In this work we focus on the chaotic diffusion in the phase space of a triaxial potential resembling an elliptical galaxy. The transport process is studied in two different action-like starting spaces in order to cope with circulating and noncirculating orbits. Estimates of the diffusion rate obtained by means of the variance approach are discussed in detail and their limitations are exposed. After revisiting the Shannon-entropy-based method from a conceptual point of view in the framework of simple arguments taken from the information theory, we apply it to measure changes in the unperturbed actions or integrals of motion of the system for different sets of small ensembles of random initial conditions. For such sets of ensembles, estimates of the Lyapunov times are also provided. The results show that, within the chaotic component of the phase space, the Lyapunov times are shorter than any physical time scale as the Hubble time, but the diffusion times are much larger than the latter. Thus, we conclude that stable chaos dominates the dynamics of realistic galactic models.

The periodic motions of a material point are studied on the assumption that, throughout the motion, the point remains on a fixed absolutely smooth surface (in an ellipsoidal bowl), which is part of the surface of a triaxial ellipsoid. The motion occurs in a uniform field of gravity, and the largest axis of the ellipsoid is directed along the vertical.
Cases are considered where the motion of the point occurs along one of the principal sections of the surface in the neighborhood of a stable equilibrium at the lowest point of the bowl. An analytical representation of the corresponding periodic motions is obtained up to terms of degree five inclusive with respect to the magnitude of perturbation of the point from the equilibrium. The stability of these periodic motions is investigated.

This paper addresses the problem of the Chaplygin sleigh moving on an inclined plane under the action of periodic controls. Periodic controls are implemented by moving point masses. It is shown that, under periodic oscillations of one point mass in the direction perpendicular to that of the knife edge, for a nonzero initial velocity there exists a motion with acceleration or a uniform motion (on average per period) in the direction opposite to that of the largest descent. It is shown that adding to the system two point masses which move periodically along some circle enables a period-averaged uniform motion of the system from rest.

Examples of mechanical systems subjected to unilateral holonomic constraints are considered. It is assumed that the boundary of the area of unconstrained motion has singularities. Possible ways of resolving the singularities based on knowledge about the mechanical origin of the constraints are indicated.

This paper presents an analysis of nonlinear oscillations of a near-autonomous two-degreeof- freedom Hamiltonian system, $2\pi$-periodic in time, in the neighborhood of a trivial equilibrium. It is assumed that in the autonomous case, for some set of parameters, the system experiences a multiple parametric resonance for which the frequencies of small linear oscillations in the neighborhood of the equilibrium are equal to two and one. It is also assumed that the Hamiltonian of perturbed motion contains only terms of even degrees with respect to perturbations, and its nonautonomous perturbing part depends on odd time harmonics. The analysis is performed in a small neighborhood of the resonance point of the parameter space. A series of canonical transformations is made to reduce the Hamiltonian of perturbed motion to a form whose main (model) part is characteristic of the resonance under consideration and the structure of nonautonomous terms. Regions of instability (regions of parametric resonance) of the trivial equilibrium are constructed analytically and graphically. A solution is presented to the problem of the existence and bifurcations of resonant periodic motions of the system which are analytic in fractional powers of a small parameter. As applications, resonant periodic motions of a double pendulum are constructed. The nearly constant lengths of the rods of the pendulum are prescribed periodic functions of time. The problem of the linear stability of these motions is solved.

In this paper, the conditions of nonregular precession with a constant ratio of the velocities of precession and proper rotation for a gyrostat in the superposition of two homogeneous and one axisymmetric field are obtained. The case where the gyrostat has axial dynamical symmetry and the proper rotation axis coincides with the body’s symmetry axis is singled out. It is shown that in the case where the gyrostatic momentum is collinear to the symmetry axis, nonregular precession is possible with a precession velocity equal to, twice as large as, or twice as small as the proper rotation velocity. In each of these cases, the condition expressing the ratio of the axial and equatorial inertia moments of the body in terms of the nutation angle coincides with the corresponding condition obtained earlier for the nonregular precession of a solid in three homogeneous fields. In the particular case of the gyrostat’s spherical symmetry, when the precession speed is half or twice as large as its proper rotation speed, the cosine of the nutation angle is equal to one fourth; at equal speeds, the nutation angle should be equal to sixty degrees. The sets of admissible positions of the forces’ centers for the general case of nonorthogonal fields are found. The precession of a gyrostat whose gyrostatic momentum is deflected from the symmetry axis is considered. The possibility of nonregular precession is shown for the case where the precession velocity is twice as large as the proper rotation velocity. The solution is expressed in terms of elementary functions. The rotation of the gyrostat is either periodic or the rotation velocity tends to zero and the carrier body of the gyrostat approaches the equilibrium position.

