Existence of \(T/k\)-periodic solutions of a nonlinear nonautonomous system whose matrix has a multiple eigenvalue
type: Item
instance of: scholarly article
title: Existence of \(T/k\)-periodic solutions of a nonlinear nonautonomous system whose matrix has a multiple eigenvalue
zbMATH Open document ID: 1472.34081
DOI: 10.1134/S0001434621030238
author: Yanyan Li
published in: Mathematical Notes
publication date: 2021-07-08T00:00:00Z
review text: The author considers the \(n\)-dimensional system of differential equations of the following form \[ \dot{Y} = AY + BF(\sigma) + Kf(t). \] It is assumed that the matrix \(A\) has real nonzero eigenvalues, among which there are at least one positive and one multiple eigenvalues, the vectors \(B\) and \(K\) are nonzero, and the function \(F(\sigma),\)\,\(\sigma = (C,Y)\) describes a nonideal two-position relay with two output values. The perturbation function \(f(t)\) is \(T\)-periodic with \(T = 2\pi/\omega,\,\omega >0\) of the form \[ f(t) = f_0 +f_1\sin(\omega t + \varphi_1) + f_2\sin (2\omega t + \varphi_2). \] Necessary conditions for the existence of a \(T/k\)-periodic solution of the system with \(k \in \mathbb{N}\) having two switching points in the phase space are studied. An existence theorem for a \(T\)-periodic solution is proved. A numerical example is presented.
Mathematics Subject Classification ID: 34C25
Mathematics Subject Classification ID: 37C60
zbMATH DE Number: 7369730
zbMATH Keywords: system of ordinary differential equations
zbMATH Keywords: relay nonlinearity with hysteresis
zbMATH Keywords: periodic perturbation function
zbMATH Keywords: multiple eigenvalue
zbMATH Keywords: canonical transformation
zbMATH Keywords: Jordan matrix
zbMATH Keywords: periodic solution
zbMATH Keywords: switching points
reviewed by: Valerii V. Obukhovskij
MaRDI profile type: Publication
cites work: Q4085573
cites work: Q3921659
cites work: Q3994579
cites work: Existence and computation of stable modes in relay systems
cites work: Dynamics of controlled systems described by parabolic equations with hysteresis nonlinearities
cites work: Mathematical Models for Hysteresis
cites work: Differential models of hysteresis
cites work: Hysteresis and phase transitions
cites work: On necessary conditions for the existence of periodic solutions in a dynamic system with discontinuous nonlinearity and external periodic influence
cites work: Q4966482
cites work: Ten issues about hysteresis
cites work: Identification of extended Hammerstein systems with hysteresis-type input nonlinearities described by Preisach model
cites work: On existence conditions for a two-point oscillating periodic solution in an non-autonomous relay system with a Hurwitz matrix
cites work: Existence of periodic solutions to automatic control system with relay nonlinearity and sinusoidal external influence
cites work: Q5352115
cites work: On uniqueness and properties of periodic solution of second-order nonautonomous system with discontinuous nonlinearity
cites work: Periodic solutions of a system of differential equations with hysteresis nonlinearity in the presence of eigenvalue zero
cites work: Existence of periodic modes in automatic control system with a three-position relay
full work available at URL: s0001434621030238
OpenAlex ID: W3177780917
Recommended article: Q4869105
Recommended article: Existence and multiplicity of periodic solution for non-autonomous second-order systems with linear nonlinearity
Recommended article: Existence and multiplicity of periodic solutions for nonautonomous second order systems
Recommended article: Q3124100
Recommended article: ON THE MULTIPLICITY OF SOLUTIONS FOR NON-LINEAR PERIODIC PROBLEMS WITH THE NON-LINEARITY CROSSING SEVERAL EIGENVALUES
Recommended article: Q4878052
Recommended article: Existence of Solutions and of Multiple Solutions for Nonlinear Nonsmooth Periodic Systems
Recommended article: Q5368378
Recommended article: Existence of positive periodic solutions for \(n\)-dimensional nonautonomous system
Recommended article: Q4879744
description: scientific article; zbMATH DE number 7369730