Vol. 17, no. 3
Vol. 17, no. 3, 2021
Markeev A. P.
Abstract
An orbital gravitational dipole is a rectilinear inextensible rod of negligibly small mass which
moves in a Newtonian gravitational field and to whose ends two point loads are fastened. The
gravitational dipole is mainly designed to produce artificial gravity in a neighborhood of one of
the loads. In the nominal operational mode on a circular orbit the gravitational dipole is located
along the radius vector of its center of mass relative to the Newtonian center of attraction. The main purpose of this paper is to investigate nonlinear oscillations of the gravitational dipole in a neighborhood of its nominal mode. The orbit of the center of mass is assumed to be circular or elliptic with small eccentricity. Consideration is given both to planar and arbitrary spatial deviations of the gravitational dipole from its position corresponding to the nominal mode. The analysis is based on the classical Lyapunov and Poincaré methods and the methods of Kolmogorov – Arnold – Moser (KAM) theory. The necessary calculations are performed using computer algorithms. An analytic representation is given for conditionally periodic oscillations. Special attention is paid to the problem of the existence of periodic motions of the gravitational dipole and their Lyapunov stability, formal stability (stability in an arbitrarily high, but finite, nonlinear approximation) and stability for most (in the sense of Lebesgue measure) initial conditions. 
Reutov V. P., Rybushkina G. V.
Abstract
The transition to dynamical chaos and the related lateral (crossflow) transport of a passive
scalar in the reverse annular jet flow generating two chains of wavevortex structures are studied.
The quasigeostrophic equations for the barotropic (quasitwodimensional) flow written in
polar coordinates with allowance for the betaeffect and external friction are solved numerically
using a pseudospectral method. The critical parameters of the equilibrium flow with a complex
“twohump” azimuth velocity profile facilitating a faster transition to the complex dynamics are
determined. Two regular multiharmonic regimes of wave generation are revealed with increasing
flow supercriticality before the onset of Eulerian chaos. The occurrence of the complex flow
dynamics is confirmed by a direct calculation of the largest Lyapunov exponent. The evolution
of streamline images is analyzed by making video, thereby chains with single and composite
structures are distinguished. The wavenumberfrequency spectra confirming the possibility of
chaotic transport of the passive scalar are drawn for the basic regimes of wave generation. The
power law exponents for the azimuth particle displacement and their variance, which proved
the occurrence of the anomalous azimuth transport of the passive scalar, are determined. Lagrangian
chaos is studied by computing the finitetime Lyapunov exponent and its distribution
function. The internal chain (with respect to the annulus center) is found to be totally subject
to Lagrangian chaos, while only the external chain boundary is chaotic. It is revealed that the
crossflow transport occurs only in the regime of Eulerian dynamical chaos, since there exists a
barrier to it in the multiharmonic regimes. The images of fluid particles confirming the presence
of lateral transport are obtained and their quantitative characteristics are determined.

da Silva V. R., Menin O. H.
Abstract
The rapid spread of SARSCoV2/COVID19 in the first months of 2020 overburdened health
systems worldwide. The absence of vaccines led public authorities to respond to the pandemic
by adopting nonpharmaceutical interventions, mainly social distancing policies. Yet concerns
have been raised on the economic impact of such measures. Considering the impracticability of
conducting controlled experiments to assess the effectiveness of such interventions, mathematical
models have played an essential role in helping decision makers. Here we present a simple
modified SIR (susceptibleinfectiousrecovered) model that includes social distancing and two
extra compartments (hospitalized and dead due to the disease). Our model also incorporates
the potential increase in the mortality rate due to the health system saturation. Results from
numerical experiments corroborate the striking role of social distancing policies in lowering and
delaying the epidemic peak, thus reducing the demand for intensive health care and the overall
mortality. We also probed into optimal social distancing policies that avoid the health system
saturation and minimize the economic downturn.

Lamzoud K., Assoudi R., Bouisfi F., Chaoui M.
Abstract
We present here an analytical calculation of the hydrodynamic interactions between a smooth
spherical particle held fixed in a Poiseuille flow and a rough wall. By the assumption of a low
Reynolds number, the flow around a fixed spherical particle is described by the Stokes equations.
The surface of the rigid wall has periodic corrugations, with small amplitude compared with the
sphere radius. The asymptotic development coupled with the Lorentz reciprocal theorem are
used to find the analytical solution of the couple, lift and drag forces exerted on the particle,
generated by the secondorder flow due to the wall roughness. These hydrodynamic effects are
evaluated in terms of amplitude and period of roughness and also in terms of the distance between
sphere and wall.

