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N. V. Kuznetsov

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

At least 37 records · Page 2Linked to original sources
arXiv2017

Finite-time Lyapunov dimension and hidden attractor of the Rabinovich system

The Rabinovich system, describing the process of interaction between waves in plasma, is considered. It is shown that the Rabinovich system can exhibit a {hidden attractor} in the case of multistability as well as a classical {self-excited attractor}. The hidden attractor in this system can be localized by analytical-numerical methods based on the {continuation} and {perpetual points}. For numerical study of the attractors' dimension the concept of {finite-time Lyapunov dimension} is developed. A conjecture on the Lyapunov dimension of self-excited attractors and the notion of {exact Lyapunov dimension} are discussed. A comparative survey on the computation of the finite-time Lyapunov exponents by different algorithms is presented and an approach for a reliable numerical estimation of the finite-time Lyapunov dimension is suggested. Various estimates of the finite-time Lyapunov dimension for the hidden attractor and hidden transient chaotic set in the case of multistability are given.

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arXiv2017

Scenario of the Birth of Hidden Attractors in the Chua Circuit

Recently it was shown that in the dynamical model of Chua circuit both the classical selfexcited and hidden chaotic attractors can be found. In this paper the dynamics of the Chua circuit is revisited. The scenario of the chaotic dynamics development and the birth of selfexcited and hidden attractors is studied. It is shown a pitchfork bifurcation in which a pair of symmetric attractors coexists and merges into one symmetric attractor through an attractormerging bifurcation and a splitting of a single attractor into two attractors. The scenario relating the subcritical Hopf bifurcation near equilibrium points and the birth of hidden attractors is discussed.

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arXiv2017

Solution of the Gardner problem on the lock-in range of phase-locked loop

The lock-in frequency and lock-in range concepts were introduced in 1966 by Floyd Gardner to describe the frequency differences of phase-locked loop based circuit for which the loop can acquire lock within one beat, i.e. without cycle slipping. These concepts became popular among engineering community and were given in various engineering publications. However rigorous mathematical explanations these concepts turned out to be a challenging task, thus, in the 2nd edition of Gardner's well-known work, Phaselock Techniques, he wrote that "despite its vague reality, lock-in range is a useful concept" and posed the problem "to define exactly any unique lock-in frequency". In this paper an effective solution for Gardner's problem on the unique definition of the lock-in frequency and lock-in range is discussed. The lock-in range and lock-in frequency computation is explained on the example of classical second-order PLL with lead-lag and active proportional-integral filters. The obtained results can also be used for the lock-in range computation of such PLL-based circuits as two-phase PLL, two-phase Costas loop, BPSK Costas loop, and optical Costas loop, used in intersatellite communication.

math.DS↗
arXiv2017

Localization of hidden Chua attractors by the describing function method

In this paper the Chua circuit with five linear elements and saturation non-linearity is studied. Numerical localization of self-excited attractor in the Chua circuit model can be done by computation of trajectory with initial data in a vicinity of an unstable equilibrium. For a hidden attractor its basin of attraction does not overlap with a small vicinity of equilibria, so it is difficult to find the corresponding initial data for localization. This survey is devoted to the application of describing function method for localization of hidden periodic and chaotic attractors in the Chua model. We use a rigorous justification of the describing function method, based on the method of small parameter, to get the initial data for the visualization of the hidden attractors. A new configuration of hidden Chua attractors is presented.

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arXiv2017

Optical Costas loop: pull-in range estimation and hidden oscillations

In this work we consider a mathematical model of the optical Costas loop. The pull-in range of the model is estimated by analytical and numerical methods. Difficulties of numerical analysis, related to the existence of so-called hidden oscillations in the phase space, are discussed.

math.DS↗
arXiv2017

Nonlinear analysis of PLL by the harmonic balance method

In this paper we discuss the application of the harmonic balance method for the global analysis of the classical phase-locked loop (PLL) circuit. The harmonic balance is non rigorous method, which is widely used %,often without rigorous justification, for the computation of periodic solutions and the checking of global stability. The proof of the absence of periodic solutions is a key step to establish the global stability of PLL and estimate the pull-in range (which is an interval of the frequency deviations such that any solution tends to one of the equilibria). The advantages and limitations of the study of the classical PLL with lead-lag filter using the harmonic balance method is discussed.

