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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. Keywords: nonlinear oscillations, resonance, stability, canonical transformations Citation: Markeev A. P., On the Dynamics of a Gravitational Dipole, Rus. J. Nonlin. Dyn., 2021, Vol. 17, no. 3, pp. 247-261 DOI:10.20537/nd210301
 Reutov V. P.,  Rybushkina G. V. Abstract The transition to dynamical chaos and the related lateral (cross-flow) transport of a passive scalar in the reverse annular jet flow generating two chains of wave-vortex structures are studied. The quasi-geostrophic equations for the barotropic (quasi-two-dimensional) flow written in polar coordinates with allowance for the beta-effect and external friction are solved numerically using a pseudospectral method. The critical parameters of the equilibrium flow with a complex “two-hump” 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 wavenumber-frequency 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 finite-time 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 cross-flow 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. Keywords: barotropic jet flow, chains of wave structures, Eulerian and Lagrangian chaos, cross-flow chaotic transport Citation: Reutov V. P.,  Rybushkina G. V., Dynamical Chaos and Lateral Transport of a Passive Scalar in the Annular Reverse Jet Flow, Rus. J. Nonlin. Dyn., 2021, Vol. 17, no. 3, pp. 263-274 DOI:10.20537/nd210302
 da Silva V. R.,  Menin O. H. Abstract The rapid spread of SARS-CoV-2/COVID-19 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 (susceptible-infectious-recovered) 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. Keywords: epidemiology, infectious diseases, SARS-CoV-2, mathematical modeling, computational simulation, differential equations Citation: da Silva V. R.,  Menin O. H., Modified SIR Compartmental Epidemic Model with Social Distancing and Hospital Saturation Applied to the COVID-19 Pandemic, Rus. J. Nonlin. Dyn., 2021, Vol. 17, no. 3, pp. 275-287 DOI:10.20537/nd210303
 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 second-order 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. Keywords: lift force, drag force, rough wall, Stokes equations, Poiseuille flow, asymptotic development, Lorentz reciprocal theorem Citation: Lamzoud K.,  Assoudi R.,  Bouisfi F.,  Chaoui M., A Sphere Held Fixed in a Poiseuille Flow near a Rough Wall, Rus. J. Nonlin. Dyn., 2021, Vol. 17, no. 3, pp. 289-306 DOI:10.20537/nd210304
 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 quasi-periodic and chaotic spiking, both regular and chaotic bursting regimes, in which the number of spikes per burst varied greatly depending on the control parameter. Keywords: coupled neurons, synchronization, chaos, Hindmarsh –Rose, bursting Citation: Garashchuk I. R., Asynchronous Chaos and Bifurcations in a Model of Two Coupled Identical Hindmarsh – Rose Neurons, Rus. J. Nonlin. Dyn., 2021, Vol. 17, no. 3, pp. 307-320 DOI:10.20537/nd210305
 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 two-dimensional 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. Keywords: Smale diagram, $(A,B)$-diffeomorphism, $\Omega$-conjugacy Citation: Barinova M. K.,  Gogulina E. Y.,  Pochinka O. V., Omega-classification of Surface Diffeomorphisms Realizing Smale Diagrams, Rus. J. Nonlin. Dyn., 2021, Vol. 17, no. 3, pp. 321-334 DOI:10.20537/nd210306
 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 one-dimensional 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 one-dimensional 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 two-dimensional torus $\mathbb{T}^2$ or a two-dimensional sphere $\mathbb{S}^2$. Keywords: $A$-endomorphism, regular lamination, attractor, repeller, strictly invariant set Citation: Grines V. Z.,  Zhuzhoma E. V., Cantor Type Basic Sets of Surface $A$-endomorphisms, Rus. J. Nonlin. Dyn., 2021, Vol. 17, no. 3, pp. 335-345 DOI:10.20537/nd210307
 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 real-time compensation of compliance errors, that is, the deviation of the real end-effector 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 Keywords: robot stiffness, Jacobian evaluation, identification Citation: Mikhel S. K.,  Klimchik A. S., Stiffness Model Reduction for Manipulators with Double Encoders: Algebraic Approach, Rus. J. Nonlin. Dyn., 2021, Vol. 17, no. 3, pp. 347-360 DOI:10.20537/nd210308

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