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This paper is devoted to the study of the motion of a platform maintained on an aircraft in a predetermined position by a system of gyroscopes, which does follows oscillations of the aircraft. Such systems are used for determining the angular position of the aircraft. We briefly review key results on this issue without detailed proofs.

We establish the integrability of homogeneous (in some variables) dynamical systems with dissipation in the case of an arbitrary odd order and thereby generalize the results earlier obtained by the author in particular cases of such systems.

The integrability of certain classes of homogeneous dynamical systems on the tangent bundles of three-dimensional manifolds is shown. The force fields involved in the systems lead to dissipation of variable sign and generalize previously considered fields.

The so-called pendulum-like systems arise in dynamics of a rigid body in a non-conservative field, in the theory of oscillations, and in theoretical physics. In this article, the methods of analysis are described which allow us to generalize the previous results. Herewith, we deal with some qualitative questions of the theory of ordinary differential equations whose solution facilitates studying some dynamical systems. In result of investigating more general classes of systems, we show that these general systems possess the already known family of nonequivalent phase portraits. We also deal with the aspect of integrability.

In this paper, we examine the problem on the spatial free deceleration of a rigid body in a resistive medium under the assumption that the interaction of the homogeneous axisymmetric body with the medium is concentrated on the frontal part of the surface, which has the shape of a flat circular disk. In earlier works of the author, under the simplest assumptions on interaction forces, the impossibility of oscillations with bounded amplitude was proved. Note that an exact analytic description of forces and moments of the body-medium interaction is unknown, so we use the method of “embedding” of the problem into a wider class of problems; this allows one to obtain a sufficiently complete qualitative description of the motion of the body. For dynamical systems considered, we obtain particular solutions and families of phase portraits of quasi-velocities in the three-dimensional space that consist of countable sets of nonequivalent portraits with different nonlinear qualitative properties.

In this paper, we prove the integrability of certain classes of dynamical systems on the tangent bundles of four-dimensional manifolds (systems with four degrees of freedom). The force field considered possessed so-called variable dissipation; they are generalizations of fields studied earlier. This paper continues earlier works of the author devoted to systems on the tangent bundles of two- and three-dimensional manifolds.

We consider the problem of differential diagnostics of the functional status of control objects that have a module structure and a finite set of possible malfunctions and reduce this problem to two separate problems that can be solved separately: the inspection problem (ascertainment of malfunction criteria in the system) and the diagnostic problem (search of malfunctions). We prove the feasibility of solution of the problem of differential diagnostics in the cases of precise and inaccurate trajectory measurements at the level of mathematical models.

The integrability of certain classes of homogeneous dynamical systems on the tangent bundles of four-dimensional manifolds is shown. The force fields involved in the systems lead to dissipation of variable sign and generalize previously considered fields.

This paper shows the integrability of certain classes of odd-order dynamical systems that are homogeneous with respect to some of the variables and in which a system on the tangent bundle of smooth manifolds is distinguished. In this case, the force fields have dissipation of different signs and generalize previously considered cases.

In this paper, we prove the explicit integrability of certain classes of dynamical systems on the tangent bundles of 2- and 3-dimensional spheres in the case where the forces are fields with so-called variable dissipation.

В работе получены комбинаторные теоремы, относящиеся к теории коллективного выбора. Эти теоремы описывают достаточно общие условия, при которых задача о сохранении произвольным правилом агрегирования $f$ множества предпочтений $\mathcal{D}$ и задача о совместимости множества предпочтений $\mathcal{D}$ с парой $(f,\mathcal{C})$ могут быть сведены к аналогичным задачам для двух конкретных правил агрегирования: правила большинства maj и правила «считалочки» cog. Результаты получены в рамках клонового подхода к теории коллективного выбора, который был предложен С. Шелахом и разработан авторами.

We establish the integrability for a class of homogeneous (with respect to a part of variables) dynamic systems of odd (seventh and ninth) order, admitting separation of a system on the tangent bundle of smooth manifolds. Bibliography: 13 titles.

