Vol 52, No 4 (2019)

This study presents the definition of an abstract homogeneous dimensional space with a finite compactness index, the definition of the Hausdorff–Besicovitch dimension spectrum of such a space, a theorem on the Hausdorff–Besicovitch spectrum values for its subspaces, and a number of results related to these concepts. Estimates are given for the dimension of sets that allow a mapping of a contracting type onto themselves. These estimates are an abstract version of the results close to the Douady–Oesterle theorem on the dimension of attractors of smooth dynamical systems in Euclidean spaces.

Chaotic signals and systems are widely used in image encryption, secure communications, weak signal detection, and radar systems. Many researchers have focused in recent years on the development of systems with an infinite number of coexisting chaotic attractors. We propose some approaches in this work to the generation of self-reproducing systems with an infinite number of coexisting self-excited or hidden chaotic attractors with the same Lyapunov exponents based on mathematical models of Lurie systems. The proposed approach makes it possible to generate extremely multistable systems using numerous known examples of the existence of chaotic attractors in Lurie systems without resorting to an exhaustive computer search. We illustrate the proposed methods by constructing extremely multistable systems with 1-D and 2-D grids of hidden chaotic attractors using the generalized Chua system, in which hidden attractors were first discovered by G.A. Leonov and N.V. Kuznetsov.

We continue in this paper to study one of the models of a discrete analog of the Renyi problem, also known as the parking problem. Suppose that n and i are integers satisfying n ≥ 0 and 0 ≤ i ≤ n – 1. We place an open interval (i, i + 1) in the segment [0, n] with i being a random variable taking values 0, 1, 2, …, n – 1 with equal probability for all n ≥ 2. If n < 2, then we say that the interval does not fit. After placing the first interval, two free segments [0, i] and [i + 1, n] are formed and independently filled with intervals of unit length according to the same rule, and so on. At the end of the process of filling the segment [0, n] with intervals of unit length, the distance between any two adjacent unit intervals does not exceed one. Suppose now that X_n is the number of unit intervals placed. In our earlier work published in 2018, we studied the asymptotic behavior of the first moments of random variable X_n. In contrast to the classical case, the exact expressions for the expectation, variance, and third central moment were obtained. The asymptotic behavior of all central moments of random variable X_n is investigated in this paper and the asymptotic normality for X_n is proved.

Systems of linear algebraic equations (SLAEs) are considered in this work. If the matrix of a system is nonsingular, a unique solution of the system exists. In the singular case, the system can have no solution or infinitely many solutions. In this case, the notion of a normal solution is introduced. The case of a nonsingular square matrix can be theoretically regarded as good in the sense of solution existence and uniqueness. However, in the theory of computational methods, nonsingular matrices are divided into two categories: ill-conditioned and well-conditioned matrices. A matrix is ill-conditioned if the solution of the system of equations is practically unstable. An important characteristic of the practical solution stability for a system of linear equations is the condition number. Regularization methods are usually applied to obtain a reliable solution. A common strategy is to use Tikhonov’s stabilizer or its modifications or to represent the required solution as the orthogonal sum of two vectors of which one vector is determined in a stable fashion, while seeking the second one requires a stabilization procedure. Methods for numerically solving SLAEs with positive definite symmetric matrices or oscillation-type matrices using regularization are considered in this work, which lead to SLAEs with reduced condition numbers.

The linear generalized Kalman–Bucy filter problem has been studied in this work. A sum of a useful signal and a noise is an observed process. A signal and a noise are independent stationary auto-regressive processes with orders exceeding one. The filter estimates a signal, using an observed process. Two algorithms of filter are considered: a recurrent one and a direct one. Within the recurrent one, to find the next estimate of a signal, we use the current observation and several previous filter estimates. The direct algorithm uses all previous observations directly. The errors of estimates are found for both algorithms. The advantages and disadvantages of both algorithms are discussed in this paper. Calculations at the recurrent algorithm depend on the observation time. The direct algorithm is reduced to a linear algebraic system such that its order increases in time. On the other hand, the direct algorithm always converges in time, while this is not guaranteed for the recurrent algorithm. Numerical examples are given here.

The problem of plane elasticity on the stress state of strip S of constant width 2c whose opposite borders are loaded by concentrated forces is analyzed. An analytical solution to this problem is found via methods in the theory of functions of a complex variable. The stress components at an arbitrary point of the strip are defined in terms of two regular functions, Φ(z) and Ψ₁(z). To find these functions, the conformal mapping of domain S onto lower half-plane ζ is used. The half-plane problem is solved via classical technique based on Cauchy-type integrals. Exact analytical expressions for functions Φ(ζ) and Ψ₁(ζ) are obtained, which are then converted by the inverse conformal transformation into the required formulas for Φ(z) and Ψ₁(z). Since functions Ψ₁(z) and Φ'(z) were found to be related, the stresses in strip S were specified by function Φ(z) and its derivative Φ'(z). Graphs of normal and shear stresses on the lines parallel to the borders of the strip are presented. The stresses along the axis of the strip are compared to Filon’s data. The solution satisfies the differential equilibrium equations, boundary conditions, and the continuity equation.

This paper is devoted to studying the motion of non-holonomic systems with higher-order constraints. The problem of the motion of such systems is formulated as the generalized Chebyshev problem. This refers to the problem in which the solution to a system of equations of motion should simultaneously satisfy an auxiliary system of higher-order (n\( \geqslant \) 3) differential equations. Two theories are constructed to study the motion of these systems. In the first, a joint system of differential equations for the unknown generalized coordinates and Lagrange multipliers is constructed. In the second theory, the equations of motion are derived by applying the generalized Gauss principle. The higher-order constraints are considered the program constraints in this investigation. Thus, the problem of finding the control satisfying the program given in the form of auxiliary system of differential equations linear in the (n\( \geqslant \) 3)-order derivatives of the sought generalized coordinates is formulated. A novel class of control problems is therefore introduced into consideration. Several examples are provided of solving the real mechanical problems formulated as the generalized Chebyshev problems. The paper is a review of the research performed for many years at the Department of Theoretical and Applied Mechanics of St. Petersburg University.