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N. V. Kuznetsov

Publications and source records attributed to N. V. Kuznetsov.

49 records · Page 3Linked to original sources
arXiv2016

Hidden and self-excited attractors in electromechanical systems with and without equilibria

This paper studies hidden oscillations appearing in electromechanical systems with and without equilibria. Three different systems with such effects are considered: translational oscillator-rotational actuator, drilling system actuated by a DC-motor and drilling system actuated by induction motor. We demonstrate that all three systems experience hidden oscillations in sense of mathematical definition. But from physical point of view in certain cases it is quite easy to localize there oscillations.

math.DS↗
arXiv2015

Hidden attractors in fundamental problems and engineering models

Recently a concept of self-excited and hidden attractors was suggested: an attractor is called a self-excited attractor if its basin of attraction overlaps with neighborhood of an equilibrium, otherwise it is called a hidden attractor. For example, hidden attractors are attractors in systems with no equilibria or with only one stable equilibrium (a special case of multistability and coexistence of attractors). While coexisting self-excited attractors can be found using the standard computational procedure, there is no standard way of predicting the existence or coexistence of hidden attractors in a system. In this plenary survey lecture the concept of self-excited and hidden attractors is discussed, and various corresponding examples of self-excited and hidden attractors are considered.

nlin.CD↗
arXiv2015

Limitations of PLL simulation: hidden oscillations in MatLab and SPICE

Nonlinear analysis of the phase-locked loop (PLL) based circuits is a challenging task, thus in modern engineering literature simplified mathematical models and simulation are widely used for their study. In this work the limitations of numerical approach is discussed and it is shown that, e.g. hidden oscillations may not be found by simulation. Corresponding examples in SPICE and MatLab, which may lead to wrong conclusions concerning the operability of PLL-based circuits, are presented.

cs.OH↗
arXiv2015

The Lyapunov dimension formula for the global attractor of the Lorenz system

The exact Lyapunov dimension formula for the Lorenz system has been analytically obtained first due to G.A.Leonov in 2002 under certain restrictions on parameters, permitting classical values. He used the construction technique of special Lyapunov-type functions developed by him in 1991 year. Later it was shown that the consideration of larger class of Lyapunov-type functions permits proving the validity of this formula for all parameters of the system such that all the equilibria of the system are hyperbolically unstable. In the present work it is proved the validity of the formula for Lyapunov dimension for a wider variety of parameters values, which include all parameters satisfying the classical physical limitations. One of the motivation of this work is the possibility of computing a chaotic attractor in the Lorenz system in the case of one unstable and two stable equilibria.

math.DS↗
arXiv2015

Limitations of the classical phase-locked loop analysis

Nonlinear analysis of the classical phase-locked loop (PLL) is a challenging task. In classical engineering literature simplified mathematical models and simulation are widely used for its study. In this work the limitations of classical engineering phase-locked loop analysis are demonstrated, e.g., hidden oscillations, which can not be found by simulation, are discussed. It is shown that the use of simplified dynamical models and the application of simulation may lead to wrong conclusions concerning the operability of PLL-based circuits.

math.DS↗
arXiv2015

A dynamical model of Sayano-Shushenskaya hydropower plant: stability, oscillations, and accident

This work is devoted to the construction and study of a mathematical model of hydropower unit, consisting of synchronous generator, hydraulic turbine, and speed governor. It is motivated by the accident happened on the Sayano-Shushenskaya hydropower plant in 2009 year. Parameters of the Sayano-Shushenskaya hydropower plant were used for modeling the system. Oscillations in zones, which were not recommended for operation, were found. The obtained results are consistent with the full-scale test results carried out for hydropower units of the Sayano-Shushenskaya hydropower plant in 1988.

math.DS↗
arXiv2015

Homoclinic orbits, and self-excited and hidden attractors in a Lorenz-like system describing convective fluid motion

In this tutorial, we discuss self-excited and hidden attractors for systems of differential equations. We considered the example of a Lorenz-like system derived from the well-known Glukhovsky--Dolghansky and Rabinovich systems, to demonstrate the analysis of self-excited and hidden attractors and their characteristics. We applied the fishing principle to demonstrate the existence of a homoclinic orbit, proved the dissipativity and completeness of the system, and found absorbing and positively invariant sets. We have shown that this system has a self-excited attractor and a hidden attractor for certain parameters. The upper estimates of the Lyapunov dimension of self-excited and hidden attractors were obtained analytically.

nlin.CD↗
arXiv2015

A short survey on nonlinear models of the classic Costas loop: rigorous derivation and limitations of the classic analysis

Rigorous nonlinear analysis of the physical model of Costas loop --- a classic phase-locked loop (PLL) based circuit for carrier recovery, is a challenging task. Thus for its analysis, simplified mathematical models and numerical simulation are widely used. In this work a short survey on nonlinear models of the BPSK Costas loop, used for pre-design and post-design analysis, is presented. Their rigorous derivation and limitations of classic analysis are discussed. It is shown that the use of simplified mathematical models, and the application of non rigorous methods of analysis (e.g., simulation and linearization) may lead to wrong conclusions concerning the performance of the Costas loop physical model.

math.DS↗
arXiv2015

Estimation of Lyapunov dimension for the Chen and Lu systems

Nowadays various estimates of Lyapunov dimension of Lorenz-like systems attractors are actively developed. Within the frame of this study the question arises whether it is possible to obtain the corresponding estimates of dimension for the Chen and Lu systems using the reduction of them to the generalized Lorenz system. In the work (Chen and Yang, 2013) Leonov's method was applied for the estimation of Lyapunov dimension, and as a consequence the Lyapunov dimension of attractors of the Chen and Lu systems with the classical parameters was estimated. In the present work an inaccuracy in (Chen and Yang, 2013) is corrected and it is shown that the revised domain of parameters, where the estimate of Lyapunov dimension is valid, does not involve the classical parameters of the Chen and Lu systems.

nlin.CD↗
arXiv2014

Homoclinic orbit and hidden attractor in the Lorenz-like system describing the fluid convection motion in the rotating cavity

In this paper a Lorenz-like system, describing the process of rotating fluid convection, is considered. The present work demonstrates numerically that this system, also like the classical Lorenz system, possesses a homoclinic trajectory and a chaotic self-excited attractor. However, for considered system, unlike the classical Lorenz one, along with self-excited attractor a hidden attractor can be localized. Analytical-numerical localization of hidden attractor is presented.

nlin.CD↗
arXiv2014

Differences and similarities in the analysis of Lorenz, Chen, and Lu systems

Currently it is being actively discussed the question of the equivalence of various Lorenz-like systems and the possibility of universal consideration of their behavior, in view of the possibility of reduction of such systems to the same form with the help of various transformations. In the present paper the differences and similarities in the analysis of the Lorenz, the Chen and the Lu systems are discussed and it is shown that the Chen and the Lu systems are valuable for the development of new methods for the analysis of chaotic systems.

nlin.CD↗
arXiv1998

The true order of the riemann zeta--function

For the Riemann zeta-function on the critical line the terminal estimate have been proved, which had been conjectured by Lindelöf at the beginning of this Centure. The proof is based on the authors relations which connect the bilinear forms of the eigenvalues of the Hecke operators with sums of the Kloosterman sums. By the way, it is proved that for the Hecke series (which are associated with the eigenfunctions of the automorphic Laplacian) the natural analogue of the Lindelöf conjecture is true also. \noindent Bibl. 14.

math.NT↗