Tag Archives: stability


UDC 517.927

O.R. Balaban

The paper considers evolutionary problems underlying the mathematical description of oscillatory and hydrodynamic processes in network-like objects (waveguides, hydraulic networks, etc.). The main attention is paid to the analysis of the properties of the elliptic operator (the one-dimensional Laplace operator) with distributed parameters on the network, establishing the spectral completeness of the system of eigenfunctions in the class of square-integrable functions. Conditions are obtained that guarantee Neumann stability (spectral stability) of difference schemes for evolutionary problems; a solution to the moment method control problem is presented. The methods for studying evolutionary problems are based on the properties of a positive definite elliptic operator: a system of eigenfunctions forms a basis in the space of functions summable with a square; series in the system of eigenfunctions admit a priori estimates of the solutions of the evolutionary problem; approximation of an elliptic operator reduces it to a finite-dimensional operator in a finite-dimensional space of grid functions with a natural Euclidean norm, which (a finite-dimensional operator) approximates the original with any predetermined accuracy in the sense of the norm of the space of functions summable squared. For evolutionary problems, an explicit first-order approximation scheme on the graph grid (parabolic system) and an explicit second-order approximation scheme (hyperbolic system) are used. The oscillatory properties of the obtained operators are established, similar to the classical oscillatory properties. For difference schemes of parabolic and hyperbolic systems of equations, conditions are obtained that guarantee countable spectral stability (stability in the sense of Neumann) and, therefore, the possibility of obtaining analogues of A.F. Filippova on the convergence of difference schemes in terms of approximation steps of a graph grid. To illustrate the applicability of the approach used, the control problem is considered – the translation of evolutionary systems of parabolic and hyperbolic types from given initial to given final states; conditions are obtained that guarantee the controllability of the systems under study.

Keywords: laplace operator on a graph, evolution problems, approximation, difference schemes, stability, convergence, method of moments.

Full text:


UDC 531.391

O.V. Lyubimtsev, O.L. Lyubimtseva

The problems of the dynamics and stability of vibro-impact systems today constitute an independent section of the applied theory of oscillations. The interest in these problems is primarily due to the wide use in practice of machines and technologies that use systematic shock interactions as the basis of work processes. Vibrating hammers, vibro-impact tools, shock absorbers, disc brakes, machines for vibro-impact testing, devices for vibrotransport of piece and bulk cargo, vibroseparation, volumetric vibro-processing – this is not a complete list, which gives an idea of the diversity of technological uses of vibro-impact systems and range of issues requiring the application of the theory of these systems. Vibro-impact systems, as compared with conventional oscillatory systems, have additional parameters that characterize for one-dimensional systems, the gaps in shock pairs and the coefficients of restoring the speed upon impact. Previously, one of the authors found conditions for the existence and stability of periodic motions of a body moving horizontally using a belt mechanism due to the force of dry friction located inside the container, which performs straight-line harmonic oscillations. This model and its particular cases reflect the dynamics of both systems with shock interactions and systems with friction. We also note that some non-autonomous systems with one degree of freedom are inherent in some properties of multidimensional systems. In this paper, we study the evolution of periodic motions with impacts depending on one of the parameters (the other parameters are assumed to be fixed) and a general analysis of the period doubling bifurcation for periodic motions with two impacts is carried out.

Keywords: : dynamic system, point mapping, periodic motion, stability.

Full text:


UDC 519. 72

V.I. Sumin, O.V. Isaev, M.V. Skulkov

With the purpose of ensuring highly reliable information processing in practice at assessment of stability of functioning of the security systems considered from positions of information structures in the conditions of negative impacts, it is necessary not only to develop new models and algorithms of steady interaction of elements of the specified information sets, but also to consider requirements to efficiency of the information processes proceeding at the same time. Increase in amount of negative impacts on elements of security complexes demands modernization of systems of parrying of negative impacts and also carrying out the analysis of stability of functioning of this sort information structures. The improvement of information structures and optimization of information processes made on the basis of development of adequate models of functioning of systems of complex safety of objects of special importance in the conditions of factors of external influences is a scientific and technical task relevant now which implementation will allow to minimize lag of rates of development of the security equipment and technologies from dynamically improved instruments of destabilization of elements of systems of protection of objects of special importance. Development and the solution of adequate mathematical model of interaction of information structures of security systems and negative impacts are intended to describe dynamics of evolution of their elements on the phase plane of space and taking into account integrated representation of stability conditions and also the second method of Lyapunov to create mathematical model of steady management of information process of interaction of elements of the specified information sets. The task of the analysis of phase portraits of a condition of security complexes as information systems is connected with a research of the attractors representing areas (the phase vicinities) of space consisting of set of concentric circles in the form of a set of the points which are attracting trajectories of evolution of elements of information structures of security systems and indicating areas of their steady functioning.

Keywords: attractor, algorithm, dynamic system, stability, efficiency, information system, information structure, information process, information set, interaction model, management.