We consider consecutive aggregation procedures for individual preferences 𝔠 ∈ ℭr (A) on a set of alternatives A, |A| ≥ 3: on each step, the participants are subject to intermediate collective decisions on some subsets B of the set A and transform their a priori preferences according to an adaptation function 𝒜. The sequence of intermediate decisions is determined by a lot J, i.e., an increasing (with respect to inclusion) sequence of subsets B of the set of alternatives. An explicit classification is given for the clones of local aggregation functions, each clone consisting of all aggregation functions that dynamically preserve a symmetric set 𝔇 ⊆ ℭr (A) with respect to a symmetric set of lots 𝒥. On the basis of this classification, it is shown that a clone ℱ of local aggregation functions that preserves the set ℜ2 (A) of rational preferences with respect to a symmetric set 𝒥 contains nondictatorial aggregation functions if and only if 𝒥 is a set of maximal lots, in which case the clone ℱ is generated by the majority function. On the basis of each local aggregation function f, lot J, and an adaptation function 𝒜, one constructs a nonlocal (in general) aggregation function fJ,A that imitates a consecutive aggregation procesure. If f dynamically preserves a set 𝔇 ⊆ ℭr (A) with respect to a set of lots 𝒥, then the aggregation function fJ,A preserves the set 𝔇 for each lot J ∈ 𝒥. If 𝔇 = ℜ2(A), then the adaptation function can be chosen in such a way that in any profile c ∈ (ℜ2(A))n, the Condorcet winner (if it exists) would coincide with the maximal element with respect to the preferences fJ,A (c) for each maximal lot J and f that dynamically preserves the set of rational preferences with respect to the set of maximal lots.

In many problems of dynamics, one has to deal with mechanical systems whose configurational spaces are two- or three-dimensional manifolds. For such a system, the phase space naturally coincides with the tangent bundle of the corresponding manifold. Thus, the problem of a flow past a (four-dimensional) pendulum on a (generalized) spherical hinge leads to a system on the tangent bundle of a two- or threedimensional sphere whose metric has a particular structure induced by an additional symmetry group. In such cases, dynamical systems have variable dissipation, and their complete list of first integrals consists of transcendental functions in the form of finite combinations of elementary functions. Another class of problems pertains to a point moving on a two- or three-dimensional surface with the metric induced by the encompassing Euclidean space. In this paper, we establish the integrability of some classes of dynamical systems on tangent bundles of two- and three-dimensional manifolds, in particular, systems involving fields of forces with variable dissipation and of a more general type than those considered previously.

In the proposed cycle of work, we begin the study of the motion of an aircraft which is described bynonlinear ordinary differential equations. Based on these equations, the probable malfunctions in the motioncontrol system are classified, the concepts of reference malfunctions and their neighborhoods are introduced,the mathematical modeling of these malfunctions and their neighborhoods is carried out, the concept ofdiagnostic space is introduced, and the mathematical structure of this space is defined. Proposed work isthe first in the cycle, therefore, the classification of malfunctions is given. This activity is also a preparatorypart of the diagnostic problem, which can be represented in the form of two successively solved problems,i.e., control problem, that is the problem of determining the presence of a malfunction in the system, anddiagnostic problem, that is the recognition problem of malfunction specification. This activity is just anillustration of the proposed approach.

We establish the integrability of some homogeneous (with respect to a part of variables) dynamical systems of odd (third and fifth) order. In the class of such systems, we extract a system on the tangent bundle of smooth manifolds. The force field is separated into the internal (conservative) force field and the external force field with alternating dissipation. The external force field is introduced by using a certain unimodular transformation and generalizes the cases studied by the author earlier.

We study the relative structural stability (relative roughness) of dynamical systems in some subspace of the space of all dynamical systems; moreover, the space of deformations of (dynamical) systems does not coincide with the space of all admissible deformations. We give some examples of relatively rough systems and relatively nonrough systems of different degree of nonroughness arising in rigid body dynamics and oscillation theory.

