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T. N. Mokaev

Publications and source records attributed to T. N. Mokaev.

16 recordsLinked to original sources
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

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

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↗
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

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.

nlin.CD↗
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.

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.

nlin.CD↗
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↗
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↗