0000000001246934
AUTHOR
Gennady A. Leonov
showing 40 related works from this author
Analytic Exact Upper Bound for the Lyapunov Dimension of the Shimizu–Morioka System
2015
In applied investigations, the invariance of the Lyapunov dimension under a diffeomorphism is often used. However, in the case of irregular linearization, this fact was not strictly considered in the classical works. In the present work, the invariance of the Lyapunov dimension under diffeomorphism is demonstrated in the general case. This fact is used to obtain the analytic exact upper bound of the Lyapunov dimension of an attractor of the Shimizu–Morioka system. peerReviewed
Nonlinear Analysis of Phase-locked Loop-Based Circuits
2013
Main problems of simulation and mathematical modeling of high-frequency signals for analog Costas loop and for analog phase-locked loop (PLL) are considered. Two approachers which allow to solve these problems are considered. In the first approach, nonlinear models of classical PLL and classical Costas loop are considered. In the second approach, engineering solutions for this problems are described. Nonlinear differential equations are derived for both approaches.
Simulation of PLL with impulse signals in MATLAB: Limitations, hidden oscillations, and pull-in range
2015
The limitations of PLL simulation are demonstrated on an example of phase-locked loop with triangular phase detector characteristic. It is shown that simulation in MatLab may not reveal periodic oscillations (e.g. such as hidden oscillations) and thus may lead to unreliable conclusions on the width of pull-in range.
Hidden attractors on one path : Glukhovsky-Dolzhansky, Lorenz, and Rabinovich systems
2017
In this report, by the numerical continuation method we visualize and connect hidden chaotic sets in the Glukhovsky-Dolzhansky, Lorenz and Rabinovich systems using a certain path in the parameter space of a Lorenz-like system.
Nonlinear analysis of classical phase-locked loops in signal's phase space
2014
Abstract Discovery of undesirable hidden oscillations, which cannot be found by the standard simulation, in phase-locked loop (PLL) showed the importance of consideration of nonlinear models and development of rigorous analytical methods for their analysis. In this paper for various signal waveforms, analytical computation of multiplier/mixer phase-detector characteristics is demonstrated, and nonlinear dynamical model of classical analog PLL is derived. Approaches to the rigorous nonlinear analysis of classical analog PLL are discussed.
Analytical-Numerical Localization of Hidden Attractor in Electrical Chua’s Circuit
2013
Study of hidden oscillations and hidden chaotic attractors (basin of attraction of which does not contain neighborhoods of equilibria) requires the development of special analytical-numerical methods. Development and application of such methods for localization of hidden chaotic attractors in dynamical model of Chua’s circuit are demonstrated in this work.
IWCFTA2012 Keynote Speech I - Hidden attractors in dynamical systems: From hidden oscillation in Hilbert-Kolmogorov, Aizerman and Kalman problems to …
2012
Summary form only given. In this survey an attempt is made to reflect the current trends in the synthesis of analytical and numerical methods to develop efficient analytical-numerical methods, based on harmonic linearization, applied bifurcation theory and numerical methods, for searching hidden oscillations.
Hidden oscillations in aircraft flight control system with input saturation
2013
Abstract The presence of actuator saturation can dramatically degrade the system performance. Since the feedback loop is broken when the actuator saturates the unstable modes of the regulator may then drift to undesirable values. The consequences are that undesired nonlinear oscillations appear and that the settling time may unacceptably increase. Rigorous analysis of nonlinear aircraft models is a very difficult task, that is why a numerical simulation is often used as an analysis and design tool. In this paper difficulties of numerical analysis related to the existence of hidden oscillations in the aircraft control system are demonstrated.
Tutorial on dynamic analysis of the Costas loop
2016
Abstract Costas loop is a classical phase-locked loop (PLL) based circuit for carrier recovery and signal demodulation. The PLL is an automatic control system that adjusts the phase of a local signal to match the phase of the input reference signal. This tutorial is devoted to the dynamic analysis of the Costas loop. In particular the acquisition process is analyzed. Acquisition is most conveniently 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 all these parameters have been derived (many of them are approximations, some even crude approximations), this has not…
Hidden oscillations in stabilization system of flexible launcher with saturating actuators
2013
Abstract In the paper the attitude stabilization system of the unstable flexible launcher with saturating input is considered. It is demonstrated that due to actuator saturation the system performance can significantly degrade. The analytical-numerical method is applied to demonstrate possibility of hidden oscillations and localize their attractor.
