Том 214, № 6 (2016)

Some possibilities of the author’s approach to the problem of nonintegrability of multidimensional systems related to the splitting of multidimensional separatrices and branching of solutions in the complex domain are discussed, using the example of the problem of the motion of a spherical pendulum with a suspension point performing small spatial periodic oscillations. Previous results are briefly reproduced and their generalizations are discussed, which are based on the calculation of a perturbation of the linear part of the Poincaré map at a hyperbolic point. We have succeeded in obtaining weaker conditions of nonintegrability, since this perturbation, generally speaking, violates the scalar nature of the restrictions of the linear part of the map to its two-dimensional expanding and contracting invariant subspaces. However, these conditions are expressed in terms of some repeated integrals because one must work in the second order of the perturbation theory. It is shown that in the case where the acceleration of the suspension point is represented by a function of complex time uni-valued over punctured vicinities of some isolated singularities, the nonintegrability conditions can be reduced to very simple ones in terms of certain local quantities associated with these singularities. The approach developed can be useful in problems where an unperturbed system possesses a symmetry leading to a degeneration, like the scalar nature of the restrictions of the linear part of the Poincaré map to its invariant subspaces.

Category theory is applied to the problem of representing heterogeneous software engineering technologies in a unified form suitable for their integration and coordination within the common software systems engineering cycle. Special attention is paid to modern technologies such as model-driven engineering and aspect-oriented programming. Universal category-theoretic semantic models of these technologies are constructed. A novel method of separation of concerns by explicating the aspectual structure of formal models of programs is proposed. We construct and analyze formal technologies (architecture schools) for designing technologies that comprise a mathematical basis for model-driven engineering.

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.