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N. V. Kuznetsov

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

At least 19 recordsLinked to original sources
arXiv2025

Divergence Method to Stability Study of Andronov-Vyshnegradsky Problem. Hidden Oscillations

The classical Andronov-Vyshnegradsky problem, which deals with locating regions of stability and oscillations in control systems with a Watt regulator, is solved using a divergence method for studying the stability of dynamic systems. This system is studied both with and without the self-regulation effect. The exact value of the hidden boundary of the global stability region is obtained. The stability criteria for a system with a Watt regulator are also presented in the context of the solvability of a linear matrix inequality. Computer modelling shows that the system exhibits hidden oscillations when the self-regulation effect is present and when it is not. The conditions for computing the hidden boundary of global stability are determined by three parameters in the Watt regulator model.

eess.SY↗
arXiv2022

Model-free prediction of multistability using echo state network

In the field of complex dynamics, multistable attractors have been gaining a significant attention due to its unpredictability in occurrence and extreme sensitivity to initial conditions. Co-existing attractors are abundant in diverse systems ranging from climate to finance, ecological to social systems. In this article, we investigate a data-driven approach to infer different dynamics of a multistable system using echo state network (ESN). We start with a parameter-aware reservoir and predict diverse dynamics for different parameter values. Interestingly, machine is able to reproduce the dynamics almost perfectly even at distant parameters which lie considerably far from the parameter values related to the training dynamics. In continuation, we can predict whole bifurcation diagram significant accuracy as well. We extend this study for exploring various dynamics of multistable attractors at unknown parameter value. While, we train the machine with the dynamics of only one attarctor at parameter $p$, it can capture the dynamics of co-existing attractor at a new parameter value $p+Δp$. Continuing the simulation for multiple set of initial conditions, we can identify the basins for different attractors. We generalize the results by applying the scheme on two distinct multistable systems.

nlin.AO↗
arXiv2022

Forecasting and control in overlapping generations model: chaos stabilization via artificial intelligence

Irregular, especially chaotic, behavior is often undesirable for economic processes because it presents challenges for predicting their dynamics. In this situation, control of such a process by its mathematical model can be used to suppress chaotic behavior and to transit the system from irregular to regular dynamics. In this paper, we have constructed an overlapping generations model with a control function. By applying evolutionary algorithms we showed that in the absence of control, both regular and irregular behavior (periodic and chaotic) could be observed in this model. We then used the synthesis of control by the Pyragas control method with two control parameters to solve the problem of controlling the irregular behavior of the model. We solved a number of optimization problems applying evolutionary algorithms to select control parameters in order to ensure stability of periodic orbits. We compared qualitative and quantitative characteristics of the model's dynamics before and after applying control and verified the results obtained using simulation. We thus demonstrated that artificial intelligence technologies (in particular, evolutionary algorithms) combined with the Pyragas control method are well suited for in-depth analysis and stabilization of irregular dynamics in the model considered in this paper.

physics.soc-ph↗
arXiv2021

Nonlinear analysis of charge-pump phase-locked loop: the hold-in and pull-in ranges

In this paper a fairly complete mathematical model of CP-PLL, which reliable enough to serve as a tool for credible analysis of dynamical properties of these circuits, is studied. We refine relevant mathematical definitions of the hold-in and pull-in ranges related to the local and global stability. Stability analysis of the steady state for the charge-pump phase locked loop is non-trivial: straight-forward linearization of available CP-PLL models may lead to incorrect conclusions, because the system is not smooth near the steady state and may experience overload. In this work necessary details for local stability analysis are presented and the hold-in range is computed. An upper estimate of the pull-in range is obtained via the analysis of limit cycles. The study provided an answer to Gardner's conjecture on the similarity of transient responses of CP-PLL and equivalent classical PLL and to conjectures on the infinite pull-in range of CP-PLL with proportionally-integrating filter.

eess.SP↗
arXiv2021

The birth of the global stability theory and the theory of hidden oscillations

The first mathematical problems of the global analysis of dynamical models can be traced back to the engineering problem of the Watt governor design. Engineering requirements and corresponding mathematical problems led to the fundamental discoveries in the global stability theory. Boundaries of global stability in the space of parameters are limited by the birth of oscillations. The excitation of oscillations from unstable equilibria can be easily analysed, while the revealing of oscillations not connected with equilibria is a chalfilenging task being studied in the theory of hidden oscillations. In this survey, a brief history of the first global stability criteria development and corresponding counterexamples with hidden oscillations are discussed.

