 EDITORINCHIEF
 DEPUTY EDITORINCHIEF
 Editorial board

 Passed Away
Condolence note
Professor Alexey V. Borisov, EditorinChief of the journal RCD, suddenly passed away at the age of 55 on January 24, 2021.

Volume 26, Number 4
Markeev A. P.
On the Metric Stability and the Nekhoroshev Estimate of the Velocity of Arnold Diffusion in a Special Case of the Threebody Problem
Abstract
A study is made of the stability of triangular libration points in the nearlycircular
restricted threebody problem in the spatial case. The problem of stability for most (in the sense
of Lebesgue measure) initial conditions in the planar case has been investigated earlier. In the
spatial case, an identical resonance takes place: for all values of the parameters of the problem
the period of Keplerian motion of the two main attracting bodies is equal to the period of
small linear oscillations of the third body of negligible mass along the axis perpendicular to the
plane of the orbit of the main bodies. In this paper it is assumed that there are no resonances
of the planar problem through order six. Using classical perturbation theory, KAM theory
and algorithms of computer calculations, stability is proved for most initial conditions and the
Nekhoroshev estimate of the time of stability is given for trajectories starting in an addition to
the abovementioned set of most initial conditions.

Kourliouros K.
Sections of Hamiltonian Systems
Abstract
A section of a Hamiltonian system is a hypersurface in the phase space of the
system, usually representing a set of onesided constraints (e. g., a boundary, an obstacle or
a set of admissible states). In this paper we give local classification results for all typical
singularities of sections of regular (nonsingular) Hamiltonian systems, a problem equivalent
to the classification of typical singularities of Hamiltonian systems with onesided constraints.
In particular, we give a complete list of exact normal forms with functional invariants, and
we show how these are related/obtained by the symplectic classification of mappings with
prescribed (Whitneytype) singularities, naturally defined on the reduced phase space of the
Hamiltonian system.

Pochinka O. V., Zinina S. K.
Construction of the Morse – Bott Energy Function for Regular Topological Flows
Abstract
In this paper, we consider regular topological flows on closed $n$manifolds. Such
flows have a hyperbolic (in the topological sense) chain recurrent set consisting of a finite number
of fixed points and periodic orbits. The class of such flows includes, for example, Morse – Smale
flows, which are closely related to the topology of the supporting manifold. This connection is
provided by the existence of the Morse – Bott energy function for the Morse – Smale flows. It
is well known that, starting from dimension 4, there exist nonsmoothing topological manifolds,
on which dynamical systems can be considered only in a continuous category. The existence of
continuous analogs of regular flows on any topological manifolds is an open question, as is the
existence of energy functions for such flows. In this paper, we study the dynamics of regular
topological flows, investigate the topology of the embedding and the asymptotic behavior of
invariant manifolds of fixed points and periodic orbits. The main result is the construction of
the Morse – Bott energy function for such flows, which ensures their close connection with the
topology of the ambient manifold.

Balabanova N., Montaldi J.
Twobody Problem on a Sphere in the Presence of a Uniform Magnetic Field
Abstract
We investigate the motion of one and two charged nonrelativistic particles on a
sphere in the presence of a magnetic field of uniform strength. For one particle, the motion
is always circular, and determined by a simple relation between the velocity and the radius
of motion. For two identical particles interacting via a cotangent potential, we show there are
two families of relative equilibria, called Type I and Type II. The Type I relative equilibria
exist for all strengths of the magnetic field, while those of Type II exist only if the field is
sufficiently strong. The same is true if the particles are of equal mass but opposite charge. We
also determine the stability of the two families of relative equilibria.

Zhuravlev N. B., Rossovsky L. E.
Spectral Radius Formula for a Parametric Family of Functional Operators
Abstract
The conditions for the unique solvability of the boundaryvalue problem for a functional differential equation with shifted and compressed arguments are expressed via the spectral radius formula for the corresponding class of functional operators. The use of this formula involves calculation of certain type limits, which, even in the simplest cases, exhibit an amazing ``chaotic'' dependence on the compression ratio.
For example, it turns out that the spectral radius of the operator
$$
L_2(\mathbb R^n)\ni u(x)\mapsto u(p^{1}x+h)u(p^{1}xh)\in L_2(\mathbb R^n),\quad p>1,\quad h\in\mathbb R^n,
$$
is equal to $2p^{n/2}$ for transcendental values of $p$, and depends on the coefficients of the minimal polynomial for $p$ in the case where $p$ is an algebraic number. In this paper, we study this dependence.
The starting point is the wellknown statement that, given a velocity vector with rationally
independent coordinates, the corresponding linear flow is minimal on the torus, i.e., the
trajectory of each point is everywhere dense on the torus. We prove a version of this statement that helps to control the behavior of trajectories also in the case of rationally dependent velocities. Upper and lower bounds for the spectral radius are obtained for various cases of the coefficients of the minimal polynomial for $p$. The main result of the paper is the exact formula of the spectral radius for rational (and roots of any degree of rational) values of~$p$.

Andrade J., Vidal C., Sierpe C.
Stability of the Relative Equilibria in the Twobody Problem on the Sphere
Abstract
We consider the 2body problem in the sphere $\mathbb{S}^2$. This problem is modeled by a Hamiltonian system with $4$ degrees of freedom and, following the approach given in [4], allows us to reduce the study to a system of $2$ degrees of freedom. In this work we will use theoretical tools such as normal forms and some nonlinear stability results on Hamiltonian systems for demonstrating a series of results that will correspond to the open problems proposed in [4] related to the nonlinear stability of the relative equilibria. Moreover, we study the existence of Hamiltonian pitchfork and centersaddle bifurcations.

Shibayama M., Yamada J.
Nonintegrability of the Reduced Planar Threebody Problem with Generalized Force
Abstract
We consider the planar threebody problem with generalized potentials. Some
nonintegrability results for these systems have been obtained by analyzing the variational
equations along homothetic solutions. But we cannot apply it to several exceptional cases. For
example, in the case of inversesquare potentials, the variational equations along homothetic
solutions are solvable. We obtain sufficient conditions for nonintegrability for these exceptional
cases by focusing on some particular solutions that are different from homothetic solutions.

Biswas A., Kara A. H., Ekici M., Zayed E., Alzahrani A. K., Belic M. R.
Conservation Laws for Solitons in Magnetooptic Waveguides with Anticubic and Generalized Anticubic Nonlinearities
Abstract
This paper implements a multiplier approach to exhibit conservation laws in
magnetooptic waveguides that maintain anticubic as well as generalized anticubic forms of the
nonlinear refractive index. Three conservation laws are retrieved for each form of nonlinearity.
They are power, linear momentum and Hamiltonian. The conserved quantities are computed
from their respective densities.
