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 solutions of the considered systems in the critical case. Using the ideas of the center manifold method, we show the existence of the so-called critical manifold that is positively invariant for trajectories of the initial system. We establish that the asymptotics for solutions of the system on this manifold defines the asymptotics for all solutions of the initial system. In the first part of this work, we propose an algorithm for an approximate construction of the critical manifold. Moreover, we establish the unique solvability for auxiliary algebraic problems that occur within the algorithm implementation.

The problem of existense and stability of continuous wave (CW) solutions R exp(iΛt) of a Fourier Domain Mode Locking laser model is studied. This model consists of two differential equations with delay. The delay is sufficiently large. It is nessesary for the existense of CW solutions of this model that parameters determining the ”main part” of solution must lie on a certain curve (Γ(κ, g0)). Sufficient conditions of stability of CW solutions for all sufficiently large values of delay are found. The location of stability regions on Γ(κ, g0) is studied. In the case of zero linewidth enhancement factor α for all values of parameters of the linear attenuation factor per cavity round trip κ and the linear unsaturated gain parameter g0 the number of stability regions and their boundaries on Γ(κ, g0) are found analytically. The comparison of location of stability regions on Γ(κ, g0) in tha case of zero α and nonzero α is made.

In the paper, a neural network model based on three McCulloch–Pitts adder neurons is considered. Previously, the features of the model dynamics were numerically analyzed and the stability of various dynamic modes of the network with small changes of the current state was studied, including the detection of parameters specific for the chaotic system dynamics. In the paper, the stability problem of the neural network equilibrium states (steady operating mode) is considered for different values of feedback synaptic weight coefficients. An analytic proof of the zero equilibrium state instability for the corresponding dynamic system for all parameter values of the neural network is presented as the main result of the article.

In the article, it is established the asymptotic equivalence of dynamics of the neural networks consisting of the impulse neurons and the neural networks built of the neural cellular automata of different kinds (autogenerators and detectors). Such an equivalence takes place for appropriate parameters values and non-intersection of input impacts for each neuron. In the first chapter, we describe the model of impulse autogenerator neuron and two different models of impulse detector neuron. For these models, we prove statements about duration of the latent period for a neuron under impact. In the second chapter, we describe the model of a neural cellular automaton with autogenerator dynamics and a modification of this model with detector dynamics. For these models, we also prove statements about the latent period for a neural automaton under impact. In the third chapter, we prove some statements about asymptotic equivalence of the impulse neurons and neural cellular automata of different kinds.

In this article, we continue the study of the properties acquired by the differentiation operator Λ with spreading beyond the space W₁¹. The study is conducted by introducing the family of spaces Yp¹ , 0 < p < 1, having analogy with the family Wp¹, 1 ≤ p < ∞. Spaces Yp¹ are equiped with quasinorms constructed on quasinorms spaces Lp as the basis; Λ : Yp¹ → Lp. We have given a sufficient condition for a function, piecewise belonging to the space Yp¹ to be in this space (if f ∈ Yp¹ [x_i-1; x_i ], i ∈ N, 0 = x0 < x₁ < · · · < x_i < · · · < 1, then f ∈ Yp¹ [0; 1]). In other words, it is the sign when the equality: Λ(S fi) = S Λ(fi) is true. The bounded variation in the Jordan sense is closest to the sufficient condition among the classic characteristics of functions. As a corollary, it comes out that, if a function f piecewise belongs to the space of W₁¹ and has a bounded variation, f belongs to each space Yp¹ , 0 < p < 1.

The problem of minimizing the error of a cubature formula on the classes of functions given by modulus of continuity for cubature formulas with fixed nodes on the boundary of gird rectangular localization domain of nodes is considered. We give the exact solution of this problem on the wide classes of functions of two variables. It was previously shown by N.P. Korneychuk that if the boundary nodes of a rectangular lattice Q_k,i = { x_k-1 ≤ x ≤ x_k , y_i-1 ≤ y ≤ y_i} are not included in the number of nodes cubature formula

Z Z (Q) f(x, y)dxdy = Xm k=1 Xn i=1 pkif(xk, yi) + Rmn(f), (1)

the formula of average rectangles is the best for classes of functions ^ω1,ω2 (Q), H^ω1_p1 (Q) and H^ω1_p2(Q) among all quadrature formulas of the form (1). It is proved that if into the number of nodes in the formula (1) all boundary nodes (such formulas are called Markov-type) are added, then for these classes of functions the best formula is trapezoids. The exact errors for all classes of functions are calculated.

In this article, we consider the approximate solution of an optimal control dot mobile problem for a system of nonlinear partial hyperbolic and ordinary differential equations with initial and boundary value conditions and a nonlinear optimality criterion. The use of the Fourier method of variables separation reduces the generalized solution of the initial-boundary value problem to the countable system of nonlinear integral equations (CSNIE). To ease the computational procedures, it is considered the corresponding shorter (truncated) system of nonlinear integral equations (SSNIE) instead of CSNIE. By the methods of successive approximations and integral inequalities, it is studied the one-value solvability of SSNIE for the fixed values of the control. It is estimated a permissible error with respect to the shorter generalized solution of the initial-boundary value problem. It is approximately calculated the nonlinear functional of quality under the known optimal operating influences.

We consider the local dynamic of the logistic equation with rapidly oscillating timeperiodic piecewise constant or piecewise linear coefficient of delay. It was shown that the averaged equation is a logistic equation with two delays in first case and logistic equation with distributed delay in second case. The criterion of equilibrium point stability was obtained in both cases. Dynamical properties of the original equation were considered in the critical case of equilibrium point of averaged equation stability problem. It was shown, that local dynamic in the critical case is defined by Lyapunov value whose sign depends on the parameters of the problem.