Middle Volga Mathematical Society journal

In this paper we consider $\Omega$-stable diffeomorphisms defined on smooth closed orientable manifolds of dimension $n\ge 3$, whose all nontrivial basic sets are either expanding attractors or contracting repellers of co-dimension 1. Due to the simple topological structure of the basins of such attractors and repellers, one can make a transition from a given dynamical system with nontrivial basic sets to a regular system which is a homeomorphism with a finite hyperbolic chain-recurrent set. It is well known that not every discrete dynamical systems has energy functions, i.e. a global Lyapunov function whose set of critical points coincides with the chain-recurrent set of the system. Counterexamples were found both among regular diffeomorphisms and among diffeomorphisms with chaotic dynamics. The main result of this paper is the proof of the fact that the topological energy functions for the original diffeomorphism and for its corresponding regular homeomorphism exist or do not exist simultaneously. Thus, numerous results obtained in the field of existence of energy functions for systems with regular dynamics, e.g., for Morse–Smale diffeomorphisms, may be applied to the study of the diffeomorphisms with expanding attractors and contracting repellers of co-dimension 1.

An analysis of existing approaches to the development of software solutions designed to solve optimal control problems is carried out, and a conclusion is drawn about the need to develop specialized numerical software systems. As a numerical method for solving optimal control problems, a parameterization method is proposed, which allows, on the basis of a unified approach, to solve optimal control problems with point or distributed delay and without delay as well. The method describes a scheme for representing a control action in the form of a generalized spline with moving nodes and subsequent reduction of the original optimal control problem with or without delay to a nonlinear programming problem with respect to the spline parameters and temporary nodes. For stated nonlinear programming problem, algorithms for calculating the first and second order derivatives of the objective function are presented. These algorithms make it possible to calculate derivatives based on solving Cauchy problems for direct and adjoint systems. This approach differs from the standard method of calculation based on difference approximation and can significantly reduce the overall amount of calculations. Based on the specifics of the parameterization method, a concept for developing a software package is proposed, and the main provisions of the development are derived. Thus, the software package offers independence in the implementation of methods for solving nonlinear programming problems and discrete schemes for solving Cauchy problems. It also offers a unified (independent of the type of optimal control problem) approach to control parameterization. The results of computational experiments carried out using the parameterization method are also presented. These results confirm the effectiveness of using a unified approach while solving of optimal control problems with point delay, distributed delay, and with no delay.

The paper considers a linear differential operator and several nonlinear differential and integro-differential operators that form the basis to the equations of vibration of a deformable plate. In the nonlinear operators, the nonlinearity of the bending moment and of the damping forces, as well as the longitudinal force arising from the elongation of the plate due to its deformation are taken into account. Basing on the proposed equations, mathematical models of the mechanical system consisting of a non-deformable pipeline connected at one end with a sensor designed to measure the pressure in the combustion chamber of an aircraft engine and at the other end with this chamber have been developed. The sensitive element of the sensor, which transmits the pressure information, is a deformable plate, whose edges are rigidly fixed. The models take into account the aerohydrodynamic effect of the working medium on this element and the temperature variation over time along the thickness of the element. Using the small parameter method, the first approximation for asymptotic equations is obtained that describes joint dynamics of the working medium in the pipeline and of the sensitive element. The study of the elastic element’s dynamics is based on the application of the Bubnov-Galerkin method and on the numerical experiments in Mathematica 12.0. A comparative analysis of solutions for linear and nonlinear models is performed. The influence of the above-mentioned nonlinearity types on the change in the value of the plate deflection is shown.

The authors simulate non-uniform stress distribution in multilayer building envelope. Its layers, made of different kinds of concrete, are tightly coupled with each other. Such constructions are of significant practical interest because their heat insulation properties may be much better than the corresponding properties of uniform slabs of the same thickness (though their toughness is still sufficient). The authors state the problem about the distribution of equivalent and maximal principal stresses in such a slab. In the framework of this problem loads and kinematic constraints imposed on the body of interest are typical for industrial and civil buildings. Numerical simulation is provided with the aid of ANSYS finite element software package. The paper discusses the reasons for non-physical behaviour demonstrated by the stress after mesh refinement and proposes the way to fix this problem. Then the authors use the approach described to find a critical load causing destruction of the slab.