When an articulated drinking straw is slid over a rod vibrated by a motor, the straw moves up and down continuously. To clarify the mechanism for this motion, a straw was created that was narrower at the center than at the ends and the mechanism for the up-and-down motion as well as its reversal at the rod's tip was examined. The angle $\psi$ between the rod and the straw is constant because they are in contact at three points, i.e., at both ends and the center. Therefore, when the straw rotates around the rod, it will move unidirectionally according to the sign of $\psi$. This velocity is determined as a function of the straw half-length and the number of rotations of the nut to which the rod is attached. As the straw rises, the rod's tip enters the straw, and the ascent stops when $\psi$ reaches zero because the rate of ascent is proportional to $\psi$. The sign of $\psi$ then reverses to start the downward motion, and the straw returns. When it reaches a reflector disk, the sign of $\psi$ reverses again and the straw rises. Experiments were conducted to measure the velocities of three straws with different lengths, and the results showed that the theoretical velocities were greater than the experimental ones. The reason for this is assumed to be that although in theory the straw does not slip at the point of contact between its center and the rod, in reality it does slip. However, the theoretical and experimental velocities decreased in similar ways with increasing straw half-length, and the agreement between them was determined by the relative error. For half-lengths below $1.0$ cm, the average agreement was approx. $80\,\%$, and for all lengths it was approx. $73\,\%$. The agreement would have been even better if it had not been for the effects of slippage between the straw and the rod, the presence of nodes and antinodes of the rod vibration, and deformation in the central portion of straw. Considering these effects, the experimental values support the validity of the mechanistic considerations of the straw motion and the theoretically determined velocity.

The paper experimentally investigates the problem of the influence of periodic vibrations of the pivot point of a physical pendulum on its nonlinear oscillations in the vicinity of a stable equilibrium position on the vertical. The vibrations are assumed to be periodic and occur in the plane of the pendulum’s motion along an elliptical trajectory. In the experimental plane of parameters: the amplitude of pendulum oscillations and the parameter characterizing the difference in the vibration intensity of the pivot point in the horizontal and vertical directions, the values at which the pendulum clock gains and delays are selected. The experiment showed that with a vibration of $7.0$ Hz, which is more intense in the horizontal direction, the oscillation period of the pendulum angle increases by $0.017$ seconds compared to the pendulum’s natural period. In contrast, with vibration more intense in the vertical direction, the period decreases by $0.0164$ seconds. The experiments were carried out on an ABB IRB $1600$ industrial robot manipulator with a developed pendulum and a reflector with a lens system for a laser tracker installed at the end effector of the robot. Tracking of the trajectory of the pendulum pivot point was carried out using an API Radian Pro laser tracker, the amplitude and frequency of pendulum oscillations were recorded using a machine vision camera and image processing methods.

We study an $n$-dimensional system of ordinary differential equations (ODEs) with a relay nonsymmetric hysteresis. Conditions under which the system of ODEs governs a dynamical system are specified. We obtain sufficient conditions for motions of the system to be recurrent or periodic. Also, we consider various configurations for the closed phase trajectories (orbits) of the motions as well as properties of fixed (switching) points on these trajectories. An example for the system of dimension 3 is given to support the obtained results.