Full text:


UDC 519.676

D.E. Orlova

Stability of the decisions accepted at maintenance of functioning of organizational-technical systems, is understood as their ability to keep the urgency in the conditions of action of various stirring factors. Methods and numerical algorithms of an estimation of stability of two classes of decisions in such systems are offered. Decisions of vertical type concern the first class “the head-subordinate” which stability is treated on Nash. As the decisions which infringement is unprofitable to infringers, whether it be the head or subordinates. The decisions of horizontal type accepted at level of interaction of subordinates which stability is treated on A.M. Lyapunov when the estimation of stability of investigated object is reduced to a question on existence of the stationary not trivial decision of system of the differential equations describing dynamics of this object concern the second class. For both classes of decisions formal conditions of maintenance of stability come to light and the algorithms are developed, allowing to establish stability level in typical situations. Algorithms are realised in integrated TURBO PASCAL environment with application of procedures and functions VISUAL BAISIC, DELPHI and C ++, focused on an application creation under control of Windows 7. Numerical experiment proves their convergence. The methods described in article can find practical application as the tool of support of decision-making at management difficult dynamic system of organizational-technical type.

Keywords: :organizational-technical system, the administrative decision, stability, system of the differential equations, algorithm.

Full text:


UDC 519.676

V.I. Novoseltsev, A.N. Noev, D.E. Orlova

The mathematical model allowing in quantitative expression to establish influence of mutual cyberattacks to economic efficiency of contestant firms is considered. The basis of model is worked out by Lotke-Voltaire’s made in the assumption the modified equations that change of economic efficiency of each firm in the absence of the competitor and, accordingly, cyberattacks, is described by the logistical equation. The qualitative method of differential calculus defines conditions at which observance, despite mutual attacks, competitors do not undergo economic bankruptcy, and continue to function in a normal mode. As the integrated indicator characterizing economic efficiency of contestant firms, the volume of the goods realized by them or the rendered services is applied. The model can be used for a substantiation of requirements to maintenance of information security of competing subjects of the modern market in the conditions of mutual cyberattacks.

Keywords: : cyberattack, mathematical model, economic efficiency, information security, stability.

Full text:


UDC 621.316.726.078

Y.A. Gelozhe, P.P. Klimenko, A.V. Maksimov

In the processing and generation of radio signals, a phase locked loop is widely used. When processing signals in demodulators, the carrier wave synchronization and clock synchronization are carried out. Digital frequency synthesizers are usually created on the basis of a phase automatic system with digital frequency dividers.
Under the conditions of formation of reference sinusoidal oscillations, it is usually necessary to obtain a low level of non-harmonic discrete spectral components localized near the carrier. These secondary components are caused by pulsations of the control voltage of the impulse-phase detector of the phase automatic system operating in the time-sampling mode. The suppression of these spectral components is carried out by including high-order low pass filters in the phase locked loop circuit. The use of such filters in the tracking demodulators increases their selectivity along the adjacent channel. In this case, there are problems with ensuring the stability of the system in the “small” and “big”.
The paper considers the work of phase locked loop, which has the property of self-organization and ensures a rapid recovery of the synchronization regime for large perturbations. This system includes a binomial low pass filter that suppresses by products of frequency synthesis and increases the selectivity of demodulators along the adjacent channel. The paper shows that this system with binomial low pass filters, including the 5th order, can satisfy the stability criterion in “big” under certain conditions.
Modeling of the phase locked loop system, which has the property of self-organization and with high-order binomial low pass filters included in its composition, has shown that the processes in it are stable at large initial detuning in frequency. The duration of the transient processes is proportional to the magnitude of the detuning in frequency.
On the basis of the obtained results, it can be asserted that the considered phase system provides an extension of the functionality in the processing and generation of radio signals.

Keywords: : phase, frequency, transient process, stability, phase detector, automatic control system.

Full text:


UDC 621.372:519.622

A.M. Pilipenko

This paper considers the problems of numerical analysis of electronic circuits in the time domain that arise when using modern circuit simulators based on SPICE. Time-domain analysis of circuits through modern electronic simulators is realized by means of Gear’s methods and the trapezoidal method. An important property of models of real electronic circuits and especially of RF circuits is simultaneous stiffness and oscillability of these models. In turn, Gear’s methods can lose stability for oscillating circuits’ analysis, because these methods are not P-stable, and the trapezoidal method has a sufficiently high error for stiff circuits’ analysis, because it is not L-stable. The aim of this paper is to develop hybrid L- and P-stable methods based on the combination of various numerical methods for solving ordinary differential equations which provide a high accuracy of numerical simulation in the time domain of stiff and oscillating circuits. Hybrid methods are built on the basis of the known Rado IIA and Lobatto IIIA methods, which are subclasses of implicit Runge-Kutta methods. Comparative analysis of the known methods and the proposed hybrid methods demonstrates high accuracy of the latter methods for time-domain simulation of stiff and oscillating circuits and systems. Hybrid methods are also effective for numerical solving differential-algebraic equations that describe arbitrary electrical circuits.

Keywords: time-domain simulation, stiff systems, oscillating circuits, implicit Runge–Kutta methods, accuracy, stability.

Full text: