In this paper we study the asymptotic integration problem in the neighborhood of infinity for a certain class of linear functional differential systems. We construct the asymptotics for the solutions of the considered systems in a critical case. In the second part of the work we establish the existence of a critical manifold for the considered class of systems and study its main properties. We also investigate the asymptotic integration problem for a reduced system. We illustrate the proposed method with an example of constructing the asymptotics for the solutions of a certain scalar delay differential equation.

We consider a scalar nonlinear differential-difference equation with two delays, which models the behavior of a single neuron. Under some additional suppositions for this equation it is applied a well-known method of quasi-normal forms. Its essence lies in the formal normalization of the Poincare – Dulac, the production of a quasi-normal form and the subsequent application of the conformity theorems. In this case, the result of the application of quasi-normal forms is a countable system of differential-difference equations, which manages to turn into a boundary value problem of the Korteweg – de Vries equation. The investigation of this boundary value problem allows to make the conclusion about the behavior of the original equation. Namely, for a suitable choice of parameters in the framework of this equation it is implemented the buffer phenomenon consisting in the presence of the bifurcation mechanism for the birth of an arbitrarily large number of stable cycles.

The problem of constructing and classifying elliptic solutions of nonlinear differential equations is studied. An effective method enabling one to find an elliptic solution of an autonomous nonlinear ordinary differential equation is described. The method does not require integrating additional differential equations. Much attention is paid to the case of elliptic solutions with several poles inside a parallelogram of periods. With the help of the method we find elliptic solutions up to the fourth order inclusively of an ordinary differential equation with a number of physical applications. The method admits a natural generalization and can be used to find elliptic solutions satisfying systems of ordinary differential equations.

Dynamical properties of a logistic equation with delay and delay control are studied by asymptotic methods. It is shown that effective control of characteristics of relaxation cycle is possible. A new method for studying the dynamics in the case of suffitiently large delay control coeffitient is worked out. It is found that the original problem of the dynamics of equations with delays is reduced to the problem of non-local dynamics of special nonlinear boundary value problems of parabolic type.

An initial-boundary problem modelling the rotation of discrete-continuum mechanical system, which consists from a solid and the rigidly connected flexible rod. To solve the problem we determine a solution notion, prove its existence, uniqueness, and continuous dependence from start conditions and parameters of the boundary task. Are resolved tasks of solution rotation from the start phase state to the finish one at a specified time moment and with the controller function norm minimum in the L∞(0, T) space and time control problem with a limited norm of controller function in the specified space. Maximum principe was formulated, and an algorithm of optimal control modelling is proposed. The moments problem is used as an investigation method.

An analysis of numerical optimization methods for solving a problem of molecular docking has been performed. Some additional requirements for optimization methods according to GPU architecture features were specified. A promising method for implementation on GPU was selected. Its implementation was described and performance and accuracy tests were performed.

Process mining is a new direction in the field of modeling and analysis of processes, where the use of information from event logs describing the history of the system behavior plays an important role. Methods and approaches used in the process mining are often based on various heuristics, and experiments with large event logs are crucial for the study and comparison of the developed methods and algorithms. Such experiments are very time consuming, so automation of experiments is an important task in the field of process mining. This paper presents the language DPMine developed specifically to describe and carry out experiments on the discovery and analysis of process models. The basic concepts of the DPMine language as well as principles and mechanisms of its extension are described. Ways of integration of the DPMine language as dynamically loaded components into the VTMine modeling tool are considered. An illustrating example of an experiment for building a fuzzy model of the process discovered from the log data stored in a normalized database is given.

In the 1980s V.A. Bondarenko found that the clique number of the graph of a polytope in many cases corresponds to the actual complexity of the optimization problem on the vertices of the polytope. For an explanation of this phenomenon he proposed the theory of direct type algorithms. This theory asserts that the clique number of the graph of a polytope is the lower bound of the complexity of the corresponding problem in the so-called class of direct type algorithms. Moreover, it was argued that this class is wide enough and includes many classical combinatorial algorithms. In this paper we present a few examples, designed to identify the limits of applicability of this theory. In particular, we describe a modification of algorithms that is quite frequently used in practice. This modification takes the algorithms out of the specified class, while the complexity is not changed. Another, much closer to reality combinatorial characteristic of complexity is the rectangle covering number of the facet-vertex incidence matrix, introduced into consideration by M. Yannakakis in 1988. We give an example of a polytope with a polynomial (with respect to the dimension of the polytope) value of this characteristic, while the corresponding optimization problem is NP-hard.

The method of collocations and least residuals (CLR), which was proposed previously for the numerical solution of two-dimensional Navier–Stokes equations governing the stationary flows of a viscous incompressible fluid, is extended here for the three-dimensional case. The solution is sought in the implemented version of the method in the form of an expansion in the basis solenoidal functions. At all stages of the CLR method construction, a computer algebra system (CAS) is applied for the derivation and verification of the formulas of the method and for their translation into arithmetic operators of the Fortran language. For accelerating the convergence of iterations a sufficiently universal algorithm is proposed, which is simple in its implementation and is based on the use of the Krylov’s subspaces. The obtained computational formulas of the CLR method were verified on the exact analytic solution of a test problem. Comparisons with the published numerical results of solving the benchmark problem of the 3D driven cubic cavity flow show that the accuracy of the results obtained by the CLR method corresponds to the known high-accuracy solutions.

A perspective model of a neuron cell — the generalized neural element (GNE) is studied in this article. The model has an universal character. It combines properties of a neuron-oscillator and a neuron-detector. In this structure cyclic sequential pulse generation elements of the ring are studied. A nonlinear mapping for mismatches between pulses of neighboring elements is constructed. We prove the existence of a fixed point of this mapping (threshold value of mismatches) and its stability in a small neighborhood of the fixed point. In doing so the existence of a stable oscillatory mode of neural activity (attractor) of a certain type is proved. The parameters of the attractor (threshold values of mismatches) can be controlled in advance, due to the choice of synaptic weights of the links in the ring.

We consider a model of neuron complex formed by a chain of diffusion coupled oscillators. Every oscillator simulates a separate neuron and is given by a singularly perturbed nonlinear differential-difference equation with two delays. Oscillator singularity allows reduction to limit system without small parameters but with pulse external action. The statement on correspondence between the resulting system with pulse external action and the original oscillator chain gives a way to demonstrate that under consistent growth of the chain node number and decrease of diffusion coefficient we can obtain in this chain unlimited growth of its coexistent stable periodic orbits (buffer phenomenon). Numerical simulations give the actual dependence of the number of stable orbits on the diffusion parameter value.