math.DS↗
arXiv2021

Analytical and numerical study of the hidden boundary of practical stability: complex versus real Lorenz systems

This work presents the continuation of the recent article "The Lorenz system: hidden boundary of practical stability and the Lyapunov dimension", published in the Nonlinear Dynamics journal. In this work, in comparison with the results for classical real-valued Lorenz system (henceforward -- Lorenz system), the problem of analytical and numerical identification of the boundary of global stability for the complex-valued Lorenz system (henceforward -- complex Lorenz system) is studied. As in the case of the Lorenz system, to estimate the inner boundary of global stability the possibility of using the mathematical apparatus of Lyapunov functions (namely, the Barbashin-Krasovskii and LaSalle theorems) is demonstrated. For additional analysis of homoclinic bifurcations in complex Lorenz system a special analytical approach by Vladimirov is utilized. To outline the outer boundary of global stability and identify the so-called hidden boundary of global stability, possible birth of hidden attractors and transient chaotic sets is analyzed.

nlin.CD↗
arXiv2021

New adaptive synchronization algorithm for a general class of hyperchaotic complex-valued systems with unknown parameters and its application to secure communication

This report aims to study adaptive synchronization between a general class of hyperchaotic complex-valued systems with unknown parameters, which is motivated by extensive application areas of this topic in nonlinear sciences (e.g., secure communications, encryption techniques, etc.). Based on the complexity of hyperchaotic dynamical systems, which may be beneficial in secure communications, a scheme to achieve adaptive synchronization of a general class of hyperchaotic complex-valued systems with unknown parameters is proposed. To verify scheme's consistency, the control functions based on adaptive laws of parameters are derived analytically, and the related numerical simulations are performed. The complex-valued Rabinovich system describing the parametric excitation of waves in a magneto-active plasma is considered as an interesting example to study this kind of synchronization. A scheme for secure communication and improving the cryptosystem is proposed; the sketch is constructed to split the message and inject some bit of information signal into parameters modulation and the other bit into the transmitter system's states, which, in turn, complicates decryption task by intruders. Meanwhile, the information signal can be accurately retrieved at the receiver side by adaptive techniques and decryption function. Different types of encrypted messages are considered for testing the robustness of the proposed scheme (e.g., plain text and gray images with diverse scales of white Gaussian noise).

math.DS↗
arXiv2020

Time-delay control for stabilization of the Shapovalov mid-size firm model

Control and stabilization of irregular and unstable behavior of dynamic systems (including chaotic processes) are interdisciplinary problems of interest to a variety of scientific fields and applications. Using the control methods allows improvements in forecasting the dynamics of unstable economic processes and offers opportunities for governments, central banks, and other policy makers to modify the behaviour of the economic system to achieve its best performance. One effective method for control of chaos and computation of unstable periodic orbits (UPOs) is the unstable delay feedback control (UDFC) approach, suggested by K. Pyragas. This paper proposes the application of the Pyragas' method within framework of economic models. We consider this method through the example of the Shapovalov model, by describing the dynamics of a mid-size firm. The results demonstrate that suppressing chaos is capable in the Shapovalov model, using the UDFC method.

nlin.CD↗
arXiv2019

A lower-bound estimate of the Lyapunov dimension for the global attractor of the Lorenz system

In this short report, for the classical Lorenz attractor we demonstrate the applications of the Pyragas time-delayed feedback control technique and Leonov analytical method for the Lyapunov dimension estimation and verification of the Eden's conjecture. The problem of reliable numerical computation of the finite-time Lyapunov dimension along the trajectories over large time intervals is discussed.

nlin.CD↗
arXiv2018

Homoclinic Bifurcations of the Merging Strange Attractors in the Lorenz-like System

In this article we construct the parameter region where the existence of a homoclinic orbit to a zero equilibrium state of saddle type in the Lorenz-like system will be analytically proved in the case of a nonnegative saddle value. Then, for a qualitative description of the different types of homoclinic bifurcations, a numerical analysis of the detected parameter region is carried out to discover several new interesting bifurcation scenarios.