Under study is the problem of spatial free deceleration of a rigid body in a resisting medium. It is assumed that a axisymmetric homogeneous body interacts with the medium only through the front part of its outer surface that has the shape of flat circular disk. Under the simplest assumptions about the impact forces from the medium, it is demonstrated that any oscillations of bounded amplitude are impossible. In this case, any precise analytical description of the force-momentum characteristics of the medium impact on the disk is missing. For this reason, we use the method of “embedding” the problem into a wider class of problems. This allows us to obtain some relatively complete qualitative description of the body motion. For the dynamical systems under consideration, it is possible to obtain particular solutions as well as the families of phase portraits in the three-dimensional space of quasi-velocities which consist of a countable set of the phase portraits that are trajectory-nonequivalent and have different nonlinear qualitative properties.

In many problems of multidimensional dynamics, systems appear whose state spaces are spheres of finite dimension. Clearly, phase spaces of such systems are tangent bundles of these spheres. In this paper, we examine nonconservative force fields in the dynamics of a multidimensional rigid body in which the system possesses a complete set of first integrals that can be expressed as finite combinations of elementary transcendental functions. We consider the case where the moment of nonconservative forces depends on the tensor of angular velocity.

We establish the integrability for some classes of dynamic systems on the tangent bundles of multidimensional manifolds. We consider the case where the force fields possess variable dissipation. An example of a four-dimensional manifold is discussed in detail.

We establish the integrability for some classes of dynamic systems on the tangent bundles of two– and three-dimensional manifolds (systems with two and three degrees of freedom). The force fields possess the so-called variable dissipation and generalize those considered in the previous publications of the author.

This paper contains a systematic exposition of some results on the equations of motion of a dynamically symmetric n-dimensional rigid body in a nonconservative field of forces. Similar bodies are considered in the dynamics of actual rigid bodies interacting with a resisting medium under the conditions of jet flow past the body with a nonconservative following force acting on the body in such a way that its characteristic point has a constant velocity, which means that the system has a nonintegrable servo-constraint.

In this review, we discuss new cases of integrable systems on the tangent bundles of finite-dimensional spheres. Such systems appear in the dynamics of multidimensional rigid bodies in non-conservative fields. These problems are described by systems with variable dissipation with zero mean. We found several new cases of integrability of equations of motion in terms of transcendental functions (in the sense of the classification of singularities) that can be expressed as finite combinations of elementary functions.

We consider a mathematical model of the influence of a medium on a rigid body with a specific shape of its surface. In this model, we take into account the additional dependence of the moment of the interaction force on the angular velocity of the body. We present a complete system of equations of motion under the quasi-stationarity conditions. The dynamical part of equations of motion forms an independent third-order system, which contains, in its turn, an independent secondorder subsystem. We ovtain a new family of phase portraits on the phase cylinder of quasi-velocities, which differs from families obtained earlier.

In this review, we discuss new cases of integrable systems on the tangent bundles of finitedimensional spheres. Such systems appear in the dynamics of multidimensional rigid bodies in nonconservative fields. These problems are described by systems with variable dissipation with zero mean. We found several new cases of integrability of equations of motion in terms of transcendental functions (in the sense of the classification of singularities) that can be expressed as finite combinations of elementary functions.

We establish the integrability of dynamical systems of some classes arising in multidimensional dynamics. The force fields under consideration possess the so-called variable dissipation with zero mean and generalize the studied earlier ones.

In this paper, we examine new cases of integrability of dynamical systems on the tangent bundle to a low-dimensional sphere, including flat dynamical systems that describe a rigid body in a nonconservative force field. The problems studied are described by dynamical systems with variable dissipation with zero mean. We detect cases of integrability of equations of motion in transcendental functions (in terms of classification of singularity) that are expressed through finite combinations of elementary functions.

