Volume 26, Number 3
Volume 26, Number 3, 2021
Ivanov A. V.
Abstract
We study a linear cocycle over the irrational rotation $\sigma_{\omega}(x) = x + \omega$ of the circle~$\mathbb{T}^{1}$. It is supposed that the cocycle is generated by a $C^{2}$map
$A_{\varepsilon}: \mathbb{T}^{1} \to SL(2, \mathbb{R})$ which depends on a small parameter $\varepsilon\ll 1$ and has the form of the Poincar\'e map corresponding to a singularly perturbed Hill equation with quasiperiodic potential. Under the assumption that the norm of the matrix $A_{\varepsilon}(x)$ is of order $\exp(\pm \lambda(x)/\varepsilon)$, where $\lambda(x)$ is a positive function, we examine the property of the cocycle to possess an exponential dichotomy (ED) with respect to the parameter~$\varepsilon$. We show that in the limit $\varepsilon\to 0$ the cocycle ``typically'' exhibits ED only if it is exponentially close to a constant cocycle.
Conversely, if the cocycle is not close to a constant one,
it does not possess ED, whereas the Lyapunov exponent is ``typically'' large.

Lara R., Bengochea A.
Abstract
In this work we introduce a planar restricted fourbody problem where a massless
particle moves under the gravitational influence due to three bodies following the figureeight
choreography, and explore some symmetric periodic orbits of this system which turns out to
be nonautonomous. We use reversing symmetries to study both theoretically and numerically a
certain type of symmetric periodic orbits of this system. The symmetric periodic orbits (initial
conditions) were computed by solving some boundaryvalue problems.

Zhu S.
Abstract
For the curved $n$body problem, we show that the set of ordinary central configurations is away from singular configurations in $\mathbb{H}^3$ with positive momentum of inertia, and away from a subset of singular
configurations in $\mathbb{S}^3$. We also show that
each of the $n!/2$ geodesic ordinary central configurations for $n$ masses has Morse index $n2$.
Then we get a direct corollary that there are at least $\frac{(3n4)(n1)!}{2}$ ordinary central
configurations for given $n$ masses if all ordinary central configurations of these masses are nondegenerate.

Kurakin L. G., Kurdoglyan A. V.
Abstract
A dynamical system with a cosymmetry is considered. V. I. Yudovich showed that
a noncosymmetric equilibrium of such a system under the conditions of the general position
is a member of a oneparameter family. In this paper, it is assumed that the equilibrium is
cosymmetric, and the linearization matrix of the cosymmetry is nondegenerate. It is shown that,
in the case of an odddimensional dynamical system, the equilibrium is also nonisolated and
belongs to a oneparameter family of equilibria. In the evendimensional case, the cosymmetric
equilibrium is, generally speaking, isolated. The Lyapunov – Schmidt method is used to study
bifurcations in the neighborhood of the cosymmetric equilibrium when the linearization matrix
has a double kernel. The dynamical system and its cosymmetry depend on a real parameter.
We describe scenarios of branching for families of noncosymmetric equilibria.

Kudryashov N. A.
Abstract
Selfsimilar reductions for equations of the Kupershmidt and Sawada – Kotera hierarchies are
considered. Algorithms for constructing a Lax pair
for equations of these hierarchies are presented. Lax pairs for ordinary differential
equations of the fifth, seventh and eleventh orders
corresponding to the Kupershmidt and the Sawada – Kotera hierarchies are given.
The Lax pairs allow us to solve these equations by means of the inverse
monodromy transform method. The application of the Painlevé test to the seventh order of the similarity reduction for the Kupershmidt hierarchy is
demonstrated. It is shown that special solutions of the similarity reductions for the Kupershnmidt and Sawada – Kotera hierarchies are determined via
the transcendents of the $K_1$ and $K_2$ hierarchies. Rational solutions of the similarity reductions of the modified Kupershmidt and Sawada – Kotera
hierarchies are given. Special polynomials associated with the selfsimilar reductions of
the Kupershmidt and Sawada – Kotera hierarchies are presented.
Rational solutions of some hierarchies are calculated by means of the Miura transformations and taking into account special polynomials.

Rocha A. A.
Abstract
Let $M$ be a closed manifold and $L$ an exact magnetic Lagrangian. In this
paper we prove that there exists a residual set $\mathcal{G}$ of $%
H^{1}\left( M;\mathbb{R}\right)$ such that the property
\begin{equation*}
{\widetilde{\mathcal{M}}}\left( c\right) ={\widetilde{\mathcal{A}}}\left(
c\right) ={\widetilde{\mathcal{N}}}\left( c\right), \forall c\in \mathcal{G},
\end{equation*}
with ${\widetilde{\mathcal{M}}}\left( c\right)$ supporting a uniquely
ergodic measure, is generic in the family of exact magnetic Lagrangians. We also prove
that, for a fixed cohomology class $c$, there exists a
residual set of exact magnetic Lagrangians such that, when this
unique
measure is supported on a periodic orbit, this orbit is hyperbolic and its
stable and unstable manifolds intersect transversally. This result is a
version of an analogous theorem, for Tonelli Lagrangians, proven in [6].

Nakamura G., Grammaticos B., Badoual M.
Abstract
We analyse the patterns of the current epidemic evolution in various countries with
the help of a simple SIR model. We consider two main effects: climate induced seasonality and
recruitment. The latter is introduced as a way to palliate for the absence of a spatial component
in the SIR model. In our approach we mimic the spatial evolution of the epidemic through a
gradual introduction of susceptible individuals.
We apply our model to the case of France and Australia and explain the appearance of two temporally wellseparated epidemic waves. We examine also Brazil and the USA, which present patterns very different from those of the European countries. We show that with our model it is possible to reproduce the observed patterns in these two countries thanks to simple recruitment assumptions. Finally, in order to show the power of the recruitment approach, we simulate the case of the 1918 influenza epidemic reproducing successfully the, by now famous, three epidemic peaks. 