A short survey on nonlinear models of the classic Costas loop: rigorous derivation and limitations of the classic analysis
2015
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 ph…
Hidden oscillations in nonlinear control systems
2011
Abstract The method of harmonic linearization, numerical methods, and the applied bifurcation theory together discover new opportunities for analysis of hidden oscillations of control systems. In the present paper new analytical-numerical algorithm for hidden oscillation localization is discussed. Counterexamples construction to Aizerman's conjecture and Kalman's conjecture on absolute stability of control systems are considered.
Finite-time Lyapunov dimension and hidden attractor of the Rabinovich system
2015
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 expon…
Convergence-based Analysis of Robustness to Delay in Anti-windup Loop of Aircraft Autopilot∗∗This work was supported by Russian Scientific Foundation…
2015
Abstract The windup phenomenon is interpreted as a consequence of the convergent property absence for system with a saturation. This makes it possible to use the frequency-domain criterion for analysis of anti-windup augmentation in the case of stable and marginally stable plants. Based on this approach, robustness of the systems with respect to time delay in the anti-windup loop is examined and the approach for an optimal choice of the static anti-windup gain is proposed. An application of the convergence-based anti-windup control strategy to aircraft flight control for the case of time-delay in the anti-windup loop is described and studied by simulations.
Prediction of Hidden Oscillations Existence in Nonlinear Dynamical Systems: Analytics and Simulation
2013
From a computational point of view, in nonlinear dynamical systems, attractors can be regarded as self-excited and hidden attractors. Self-excited attractors can be localized numerically by a standard computational procedure, in which after a transient process a trajectory, starting from a point of unstable manifold in a neighborhood of equilibrium, reaches a state of oscillation, therefore one can easily identify it. In contrast, for a hidden attractor, a basin of attraction does not intersect neighborhoods of equilibria. While classical attractors are self-excited, attractors can therefore be obtained numerically by the standard computational procedure, for localization of hidden attracto…
Analytical-numerical methods for investigation of hidden oscillations in nonlinear control systems
2011
The method of harmonic linearization, numerical methods, and the applied bifurcation the- ory together discover new opportunities for analysis of oscillations of control systems. In the present survey analytical-numerical algorithms for hidden oscillation localization are discussed. Examples of hidden attrac- tor localization in Chua's circuit and counterexamples construction to Aizerman's conjecture and Kalman's conjecture are considered.
Localization of hidden Chua's attractors
2011
Abstract The classical attractors of Lorenz, Rossler, Chua, Chen, and other widely-known attractors are those excited from unstable equilibria. From computational point of view this allows one to use numerical method, in which after transient process a trajectory, started from a point of unstable manifold in the neighborhood of equilibrium, reaches an attractor and identifies it. However there are attractors of another type: hidden attractors, a basin of attraction of which does not contain neighborhoods of equilibria . In the present Letter for localization of hidden attractors of Chuaʼs circuit it is suggested to use a special analytical–numerical algorithm.
Nonlinear dynamical model of Costas loop and an approach to the analysis of its stability in the large
2015
The analysis of the stability and numerical simulation of Costas loop circuits for high-frequency signals is a challenging task. The problem lies in the fact that it is necessary to simultaneously observe very fast time scale of the input signals and slow time scale of phase difference between the input signals. To overcome this difficult situation it is possible, following the approach presented in the classical works of Gardner and Viterbi, to construct a mathematical model of Costas loop, in which only slow time change of signal?s phases and frequencies is considered. Such a construction, in turn, requires the computation of phase detector characteristic, depending on the waveforms of th…
Synchronization of hidden chaotic attractors on the example of radiophysical oscillators
2017
In the present paper we consider the problem of synchronization of hidden and self-excited attractors in the context of application to a system of secure communication. The system of two coupled Chua models was studied. Complete synchronization was observed as for self-excited, as hidden attractors. Beside it for hidden attractors some special type of dynamic was revealed.
On differences and similarities in the analysis of Lorenz, Chen, and Lu systems
2015
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 (Algaba et al., 2013a,b, 2014b,c; Chen, 2013; Chen and Yang, 2013; Leonov, 2013a), 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. It is shown that the Chen and the Lu systems stimulate the development of new methods for the analysis of chaotic systems. Open problems are discussed.