In this paper, we consider a class of oscillatory mechanical systems described by nonlinear second-order differential equations that contain parameters with variable dissipation and possess the property of dynamic symmetry. We study properties of mathematical models of such systems depending on parameters. Special attention is paid to bifurcations and reconstructions of phase portraits and the appearance and disappearance of cycles that envelope the phase cylinder or lie entirely on its covering. We present a complete parametric analysis of system with position-viscous friction and four parameters, where the force action linearly depends on the speed. The space of parameters is split into domains in which the topological behavior of the system is preserved. For some classes of pendulum systems whose right-hand sides depend on smooth functions, a qualitative analysis is performed, i.e., the space of systems considered is split into domains of different behavior of trajectories on the phase cylinder of quasi-velocities. We also perform a systematic analysis of the problem on an aerodynamic pendulum and detect periodic modes unknown earlier; these modes correspond to cycles in the mathematical model.

In this paper, we examine the existence of transcendental first integrals for some classes of systems with symmetries. We obtain sufficient conditions of existence of first integrals of second-order nonautonomous homogeneous systems that are transcendental functions (in the sense of the theory of elementary functions and in the sense of complex analysis) expressed as finite combinations of elementary functions.

Some qualitative analysis is carried out of the rectilinear and spatial problems concerning the motion of a rigid body in a resisting medium.Anonlinearmodel is constructed of impact of the mediumon the rigid body, which takes into account the dependence of the arm of force on the reduced angular velocity of the body. Moreover, the moment of this force itself is also a function of the angle of attack. As was shown by the processing the experimental data on the motion of homogeneous circular cylinders in water, these circumstances should be taken into account in the simulation. The analysis of the plane and spatial models of the interaction of a rigid body with a medium reveal the sufficient conditions of stability of the key regime of motion, i.e., the translational rectilinear deceleration. It is also shown that, under certain conditions, both stable or unstable auto-oscillating regimes can be presented in the system.

The integrability of certain classes of dynamic systems arising in the dynamics of a multidimensional solid and in the dynamics of a point on a tangent bundle of a multidimensional sphere is shown. In this case, the force fields under consideration have the so-called variable dissipation with zero average and generalize previously considered fields.

The author constructs a nonlinear mathematical model of a plane-parallel impact of the medium on a rigid body with a front part of the outer surface shaped as a circular cone. Multivariable analysis of the dynamic equations of motion was performed. A new family of phase patterns on the phase cylinder of quasi-velocities has been obtained. This family consists of infinitely numerous topologically inequivalent phase patterns. The sufficient conditions for the stability of an important mode of motion, i.e., rectilinear translational drag, have been obtained, as well as conditions for the presence of the self-oscillatory modes in the system.

We consider a mathematical model of a plane-parallel action of a medium on a rigid body whose surface has a part which is a circular cone. We present a complete system of equations of motion under the quasi-stationarity conditions. The dynamical part of equations of motion form an independent system that possesses an independent second-order subsystem on a two-dimensional cylinder. We obtain an infinite family of phase portraits on the phase cylinder of quasi-velocities corresponding to the presence in the system of only a nonconservative pair of forces.

Many problems of multidimensional dynamics involve systems for which the spaces of states are spheres of finite dimension and the spaces of phases are the tangent bundles of such spheres. We study conservative systems and present nonconservative force fields such that the systems involving such forces possess a complete collection of first integrals that are expressed through a finite combination of elementary functions and, in general, are transcendental functions of their variables. Bibliography: 32 titles.

We construct a class of nonconservative systems of differential equations on the tangent bundle of the sphere of any finite dimension. This class has a complete set of first integrals, which can be expressed as finite combinations of elementary functions. Most of these first integrals consist of transcendental functions of their phase variables. Here the property of being transcendental is understood in the sense of the theory of functions of the complex variable in which transcendental functions are functions with essentially singular points.

This paper is a survey of integrable cases in the dynamics of a five-dimensional rigid body under the action of a nonconservative force field. We review both new results and results obtained earlier. Problems examined are described by dynamical systems with so-called variable dissipation with zero mean. The problem of the search for complete sets of transcendental first integrals of systems with dissipation is quite topical; a large number of works are devoted to it. We introduce a new class of dynamical systems that have a periodic coordinate. Due to the existence of nontrivial symmetry groups of such systems, we can prove that these systems possess variable dissipation with zero mean, which means that on the average for a period with respect to the periodic coordinate, the dissipation in the system is equal to zero, although in various domains of the phase space, either energy pumping or dissipation can occur. Based on the facts obtained, we analyze dynamical systems that appear in the dynamics of a five-dimensional rigid body and obtain a series of new cases of complete integrability of the equations of motion in transcendental functions that can be expressed through a finite combination of elementary functions.