math.DS↗
arXiv2018

Numerical analysis of dynamical systems: unstable periodic orbits, hidden transient chaotic sets, hidden attractors, and finite-time Lyapunov dimension

In this article, on the example of the known low-order dynamical models, namely Lorenz, Rossler and Vallis systems, the difficulties of reliable numerical analysis of chaotic dynamical systems are discussed. For the Lorenz system, the problems of existence of hidden chaotic attractors and hidden transient chaotic sets and their numerical investigation are considered. The problems of the numerical characterization of a chaotic attractor by calculating finite-time time Lyapunov exponents and finite-time Lyapunov dimension along one trajectory are demonstrated using the example of computing unstable periodic orbits in the Rossler system. Using the example of the Vallis system describing the El Nino-Southern Oscillation it is demonstrated an analytical approach for localization of self-excited and hidden attractors, which allows to obtain the exact formulas or estimates of their Lyapunov dimensions.

nlin.CD↗
arXiv2018

PLL and Costas loop based carrier recovery circuits for 4QAM: non-linear analysis and simulation

Design of stable carrier recovery circuits are used in many applications: wireless digital communication, optical communication, microwave devices and other applications. Quadrature Phase-Shift Keying (QPSK, 4-QAM) is used as modulation technique in many of these applications, since QPSK provides double the data rate of classic Binary PSK (BPSK) modulation. Analysis of Costas loop is a hard task because of its non-linearity. In this work we consider two well-known modifications of 4QAM Costas loop circuits and discuss a new circuit proposed by J. Ladvanszky. MATLAB Simulink models of the circuits are provided and the noise analysis is performed.

eess.SP↗
arXiv2018

A note on finite-time Lyapunov dimension of the Rossler attractor

For the Rössler system we verify Eden's conjecture on the maximum of local Lyapunov dimension. We compute numerically finite-time local Lyapunov dimensions on the Rössler attractor and embedded unstable periodic orbits. The UPO computation is done by Pyragas time-delay feedback control technique.

nlin.CD↗
arXiv2018

On the Keldysh Problem of Flutter Suppression

This work is devoted to the Keldysh model of flutter suppression and rigorous approaches to its analysis. To solve the stabilization problem in the Keldysh model we use an analog of direct Lyapunov method for differential inclusions. The results obtained here are compared with the results of Keldysh obtained by the method of harmonic balance (describing function method), which is an approximate method for analyzing the existence of periodic solutions. The limitations of the use of describing function method for the study of systems with dry friction and stationary segment are demonstrated.

nlin.CD↗
arXiv2018

The Lyapunov dimension, convergency and entropy for a dynamical model of Chua memristor circuit

For the study of chaotic dynamics and dimension of attractors the concepts of the Lyapunov exponents was found useful and became widely spread. Such characteristics of chaotic behavior, as the Lyapunov dimension and the entropy rate, can be estimated via the Lyapunov exponents. In this work an analytical approach to the study of the Lyapunov dimension, convergency and entropy for a dynamical model of Chua memristor circuit is demonstrated.

math.DS↗
arXiv2017

Finite-time and exact Lyapunov dimension of the Henon map

This work is devoted to further consideration of the Henon map with negative values of the shrinking parameter and the study of transient oscillations, multistability, and possible existence of hidden attractors. The computation of the finite-time Lyapunov exponents by different algorithms is discussed. A new adaptive algorithm for the finite-time Lyapunov dimension computation in studying the dynamics of dimension is used. Analytical estimates of the Lyapunov dimension using the localization of attractors are given. A proof of the conjecture on the Lyapunov dimension of self-excited attractors and derivation of the exact Lyapunov dimension formula are revisited.

nlin.CD↗
arXiv2017

Hidden attractors in aircraft control systems with saturated inputs

In the paper, the control problem with limitations on the magnitude and rate of the control action in aircraft control systems, is studied. Existence of hidden oscillations in the case of actuator position and rate limitations is demonstrated by the examples of piloted aircraft pilot involved oscillations (PIO) phenomenon and the airfoil flutter suppression system.

eess.SY↗