distinguishing the real and imaginary portions of the models, describing the propagation of solitons through nonlinear optical fibers, their soliton solutions are obtained using the wellorganized new Kudryashov method. It is believed that the new Kudryashov method provides an effective mathematical tool to look for soliton solutions of highorder nonlinear Schrödinger equations.
