This special issue is dedicated to the 60th anniversary of the birth of Alexey V. Borisov (27.03.1965–24.01.2021), the founder and first editor-in-chief of Russian Journal of Nonlinear Dynamics.

This paper addresses the problem of the motion a balanced ellipsoid of revolution with symmetrically truncated vertices that rolls without slipping on a plane. First integrals are found and reduction to quadratures is performed. Partial solutions to the resulting system are found. Using an analysis of bifurcation curves, bifurcations of partial solutions are investigated and a classification of all types of bifurcation diagrams depending on parameters is carried out.

In this work we construct symmetric central configurations for the $n$-body problem, for $n = 4, 5, 6, 7, 8$. Geometric arguments are used in the proof of existence of these configurations. We observe that these examples of central configurations can be proved by other means, for instance, using arguments for the existence of symmetric solutions to variational problems as is done by J. Montaldi in [10]. However, we believe that our geometrical approach will be useful in tackling other questions of central configurations of the Newtonian $n$-body problem.

The attitude dynamics of the Lagrange gyrostat is considered. A dynamical analogy of the heavy Lagrange gyrostat and the magnetic Lagrange gyrostat satellite on a circle equatorial orbit in the geomagnetic field is shown. Analytical solutions for homoclinic and heteroclinic phase trajectories are obtained. Dynamical chaos is analyzed. The practical application of the Lagrange gyrostat in space flight dynamics is presented. The results of the paper can be especially applicable for small spacecraft and nanosatellites of space remote sensing constellations.

This paper considers the motion of an elliptic foil in the presence of point vortices. For the case of a vortex pair, a bifurcation analysis of the relative equilibria (a generalization of Föppl solutions) is carried out. These solutions correspond to the motion of the system on an invariant manifold on which the dynamics is governed by a system with $\frac{3}{2}$ degrees of freedom. Using a period advance Poincaré map, a numerical analysis of the dynamics on the invariant manifold is performed for the case where the center of mass of the system moves periodically in an impulse-like manner. The occurrence of new periodic, quasiperiodic and chaotic modes of motion is demonstrated.

Relative motion in a system consisting of two massive heliocentric space stations and a small mass spacecraft equipped with a solar sail is considered. The system is approximately 1 AU from the Sun. The spacecraft is capable of coasting along a tether (called a “handrail”) connecting the stations, and the sail is oriented orthogonally to the solar rays direction. The distance between the stations is assumed to be about a few kilometers. The stations are equipped with additional solar sails that help compensate the influence of the spacecraft motion on the relative location of the stations. The conditions providing such compensation are deduced. These conditions allow one to determine the area and orientation of additional solar sails so that the distance between the stations keeps invariable, and the segment connecting the stations remains orthogonal to the solar rays after the spacecraft relocation between the most remote points permitted by the handrail. The conditions are verified numerically by comparing the solutions of the equations of motion with and without taking into account additional sails.

The proposed retrieval control combats the effect of a tether winding on the end body at the final stage of its retraction. The tether retraction rate control law providing stabilization of the tether system is developed. The stabilization angle is determined for the fastest tether retrieval. Unlike most works on the development of space tether retrieval control laws and schemes, this study considers the influence of the gravitational fields of the planet and its moon. Numerical simulation confirmed that the developed control ensures the tether retraction without a dramatic increase in the amplitude of its oscillation at the final stage.

This work is dedicated to the study of the orbital motion of an artificial satellite moving in the gravitational field of a viscoelastic planet, which in turn is moving in the gravitational field of a massive attracting center and a satellite. The planet is modeled as a homogeneous isotropic viscoelastic body, while the other celestial bodies are treated as material points. Under the influence of the gravitational fields of the attracting center and the natural satellite, tidal bulges arise in the viscoelastic body of the planet, which affect its gravitational potential and serve as a disturbing factor for the orbital motion of artificial satellites in low-flying orbits. The equations of motion for the artificial satellite are derived in terms of Delaunay canonical variables. A procedure for averaging over the “fast” angular variables is carried out. In the averaged equations, the “slow” Delaunay variables responsible for the evolution of the semimajor axis and the eccentricity of the elliptical orbit remain unchanged. The entire effect of evolution is concentrated in the change of inclination, longitude of the ascending node, and longitude of the perigee from the ascending node. With respect to the specified orbital elements, an evolutionary system of equations is derived. In the zero approximation for the dissipative terms, a closed system of second-order differential equations is identified relative to the inclination and longitude of the ascending node of the satellite’s orbit. Its stationary solutions are determined and their stability is studied. The existence of stable stationary motions for polar satellites and satellites close to equatorial ones is shown.

We deal with the problem of orbital stability of the periodic solutions of a Hamiltonian system with two degrees of freedom. We assume that third- or sixth-order resonance takes place in this system and solve the orbital stability problem in a special case of degeneracy, when it is necessary to take into account the terms up to the sixth order of the Hamiltonian expansion in the neighborhood of the periodic solution. By using methods of normal forms and KAM theory we obtain sufficient conditions of orbital stability and instability in the form of inequalities with respect to coefficients of the Hamiltonian normal form calculated up to terms of the sixth order. We also show that the above-mentioned problem of orbital stability is equivalent to the problem of Lyapunov stability of an equilibrium position of a reduced system.
We apply the above results in the problem of the orbital stability of the pendulum oscillations of a heavy rigid body with a fixed point in the case when its principal moment of inertia $A$, $B$, and $C$ satisfy the equality $A = C = 4B$.

