arXiv subjects

Alexander N. Pchelintsev

Publications and source records attributed to Alexander N. Pchelintsev.

5 recordsLinked to original sources

arXiv2022

On a High-Precision Method for Studying Attractors of Dynamical Systems and Systems of Explosive Type

The author of this article considers a numerical method that uses high-precision calculations to construct approximations to attractors of dynamical systems of chaotic type with a quadratic right-hand side, as well as to find the vertical asymptotes of solutions of systems of explosive type. A special case of such systems is the population explosion model. A theorem on the existence of asymptotes is proved. The extension of the numerical method for piecewise smooth systems is described using the Chua system as an example, as well as systems with hysteresis.

nlin.CD↗

arXiv2021

On the Poisson Stability to Study a Fourth-Order Dynamical System with Quadratic Nonlinearities

This article discusses the search procedure for the Poincaré recurrences to classify solutions on an attractor of a fourth-order nonlinear dynamical system using a previously developed high-precision numerical method. For the resulting limiting solution, the Lyapunov exponents are calculated using the modified Benettin's algorithm to study the stability of the found regime and confirm the type of attractor.

math.DS↗

arXiv2021

A Numerical-Analytical Method for Constructing Periodic Solutions of the Lorenz System

This article describes a method for constructing approximations to periodic solutions of dynamic Lorenz system with classical values of the system parameters. The author obtained a system of nonlinear algebraic equations in general form concerning of the cyclic frequency, constant terms and amplitudes of harmonics that make up harmonic approximations to the desired solutions. The initial approximation for the Newton method is selected, which converges to a solution describing a periodic solution different from the equilibrium position. The results of a computational experiment are presented. The results are verified using high-precision calculations.

math.NA↗

arXiv2020

An Accurate Numerical Method and Algorithm for Constructing Solutions of Chaotic Systems

In various fields of natural science, the chaotic systems of differential equations are considered more than 50 years. The correct prediction of the behaviour of solutions of dynamical model equations is important in understanding of evolution process and reduce uncertainty. However, often used numerical methods are unable to do it on large time segments. In this article, the author considers the modern numerical method and algorithm for constructing solutions of chaotic systems on the example of tumor growth model. Also a modification of Benettin's algorithm presents for calculation of Lyapunov exponents.

math.DS↗

arXiv2019

The harmonic balance method for finding approximate periodic solutions of the Lorenz system

We consider the harmonic balance method for finding approximate periodic solutions of the Lorenz system. When developing software that implements the described method, the math package Maxima was chosen. The drawbacks of symbolic calculations for obtaining a system of nonlinear algebraic equations with respect to the cyclic frequency, free terms and amplitudes of the harmonics, that make up the desired solution, are shown. To speed up the calculations, this system was obtained in a general form for the first time. The results of the computational experiment are given: the coefficients of trigonometric polynomials approximating the found periodic solution, the initial condition, and the cycle period. The results obtained were verified using a high-precision method of numerical integration based on the power series method and described earlier in the articles of the authors.

math.DS↗