This paper considers a nonlinear fourth-order ordinary differential equation. The study of this class of equations is conducted using an analytical approximation method based on dividing the solution domain into two parts: the region of analyticity and the vicinity of a movable singular point. This work focuses on investigating the equation in the region of analyticity and solving two problems. The first problem is a classical problem in the theory of differential equations: proving the theorem of existence and uniqueness of a solution in the region of analyticity. The structure of the solution in this region takes the form of a power series. To transition from formal series to series converging in a neighborhood of the initial conditions, a modification of the majorant method is used, which is applied in the Cauchy – Kovalevskaya theorem. This method allows determining the domain of validity of the theorem. Within this domain, error estimates for the analytical approximate solution are obtained, enabling the solution to be found with any predefined accuracy. When leaving the domain of the theorem’s validity, analytical continuation is required. To do this, it is necessary to solve the second task of the study: to study the effect of perturbation of the initial data on the structure of the analytical approximate solution.

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.

This paper is concerned with the motion of an elliptic foil in the field of a fixed point singularity. A complex potential of the fluid flow is constructed, and the forces and the torque which act on the foil from the fluid are obtained. It is shown that the equations of motion of the elliptic foil in the field of a fixed point vortex source can be represented as Lagrange – Euler equations. It is also shown that the system has an additional first integral due to the conservation of the angular momentum. An effective potential of the system under consideration is constructed. For the cases where the singularity is a vortex or a source, unstable relative equilibrium points corresponding to the circular motion of the foil around the singularity are found.

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 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.

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 develops the holomorphic regularization method of the Cauchy problem for a special type of Tikhonov system that arises in the mathematical modeling of wave solidstate gyroscope dynamics. The Tikhonov system is a system of differential equations a part of which is singularly perturbed. Unlike other asymptotic methods giving approximations in the form of asymptotically converging series, the holomorphic regularization method allows one to obtain solutions of nonlinear singularly perturbed problems in the form of series in powers of a small parameter converging in the usual sense. Also, as a result of applying the holomorphic regularization method, merged formulas for an approximate solution are deduced both in the boundary layer and outside it. These formulas allow a qualitative analysis of the approximate solution on the entire time interval including the boundary layer.
This paper consists of two sections. In Section 1, the holomorphic regularization method of the Cauchy problem for a special type of Tikhonov system is developed. The special type of Tikhonov system means the following: singularly perturbed equations are linear in the variables included in them with derivatives, the matrix of the singularly perturbed part of the system is diagonal, the remaining equations have separate linear and nonlinear parts. An algorithm for deriving an approximate solution to the Cauchy problem for the Tikhonov system of special type by using the holomorphic regularization method is presented. In Section 2, the mathematical model describing in interconnected form the mechanical oscillations of the gyroscope resonator and the electrical processes in the oscillation control circuit is considered. The algorithm for deriving an approximate solution proposed in Section 1 is used. Formulas for an approximate solution taking into account the structure of the Tikhonov system are deduced.

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 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.

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.

An analysis is made of the motion of a particle on a rotating disk equipped with rectilinear blades for different types of particle collisions with a blade. The dynamics of particle motion is considered using equations describing the dynamics of the relative motion of a particle and is studied for the cases of inelastic and elastic interaction with the blade when the coefficient of friction of the particle against the disk has a value that is the most probable from a practical point of view. The impact interaction of the particle with the blade is studied using the Routh hypothesis, which relates the normal and tangential components of the impact impulse by a dependence similar to the Amonton – Coulomb dependence between the normal and tangential components of the interaction force of two rough bodies. The dynamical coefficient of impact friction is taken to be equal to half the kinematic coefficient of friction of the particle against the disk. After an impact, depending on the values of the above-mentioned coefficients and the angle of inclination of the blade, the particle may or may not possess a velocity component tangential and (or) normal to the surface of the blade. In addition, depending on the overall dimensions of the disk and the inclination angle of the blade the moving particle can repeatedly enter into impact interaction with the blade. In this paper we consider all four patterns of motion of the particle after its first and subsequent impacts against the blade of the rotating disk, present graphs of time dependences of the projections of the velocity of the relative motion of the particle, and plot trajectories of the relative motion of the particle for two practically important cases of installation of a blade on the disk’s surface which ensure the initial collision angles equal to 0${}^\circ$ and 30${}^\circ$. The proposed model is also used to investigate the changes in the antitorque moment acting on the disk. For this purpose, in each of the cases we calculate the resulting moment of impact impulse and the impulse of friction torque throughout the motion of the particle on the disk’s surface. We present the results of numerical experiments and use them to give recommendations on the optimization of mechanical systems of this type.

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.

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 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.

Knots naturally appear in continuous dynamical systems as flow periodic trajectories. However, discrete dynamical systems are also closely connected with the theory of knots and links. For example, for Pixton diffeomorphisms, the equivalence class of the Hopf knot, which is the orbit space of the unstable saddle separatrix in the manifold $\mathbb{S}^2\times \mathbb{S}^1$, is a complete invariant of the topological conjugacy of the system. In this paper we distinguish a class of three-dimensional Morse – Smale diffeomorphisms for which the complete invariant of topological conjugacy is the equivalence class of a link in $\mathbb{S}^2\times \mathbb{S}^1$.
We prove that, if $M$ is a link complement in $\mathbb{S}^3$, or a handlebody $H_g^{}$ of genus $g\geqslant 0$, or a closed, connected, orientable 3-manifold, then the set of equivalence classes of tame links in $M$ is countable. As a corollary, we prove that there exists a countable number of equivalence classes of tame links in $\mathbb{S}^2\times \mathbb{S}^1$. It is proved that any essential link can be realized by a diffeomorphism of the class under consideration.

This paper presents a practical approach to fully automated kinematic calibration of an industrial manipulator. The approach is based on the principle of plane constraint. The electrical signal is used to fix the moment of contact between the conductive tool and the flat surface. The measurement data are manipulator configurations (joint angles) at the moment of contact. A modification of the algorithm to deal with the scaling problem is also proposed. This approach provides both high calibration accuracy and lower cost of the experimental setup compared to coordinate measuring machines (CMMs), laser trackers, and vision systems. The article examines the impact of various methods of kinematic parameterization of manipulators: the Denavit – Hartenberg agreement (DH), product of exponentials (POE), as well as the complete and parametrically continuous model (CPC) on the calibration accuracy. A comparison is made of the open-loop and the proposed closed-loop calibration methods on the Puma 560 model known in the literature. POE parameters were converted to DH and CPC to compare accuracy after calibration based on these parameterizations. The method of computing POE-CPC transformation as a solution to a certain optimization problem is proposed. The problem of identifying geometric parameters in the presence of restrictions is solved by gradient optimization methods. Experiments have been carried out on an ABB IRB 1600 industrial manipulator with an installed conductive probe and an ABB IRBP A-500 robotic positioner with a conductive metal flat surface. A technique for indirectly checking the accuracy of calibration of kinematic parameters is proposed based on a study of the accuracy of manipulation when using these parameters. A comparison is made of the manipulation accuracy when using four sets of parameters: nominal parameters obtained during factory calibration with the Leica AT901B laser tracker and two sets of parameters obtained by applying the proposed calibration method. The kinematic parameters obtained from the experiment determine more accurately the position of the manipulator TCP for part of the configuration working space, even for areas that were not used for calibration.

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.

This paper is concerned with a mechanical system consisting of a rigid body (outer body) placed on a horizontal rough plane and of an internal moving mass moving in a circle lying in a vertical plane, so that the radius vector of the point has a constant angular velocity. The interaction of the outer body and the horizontal plane is modeled by the Coulomb –Amonton law of dry friction with anisotropy (the friction coefficient depends on the direction of the body’s motion). The equation of the body’s motion is a differential equation with a discontinuous right-hand side. Based on the theory of A. F. Filippov, it is proved that, for this equation, the existence and right-hand uniqueness of the solution takes place, and that there exists a continuous dependence on initial conditions. Some general properties of the solutions are established and possible periodic regimes and their features are considered depending on the parameters of the problem. In particular, the existence of a periodic regime is proved in the case where the motion occurs without sticking of the outer body, and conditions for the existence of such a regime are shown. An analysis is made of the final dynamics of how the system reaches a periodic regime in the case where the outer body sticks twice within a period of revolution of the internal mass. This periodic regime exhibits sticking in the so-called upper and lower deceleration zones. The outer body comes twice to a stop and is at rest in these zones for some time and then continues its motion. This paper gives a complete description of the solution pattern for such motions. It is shown that, in the parameter space of the system where such a regime exists, the solution reaches this regime in finite time. All qualitatively different solutions in this case are described. In particular, special attention is devoted to the terminal motion, namely, to the solution of the system during the last period of motion of the internal mass before reaching the periodic regime. Existence regions of such solutions are found and the boundaries of the regions of initial conditions determining the qualitatively different dynamics of the system are established.

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.

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.

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.