The use of nanotechnologies has led to a significant development of nanomechanics for investigation of the mechanical behavior of various types of nanostructures and nanosystems, such as nanorods, nanowires, nanotubes, nanoribbons, nanoshells etc. Typical applications of nanomechanics include nanoelectromechanical systems (NEMS) and microelectromechanical systems (MEMS).
In this paper, which is dedicated to the memory of G.N. Kuvyrkin, the influence of the effect of spatial nonlocality on strain and stress distributions is investigated by considering the problem of deformation of a vertical cylindrical column in a gravitational field. Analytical dependences are obtained taking into account different kinematic and force boundary conditions on the upper base of the column. For the linear stress distribution over the height of the column, a considerable nonlinearity of its strained state is established and the emergence of the edge effect is revealed. This edge effect arises due to the fact that the material of the column exhibits the property of spatial nonlocality. It is noted that the largest stresses (in absolute value) decrease in the column compared to the case where the material of the column does not exhibit the property of spatial nonlocality.

The paper considers the stochastic space-fractional Kuramoto – Sivashinsky equation in the complex plane. This equation is reduced to an ordinary differential equation. For the resulting ODE, a theorem on the existence and uniqueness of the Cauchy problem in a neighborhood of the initial data is formulated and proved. For practical applications, an analytical approximate solution (a partial sum of a series) is proposed. A priori error estimates of the analytical approximate solution are provided. To extend beyond the domain of convergence of the series obtained, an analytic continuation of the approximate solution is carried out.

The paper studies the motions of a nearly autonomous, Hamiltonian system $2\pi$-periodic in time, with two degrees of freedom, in the neighborhood of a trivial equilibrium. The values of the parameters are considered near the boundary of the stability region of this equilibrium that corresponds to the zero frequency case in the limiting autonomous problem. The cases are distinguished when the other frequency is nonresonant (not equal to an integer or halfinteger number) and resonant, i. e., a multiple parametric resonance is realized in the system. The stability and instability regions (parametric resonance regions) of the trivial equilibrium of the system are obtained. The question of the existence in its neighborhood of analytic (in integer or fractional powers of a small parameter) resonant periodic motions, their number and linear stability is resolved. Earlier, similar results for the studied cases of multiple parametric resonances have been obtained in the sections of the parameter space corresponding to a fixed (resonant) value of one of the parameters. In this paper, complete neighborhoods of resonance points in the parameter space are considered. As expected, this leads to complication of both the stability diagram of the trivial equilibrium and the distribution and character of the stability of the periodic solutions under study. For each resonance case, an algorithm for analyzing these motions is developed. Differences and similarities in the properties of the system for different resonant cases are established, parallels are found with multiple resonance cases of another type, arising in the systems with another structure of the perturbing part of the Hamiltonian function.

Spherical robots are a type of robots that have a spherical shape, allowing them to move by rolling. The drive for this category of robots can be implemented in various ways. This paper proposes the use of a new electromagnetic drive for this type of robot, resembling a spherical motor in structure. It includes permanent magnets and electromagnets, and the robot’s motion is achieved through electromagnetic interaction between them.
The main goal of this work is to substantiate the fundamental possibility of controlling the motion of a spherical robot using an electromagnetic drive. To this end, first, a mathematical model of the quasi-stationary electromagnetic interaction of magnets placed on different moving shells was constructed. Second, based on this model, the dependence of the torque acting on the inner spherical shell of the robot on the angular displacement between the permanent magnet and the electromagnet was derived; using this dependence, the parameters of an effective configuration of the spherical robot with electromagnetic drive were calculated, along with the energy-optimal distribution of currents in the electromagnets. Third, using Kirchhoff’s equations, as well as results from previous works on optimal control of the motion of a spherical robot with a mechanical drive, equations for currents and voltages were derived that enable the implementation of optimal motion of the spherical robot on an uneven surface without slipping. Finally, the motions of the spherical robot on flat and bell-shaped surfaces were considered, and the dependences of currents and voltages in the electromagnets necessary for implementing the specified motions were obtained, containing information on the dynamic and frequency ranges of the control signals supplied to the electromagnets.
To track the trajectory of the spherical robot’s motion, a regulator based on linearization of the robot’s dynamics equations near the reference trajectory was used. To suppress disturbances and minimize deviations, a regulator based on the generalized $H_2$-norm was applied.

This paper describes a comprehensive approach to the development of a quadrupedal robot possessing 12 actuated degrees of freedom. The development comprises the design of the mechanical system, the creation of control electronics, and the implementation of software for motion generation. A key aspect involves the application of reinforcement learning in a physical simulator, followed by the transfer of the trained algorithms to the physical device (sim-to-real). An embedded Neural Processing Unit (NPU) is utilized to accelerate the execution of AI algorithms, such as object recognition, navigation, and motion optimization. The proposed solutions enable efficient and symmetrical locomotion, high adaptability to changing environmental conditions, and enhanced operational autonomy of the robot.