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Proposed work is the seventh work of the cycle on differential and topological diagnostics. Diagnostics ofmalfunctions in the system of indirect control of an object whose motion is described by nonlinear ordinary differential equations of the third order (the problem of B.V. Bulgakov) is considered. In the diagnostic algorithm used, the scope of control is selected, when a certain constant value is assigned to each faulty system, and according to certain rules, the numbers obtained in the process of integrating equations and characterizing the functional state of the system are compared with it. Troubleshooting is also considered in one system of direct control of the movement of the aircraft, which can be described by nonlinear differential equations of the second order. At the same time, a certain diagnostic algorithm is built and used in accordance with the previously developed methodology.

Proposed work is the sixth work of the cycle on differential and topological diagnostics. It is shown that the diagnostics in the case of trajectorial measurements corrupted by noise, which is a stochastic process of the normal white noise type with zero mean value and bounded spectrum, can be performed by using the diagnostic algorithms obtained in [5], i.e., the results of this section remain valid even in this rather general case; moreover, the diagnostic functional, which was introduced in the theorem of [5] a priori, is now obtained a posteriori.

We systematize some results on the study of the equations of spatial motion of dynamically symmetric fixed rigid bodies–pendulums located in a nonconservative force fields. The form of these equations is taken from the dynamics of real fixed rigid bodies placed in a homogeneous flow of a medium. In parallel, we study the problem of a spatial motion of a free rigid body also located in a similar force fields. Herewith, this free rigid body is influenced by a nonconservative tracing force; under action of this force, either the magnitude of the velocity of some characteristic point of the body remains constant, which means that the system possesses a nonintegrable servo constraint, or the center of mass of the body moves rectilinearly and uniformly; this means that there exists a nonconservative couple of forces in the system

Proposed activity presents next stage of the study of the problem of the plane-parallel motion of a rigid body interacting with a resistant medium through the frontal plane part of its external surface. Under constructing of the force acting of medium, we use the information on the properties of medium streamline fl w around in quasistationarity conditions (for instance, on the homogeneous circular cylinder input into the water). The medium motion is not studied, and we consider such problem in which the characteristic time of the body motion with respect to its center of masses is comparable with the characteristic time of motion of the center of masses itself.

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 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 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.

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.

Работа посвящена исследованию вопросов относительной структурной устойчивости (относительной грубости) динамических систем, рассматриваемых не на всем пространстве динамических систем, а лишь на некотором его подпространстве. При этом пространство деформаций динамических систем также не совпадает со всем пространством допустимых деформаций. В частности, рассмотрены системы дифференциальных уравнений, возникающие в динамике твердого тела и теории колебаний, в которых присутствует диссипация того или иного знака. Показана их относительная грубость, а также, при некоторых условиях, относительная негрубость различных степеней. Обсуждаются вопросы интегрируемости систем в конечных комбинациях элементарных функций.

Рассмотрена математическая модель плоскопараллельного воздействия среды на твердое тело, частью участка внешней поверхности которого является круговой конус. Приведена полная система уравнений движения в условиях квазистационарности. Получено новое семейство фазовых портретов на фазовом цилиндре квазискоростей. Также рассмотрена математическая модель пространственного воздействия среды на осесимметричное твердое тело, частью участка внешней поверхности которого является круговой конус. Исследован вопрос устойчивости по части переменных ключевого режима - пространственного прямолинейного поступательного торможения тела. Для конкретного класса тел показано, при каких инерционно-массовых характеристиках ключевой режим является устойчивым. Изучены вопросы интегрируемости.

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.

The integrability of certain classes of dynamical systems on the tangent bundle of a multidimensional manifold is shown. In this case, the force fields with variable dissipation generalize those considered earlier.

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.