Scenario of the Birth of Hidden Attractors in the Chua Circuit
2017
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.
UAV control with switched GNSS-Estimator navigation system∗∗This work was supported by Russian Scientific Foundation (project 14-21-00041) and Saint-…
2015
Abstract In the paper the switched GNSS-Estimator navigation system, recently proposed by the authors, is described and numerically studied in the framework of evaluation of the overall UAV control system accuracy.
Nonlinear Analysis of Phase-Locked Loop (PLL): Global Stability Analysis, Hidden Oscillations and Simulation Problems
2013
In the middle of last century the problem of analyzing hidden oscillations arose in automatic control. In 1956 M. Kapranov considered a two-dimensional dynamical model of phase locked-loop (PLL) and investigated its qualitative behavior. In these investigations Kapranov assumed that oscillations in PLL systems can be self-excited oscillations only. However, in 1961, N. Gubar’ revealed a gap in Kapranov’s work and showed analytically the possibility of the existence of another type of oscillations, called later by the authors hidden oscillations, in a phase-locked loop model: from a computational point of view the system considered was globally stable (all the trajectories tend to equilibria…
Nonlinear analysis of phase-locked loop
2010
Abstract New method for the rigorous mathematical nonlinear analysis of PLL systems is suggested. This method allows to calculate the characteristics of phase detectors and carry out a rigorous mathematical analysis of transient process and stability of the system.
Hidden Oscillations In The Closed-Loop Aircraft-Pilot System And Their Prevention* *This work was supported by Russian Science Foundation (project 14…
2016
Abstract The paper is devoted to studying and prevention of a special kind of oscillations-the Pilot Involved Oscillations (PIOs) which may appear in man-machine closed-loop dynamical systems. The PIO of categories II and III are defined as essentially non-linear unintended steady fluctuations of the piloted aircraft, generated due to pilot efforts to control the aircraft with a high precision. The main non-linear factor leading to the PIO is, generally, rate limitations of the aircraft control surfaces, resulting in a delay in the response of the aircraft to pilot commands. In many cases, these oscillations indicate presence of hidden, rather than self-excited attractors in the aircraft-pi…
Invariance of Lyapunov exponents and Lyapunov dimension for regular and irregular linearizations
2014
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…
The Lyapunov dimension formula for the global attractor of the Lorenz system
2015
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 …
Hidden attractor and homoclinic orbit in Lorenz-like system describing convective fluid motion in rotating cavity
2015
Abstract In this paper a Lorenz-like system, describing convective fluid motion in rotating cavity, is considered. It is shown numerically that this system, like the classical Lorenz system, possesses a homoclinic trajectory and a chaotic self-excited attractor. However, for the considered system, unlike the classical Lorenz system, along with self-excited attractor a hidden attractor can be localized. Analytical-numerical localization of hidden attractor is demonstrated.
Analytical methods for computation of phase-detector characteristics and PLL design
2011
An effective analytical methods for computation of phase detector characteristics are suggested. For high-frequency oscillators new classes of such characteristics are described. Approaches to a rigorous nonlinear analysis of PLL are discussed.
Drilling Systems: Stability and Hidden Oscillations
2013
There are many mathematical models of drilling systems Despite, huge efforts in constructing models that would allow for precise analysis, drilling systems, still experience breakdowns. Due to complexity of systems, engineers mostly use numerical analysis, which may lead to unreliable results. Nowadays, advances in computer engineering allow for simulations of complex dynamical systems in order to obtain information on the behavior of their trajectories. However, this simple approach based on construction of trajectories using numerical integration of differential equations describing dynamical systems turned out to be quite limited for investigation of stability and oscillations of these s…
Hidden Oscillations in Electromechanical Systems
2016
In this paper an electromechanical system with two different types of motor is considered. It is shown that during the spin-up, the system with DC motor may experience unwanted vibration—the Sommerfeld effect. This is a well-known effect when the motor of electromechanical system gets stuck near the resonance zone instead of reaching its nominal power. The absence of this effect is demonstrated in the system with synchronous motor. Nowadays, there are many works devoted to the study of this effect in various systems. Here we discuss the Sommerfeld effect from the point of view of localization of the so-called hidden oscillations.