A mechanical system whose phase space is the tangent bundle of a two-dimensional sphere is studied. The potential nonconservative systems describing a geodesic flow are classified. A multiparameter family of systems possessing a complete set of transcendental first integrals expressed in terms of finite combinations of elementary functions is found. Some examples illustrating the spatial dynamics of a rigid body interacting with a medium are discussed.

In this work, the integrability of some classes of dy-namic systems on tangent bundles of three-dimensional manifolds is demonstrated. The corresponding force fields possess the so-called variable dissipation and generalize those considered earlier.

The dynamic part of equations of motion is investigated for a dynamically symmetric multidimensional solid in a nonconservative force field in the presence of the following force in the case when the solid is under the action of a pair of forces. In this case, there is additional linear damping in the system. The new case of complete integrability is found in transcendental (in the sense of complex analysis) functions, which are expressed through a finite combination of elementary functions.

This paper is a review of some previous and new results on integrable cases in the dynamics of a three-dimensional rigid body in a nonconservative field of forces. These problems are stated in terms of dynamical systems with the so-called zero-mean variable dissipation. Finding a complete set of transcendental first integrals for systems with dissipation is a very interesting problem that has been studied in many publications. We introduce a new class of dynamical systems with a periodic coordinate. Since such systems possess some nontrivial groups of symmetries, it can be shown that they have variable dissipation whose mean value over the period of the periodic coordinate vanishes, although in various regions of the phase space there may be energy supply or scattering. The results obtained allow us to examine some dynamical systems associated with the motion of rigid bodies and find some cases in which the equations of motion can be integrated in terms of transcendental functions that can be expressed as finite combinations of elementary functions.

The paper presents the results of study of the motion equations for a dynamically symmetric 4D-rigid body placed in a certain non-conservative field of forces. The form of the field is taken from the dynamics of actual 2D- and 3D-rigid bodies interacting with the medium in the case when the system contains a non-conservative pair of forces forcing the center of mass of a body to move rectilinearly and uniformly. A new case of integrability is obtained for dynamic equations of body motion in a resisting medium filling a four-dimensional space under presence of a tracking force.

In this work, we consider some problems of the qualitative theory of ordinary differential equations; the study of dissipative systems, as well as variable dissipation system considered below, which, in particular, arise in the dynamics of a rigid body interacting with a medium and in the oscillation theory, depends on solutions of these problems. We consider such problems as existence and uniqueness problems for trajectories having infinitely remote points as limit sets for systems on the plane, elements of qualitative theory of monotone vector fields, and also existence problems for families of long-period and Poisson stable trajectories. In conclusion, we study the possibility of extending the Poincar´e two-dimensional topographical system and the comparison system to the many-dimensional case.

This paper studies conditions for local (in time) solvability of a qualitatively new singularllimit problem, the free (unknown) boundary problem appearing recently. In fact, there are not so many different free boundary problems, which corresponds to not so large a variety of principally different phase transitions of the first and second kinds. Therefore, the appearance of principally new problems elicits interest. This paper studies structural features of a certain problem on the basis of a certain method developed previously, precisely, the localization method [1, 3, 9].

An important role in studying the classical Cahn–Hilliard problem [5] is played by its singular-limit problem, the so-called Melin–Sikerk free boundary problem, which, at present allows one to only numerically describe the instability of the crystallization process. The purpose of this work is to prepare the material for deducing the singular-limit problem for the essentially asymmetric model [8, 21].

The problem presented below is a singular-limit problem of the extension of the Cahn-Hilliard model obtained via introducing the asymmetry of the surface tension tensor under one of the truncations (approximations) of the inner energy [2, 5, 6, 7, 8, 10, 12, 13].