math.DS↗
arXiv2017

A short survey on QPSK Costas loop mathematical models

The Costas loop is a modification of the phase-locked loop circuit, which demodulates data and recovers carrier from the input signal. The Costas loop is essentially a nonlinear control system and its nonlinear analysis is a challenging task. Thus, simplified mathematical models and their numerical simulation are widely used for its analysis. At the same time for phase-locked loop circuits there are known various examples where the results of such simplified analysis are differ substantially from the real behavior of the circuit. In this survey the corresponding problems are demonstrated and discussed for the QPSK Costas loop.

math.DS↗
arXiv2017

Hidden and self-excited attractors in Chua circuit: SPICE simulation and synchronization

Nowadays various chaotic secure communication systems based on synchronization of chaotic circuits are widely studied. To achieve synchronization, the control signal proportional to the difference between the circuits signals, adjust the state of one circuit. In this paper the synchronization of two Chua circuits is simulated in SPICE. It is shown that the choice of control signal is be not straightforward, especially in the case of multistability and hidden attractors.

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arXiv2016

The Lyapunov dimension and its estimation via the Leonov method

Along with widely used numerical methods for estimating and computing the Lyapunov dimension there is an effective analytical approach, proposed by G.A. Leonov in 1991. The Leonov method is based on the direct Lyapunov method with special Lyapunov-like functions. The advantage of this method is that it allows one to estimate the Lyapunov dimension of invariant set without local- ization of the set in the phase space and in many cases get effectively exact Lyapunov dimension formula. In this survey the invariance of Lyapunov dimension with respect to diffeomorphisms and its connection with the Leonov method are discussed. An analog of Leonov method for discrete time dynamical systems is suggested. In a simple but rigorous way, here it is presented the connection between the Leonov method and the key related works in the area: by Kaplan and Yorke (the concept of Lyapunov dimension, 1979), Douady and Oesterle (upper bounds of Hausdorff dimension via the Lyapunov dimension of maps, 1980), Constantin, Eden, Foias, and Temam (upper bounds of Hausdorff dimension via the Lyapunov exponents and dimension of dynamical systems, 1985-90), and the numerical calculation of the Lyapunov exponents and dimension.

nlin.CD↗
arXiv2016

The Lyapunov dimension and its computation for self-excited and hidden attractors in the Glukhovsky-Dolzhansky fluid convection model

Consideration of various hydrodynamic phenomena involves the study of the Navier-Stokes (N-S) equations, what is hard enough for analytical and numerical investigations since already in three-dimensional (3D) case it is a challenging task to study the limit behavior of N-S solutions. The low-order models (LOMs) derived from the initial N-S equations by Galerkin method allow one to overcome difficulties in studying the limit behavior and existence of attractors. Among the simple LOMs with chaotic attractors there are famous Lorenz system, which is an approximate model of two-dimensional convective flow and Glukhovsky-Dolzhansky model, which describes a convective process in three-dimensional rotating fluid and can be considered as an approximate model of the World Ocean. One of the widely used dimensional characteristics of attractors is the Lyapunov dimension. In the study we follow a rigorous approach for the definition of the Lyapunov dimension and justification of its computation by the Kaplan-Yorke formula, without using statistical physics assumptions. The exact Lyapunov dimension formula for the global attractors is obtained and peculiarities of the Lyapunov dimension estimation for self-excited and hidden attractors are discussed. A tutorial on numerical estimation of the Lyapunov dimension on the example of the Glukhovsky-Dolzhansky model is presented.

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arXiv2016

A Survey on Dynamic Analysis of the Costas Loop

This survey is devoted to the dynamic analysis of the Costas loop. In particular the acquisition process is analyzed in great detail. Acquision is most conventiently described by a number of frequency and time parameters such as lock-in range, lock-in time, pull-in range, pull-in time, and hold-in range. While for the classical PLL equations for all these parameters have been derived (many of them are approximations, some even crude approximations), this has not yet been carried out for the Costas loop. It is the aim of this analysis to close this gap. The paper starts with an overview on mathematical and physical models (exact and simplified) of the different variants of the Costas loop, cf. Section~1. In Sections 2--5 equations for the above mentioned key parameters are derived. Finally, the hold-in range of the Costas loop for the case where a lead-lag filter is used for the loop filter is analyzed, cf. Appendix.

eess.SY↗
arXiv2016

Hold-in, pull-in, and lock-in ranges of PLL circuits: rigorous mathematical definitions and limitations of classical theory