Harmonic Balance Method and Stability of Discontinuous Systems
2019
The development of the theory of discontinuous dynamical systems and differential inclusions was not only due to research in the field of abstract mathematics but also a result of studies of particular problems in mechanics. One of the first methods, used for the analysis of dynamics in discontinuous mechanical systems, was the harmonic balance method developed in the thirties of the twentieth century. In our work, the results of analysis obtained by the method of harmonic balance, which is an approximate method, are compared with the results obtained by rigorous mathematical methods and numerical simulation.
Theory of Differential Inclusions and Its Application in Mechanics
2017
The following chapter deals with systems of differential equations with discontinuous right-hand sides. The key question is how to define the solutions of such systems. The most adequate approach is to treat discontinuous systems as systems with multivalued right-hand sides (differential inclusions). In this work, three well-known definitions of solution of discontinuous system are considered. We will demonstrate the difference between these definitions and their application to different mechanical problems. Mathematical models of drilling systems with discontinuous friction torque characteristics are considered. Here, opposite to classical Coulomb symmetric friction law, the friction torqu…
Hidden oscillations in SPICE simulation of two-phase Costas loop with non-linear VCO
2016
Simulation is widely used for analysis of Costas loop based circuits. However it may be a non-trivial task, because incorrect choice of integration parameters may lead to qualitatively wrong conclusions. In this work the importance of choosing appropriate parameters and simulation model is discussed. It is shown that hidden oscillations may not be found by simulation in SPICE, however it can be predicted by analytical methods. peerReviewed
Hidden attractors in electromechanical systems with and without equilibria
2016
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 three systems experience hidden oscillations in sense of mathematical definition. While some of these hidden oscillations can be easily seen in natural physical experiments, the localization of others requires special efforts. peerReviewed
Computation of lock-in range for classic PLL with lead-lag filter and impulse signals
2016
For a classic PLL with square waveform signals and lead-lag filter for all possible parameters lock-in range is computed and corresponding diagrams are given. peerReviewed
Hidden Oscillations In The Closed-Loop Aircraft-Pilot System And Their Prevention
2016
The paper is devoted to studying and prevention of a special kind of oscillations-the Pilot Involved Oscillations (PIOs) which may appear in man-machine closed-loop dynamical systems. The PIO of categories II and III are defined as essentially non-linear unintended steady fluctuations of the piloted aircraft, generated due to pilot efforts to control the aircraft with a high precision. The main non-linear factor leading to the PIO is, generally, rate limitations of the aircraft control surfaces, resulting in a delay in the response of the aircraft to pilot commands. In many cases, these oscillations indicate presence of hidden, rather than self-excited attractors in the aircraft-pilot state…
Hidden attractors in dynamical systems
2016
Complex dynamical systems, ranging from the climate, ecosystems to financial markets and engineering applications typically have many coexisting attractors. This property of the system is called multistability. The final state, i.e., the attractor on which the multistable system evolves strongly depends on the initial conditions. Additionally, such systems are very sensitive towards noise and system parameters so a sudden shift to a contrasting regime may occur. To understand the dynamics of these systems one has to identify all possible attractors and their basins of attraction. Recently, it has been shown that multistability is connected with the occurrence of unpredictable attractors whi…
Computation of the lock-in ranges of phase-locked loops with PI filter
2016
In the present work the lock-in range of PLL-based circuits with proportionallyintegrating filter and sinusoidal phase-detector characteristics are studied. Considered circuits have sinusoidal phase detector characteristics. Analytical approach based on the methods of phase plane analysis is applied to estimate the lock-in ranges of the circuits under consideration. Obtained analytical results are compared with simulation results. peerReviewed
Computational Experiments with the Roots of Fibonacci-like Polynomials as a Window to Mathematics Research
2022
Fibonacci-like polynomials, the roots of which are responsible for a cyclic behavior of orbits of a second-order two-parametric difference equation, are considered. Using Maple and Wolfram Alpha, the location of the largest and the smallest roots responsible for the cycles of period p among the roots responsible for the cycles of periods 2kp (period-doubling) and kp (period-multiplying) has been determined. These purely computational results of experimental mathematics, made possible by the use of modern digital tools, can be used as a motivation for confirmation through not-yet-developed methods of formal mathematics. peerReviewed