Garashchuk I. R.
Abstract
We study a minimal network of two coupled neurons described by the Hindmarsh – Rose
model with a linear coupling. We suppose that individual neurons are identical and study
whether the dynamical regimes of a single neuron would be stable synchronous regimes in the
model of two coupled neurons. We find that among synchronous regimes only regular periodic
spiking and quiescence are stable in a certain range of parameters, while no bursting synchronous
regimes are stable. Moreover, we show that there are no stable synchronous chaotic regimes in
the parameter range considered. On the other hand, we find a wide range of parameters in
which a stable asynchronous chaotic regime exists. Furthermore, we identify narrow regions of
bistability, when synchronous and asynchronous regimes coexist. However, the asynchronous
attractor is monostable in a wide range of parameters. We demonstrate that the onset of the
asynchronous chaotic attractor occurs according to the Afraimovich – Shilnikov scenario. We
have observed various asynchronous firing patterns: irregular quasiperiodic and chaotic spiking,
both regular and chaotic bursting regimes, in which the number of spikes per burst varied greatly
depending on the control parameter.

Barinova M. K., Gogulina E. Y., Pochinka O. V.
Abstract
The present paper gives a partial answer to Smale's question
which diagrams can correspond to $(A,B)$diffeomorphisms.
Model diffeomorphisms of the twodimensional torus derived
by ``Smale surgery'' are considered, and necessary and
sufficient conditions for their topological conjugacy are
found. Also, a class $G$ of $(A,B)$diffeomorphisms on surfaces which are the connected
sum of the model diffeomorphisms is introduced. Diffeomorphisms of the class $G$ realize any connected Hasse
diagrams (abstract Smale graph). Examples of diffeomorphisms from $G$ with isomorphic labeled Smale diagrams which are not ambiently $\Omega$conjugated are constructed. Moreover, a subset $G_{*}^{} \subset G$ of diffeomorphisms for which the isomorphism class of labeled Smale diagrams is a complete invariant of the ambient $\Omega$conjugacy is singled out.

Grines V. Z., Zhuzhoma E. V.
Abstract
The paper is devoted to an investigation of the genus of an orientable closed surface $M^2$
which admits $A$endomorphisms whose nonwandering set contains a onedimensional strictly
invariant contracting repeller $\Lambda_r^{}$ with a uniquely defined unstable bundle and with
an admissible boundary of finite type. First, we prove that, if $M^2$ is a torus or a
sphere, then $M^2$ admits such an endomorphism. We also show that, if $ \Omega$ is a basic set with a uniquely defined unstable bundle of the endomorphism $f\colon M^2\to M^2$ of a closed orientable surface $M^2$ and $f$ is not a diffeomorphism, then $ \Omega$ cannot be a Cantor type expanding attractor. At last, we prove that, if $f\colon M^2\to M^2$ is an $A$endomorphism whose nonwandering set consists of a finite number of isolated periodic sink orbits and a onedimensional strictly invariant contracting repeller of Cantor type $\Omega_r^{}$ with a uniquely defined unstable bundle and such that the lamination consisting of stable manifolds of $\Omega_r^{}$ is regular, then $M^2$ is a twodimensional torus $\mathbb{T}^2$ or a twodimensional sphere $\mathbb{S}^2$.

Mikhel S. K., Klimchik A. S.
Abstract
The accuracy of the robot positioning during material processing can be improved if the
deformation under the load is taken into account. A manipulator stiffness model can be obtained
using various approaches which differ in the degree of detail and computational complexity.
Regardless of the model, its practical application requires knowledge of the stiffness parameters
of the robot components, which implies solving the identification problem. In this work, we consider a reduced stiffness model, which assumes that the manipulator links are rigid, while the joints are compliant and include both elasticities in the joints themselves and the elastic properties of the links. This simplification reduces the accuracy of the model, but allows us to identify the stiffness parameters, which makes it suitable for practical application. In combination with a double encoders measurement system, this model allows for realtime compensation of compliance errors, that is, the deviation of the real endeffector position from the calculated one due to the deformation of the robot under load. The paper proposes an algebraic approach to determining the parameters of the reduced model in a general form. It also demonstrates several steps that can be done to simplify computations. First, it introduces the backward semianalytical Jacobian computation technique, which allows reducing the number of operations for the manipulator with virtual joints. Second, it provides an algorithm for the calculation of the required intermediate matrices without explicit Jacobian calculation and using more compact expressions. To compare the proposed techniques with the experimental approach, the robot deformation under load is simulated and the tool displacement is estimated. It is shown that both approaches are equivalent in terms of accuracy. While the experimental method is easier to implement, the algebraic approach allows analyzing the contribution of each link in a reduced model of 