The terms hold-in, pull-in (capture), and lock-in ranges are widely used by engineers for the concepts of frequency deviation ranges within which PLL-based circuits can achieve lock under various additional conditions. Usually only non-strict definitions are given for these concepts in engineering literature. After many years of their usage, F.~Gardner in the 2nd edition of his well-known work, Phaselock Techniques, wrote "There is no natural way to define exactly any unique lock-in frequency" and "despite its vague reality, lock-in range is a useful concept." Recently these observations have led to the following advice given in a handbook on synchronization and communications "We recommend that you check these definitions carefully before using them." In this survey it is shown that, from a mathematical point of view, in some cases the hold-in and pull-in "ranges" may not be the intervals of values but a union of intervals and thus their widely used definitions require clarification. Rigorous mathematical definitions for the hold-in, pull-in, and lock-in ranges are given. An effective solution for the problem on the unique definition of the lock-in frequency, posed by Gardner, is suggested.

math.DS↗
arXiv2016

Lock-in range of PLL-based circuits with proportionally-integrating filter and sinusoidal phase detector characteristic

In the present work PLL-based circuits with sinusoidal phase detector characteristic and active proportionally-integrating (PI) filter are considered. The notion of lock-in range -- an important characteristic of PLL-based circuits, which corresponds to the synchronization without cycle slipping, is studied. For the lock-in range a rigorous mathematical definition is discussed. Numerical and analytical estimates for the lock-in range are obtained.

math.DS↗
arXiv2016

Dynamics of the Zeraoulia-Sprott map revisited

In the paper "Some Open Problems in Chaos Theory and Dynamics" by Zeraoulia and Sprott, the two-dimensional map (x,y) -> (-ax(1+y^2)^{-1}, x+by) was considered and the problem of analytical study of the boundedness of its attractors was formulated. In the present paper, the boundedness of its attractors is studied, the corresponding analytical estimation of absorbing set is obtained, and thus an answer to the problem is given.

nlin.CD↗
arXiv2016

A short survey on Lyapunov dimension for finite dimensional dynamical systems in Euclidean space

Nowadays there are a number of surveys and theoretical works devoted to the Lyapunov exponents and Lyapunov dimension, however most of them are devoted to infinite dimensional systems or rely on special ergodic properties of the system. At the same time the provided illustrative examples are often finite dimensional systems and the rigorous proof of their ergodic properties can be a difficult task. Also the Lyapunov exponents and Lyapunov dimension have become so widespread and common that they are often used without references to the rigorous definitions or pioneering works. The survey is devoted to the finite dimensional dynamical systems in Euclidean space and its aim is to explain, in a simple but rigorous way, the connection between the key works in the area: by Kaplan and Yorke (the concept of Lyapunov dimension, 1979), Douady and Oesterle (estimation of Hausdorff dimension via the Lyapunov dimension of maps, 1980), Constantin, Eden, Foias, and Temam (estimation of Hausdorff dimension via the Lyapunov exponents and dimension of dynamical systems, 1985-90), Leonov (estimation of the Lyapunov dimension via the direct Lyapunov method, 1991), and numerical methods for the computation of Lyapunov exponents and Lyapunov dimension. In this survey a concise overview of the classical results is presented, various definitions of Lyapunov exponents and Lyapunov dimension are discussed. An effective analytical method for the estimation of Lyapunov dimension is presented, its application to the self-excited and hidden attractors of well-known dynamical systems is demonstrated, and analytical formulas of exact Lyapunov dimension are obtained.

math.DS↗
arXiv2016

Invariance of Lyapunov exponents and Lyapunov dimension for regular and irregular linearizations

Nowadays the Lyapunov exponents and Lyapunov dimension have become so widespread and common that they are often used without references to the rigorous definitions or pioneering works. It may lead to a confusion since there are at least two well-known definitions, which are used in computations: the upper bounds of the exponential growth rate of the norms of linearized system solutions (Lyapunov characteristic exponents, LCEs) and the upper bounds of the exponential growth rate of the singular values of the fundamental matrix of linearized system (Lyapunov exponents, LEs). In this work the relation between Lyapunov exponents and Lyapunov characteristic exponents is discussed. The invariance of Lyapunov exponents for regular and irregular linearizations under the change of coordinates is demonstrated.

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