This paper proposes a mathematical model for nonlinear oscillations of a Kirchhoff plate resting on an elastic foundation with hardening cubic nonlinearity and interacting with a pulsating layer of viscous gas. The plate is the bottom of a narrow plane channel filled with the viscous gas; the upper channel wall is rigid. Within this model, the aeroelastic response and phase response of the plate to pressure pulsation at the channel ends are determined and investigated. The formulated model allows us to simultaneously study the effect on the plate vibrations of its dimensions and the material physical properties, the nonlinearity of the plate elastic foundation, the inertia of gas motion, as well as the gas compressibility and its dissipative properties. The model was developed based on the formulation and solution of the nonlinear boundary value problem of mathematical physics. The equation of plate dynamics together with the equations of viscous gas dynamics for the case of barotropic compressible medium, as well as boundary conditions at the channel ends and gas contact surfaces with the channel walls, constitute this coupled problem of aeroelasticity. The gas dynamics was considered similarly to the hydrodynamic lubrication theory, but with retention of inertial terms. Using the perturbation method, the asymptotic analysis of the aeroelasticity problem is carried out, which made it possible to linearize the equations of dynamics for the thin layer of viscous gas and solve them by the iteration method. As a result, the law of gas pressure distribution along the plate was determined and the original coupled problem was reduced to the study of a nonlinear integro-differential equation describing the aeroelastic oscillations of the plate. The use of the Bubnov – Galerkin method to study the obtained equation led us to reduce the original problem to the study of the generalized Duffing equation. The application of the harmonic balance method allowed us to determine the primary aeroelastic and phase responses of the plate in the form of implicit functions. A numerical study of these responses was carried out to evaluate the influence of the plate’s nonlinear-elastic foundation, gas motion inertia and its compressibility.

This article provides a review of the literature devoted to methods of modeling contact heat transfer, as well as the author’s method of numerical modeling and its verification on experimental data.
Based on the analysis of the literature, we draw a conclusion about the determining effect of the roughness of the contacting surfaces on the heat exchange between them. We introduce the concept of a digital double of touching surfaces, that is, a model that takes into account their roughness and allows simulating joint mechanical and thermal interaction between contacting bodies.
We propose a method of numerical simulation of contact heat transfer, the calculation results of which are in good agreement with experimental data.

This paper considers the Lorentzian optimal control problem on two-dimensional de Sitter space. Normal and abnormal optimal trajectories are studied using the Pontryagin maximum principle. Attainable sets, spheres and distance in the Lorentzian metric are computed. Killing vector fields and isometries are described.

We consider a left-invariant sub-Riemannian problem on the Lie group of rotations of a threedimensional space. We find the cut locus numerically, in fact we construct the optimal synthesis numerically, i. e., the shortest arcs. The software package nutopy designed for the numerical solution of optimal control problems is used. With the help of this package we investigate sub-Riemannian geodesics, conjugate points, Maxwell points and diffeomorphic domains of the exponential map. We describe some operating features of this software package.

We are concerned with the existence of heteroclinic orbits for singular Hamiltonian systems of second order $\ddot{q}(t) + \nabla V(t, \,q)=0 $ where $V(t,\,q)$ is periodic in $t$ and has a singularity at a point ${q=e}$. Suppose $V$ possesses a global maximum $\overline V$ on $\mathbb R \times \mathbb R ^N\setminus\{e\}$ and $V(t,\,x)= \overline{V}$ if and only if $x\in \mathcal{M}$ where $\mathcal{M}$ contains at least two points and consists only of isolated points. Under these and suitable conditions on $V$ near $q=e$ and at infinity, we show for each $a_0^{}\in \mathcal M$, the existence of at least one heteroclinic orbit joining $a_0^{}$ to $\mathcal M \setminus\{a_0^{}\}$. Two different settings are studied. For the first, the usual strong force condition of Gordon near the singularity is assumed. For the second, the potential $V$ behaves near $q=e$ like $-\frac1{|q-e|^\alpha}$ with $0<\alpha<2$ (the weak force case). In both cases the existence of heteroclinic orbits $q\colon\mathbb R \to\mathbb R^N\setminus\{e\}$ is obtained via a minimization of the corresponding action functional.

We investigate discrete-time dynamical systems generated by an infinite-dimensional nonlinear operator that maps the Banach space $l_1^{}$ to itself. It is demonstrated that this operator possesses up to seven fixed points. By leveraging the specific form of our operator, we illustrate that analyzing the operator can be simplified to a two-dimensional approach. Subsequently, we provide a detailed description of all fixed points, invariant sets for the two-dimensional operator and determine the set of limit points for its trajectories. These results are then applied to find the set of limit points for trajectories generated by the infinite-dimensional operator.