{"title":"On One Type of Oscillatory Solutions of a Nonautonomous System with Relay Hysteresis","authors":"V. V. Yevstafyeva","doi":"10.1134/s0001434624050080","DOIUrl":null,"url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p> We consider an <span>\\(n\\)</span>-dimensional system of first-order ordinary differential equations with a constant matrix having real, simple, and nonzero eigenvalues, with a discontinuous nonlinearity of two-position relay type with positive hysteresis and a continuous bounded perturbation function. We study continuous two-point oscillatory solutions with a certain period for the representative point to be returned to the switching hyperplane in the state space. When solving the Cauchy problem with initial condition at the switching point, we use the fitting method. We construct a system of transcendental equations for the switching instants and points. We prove a criterion for the existence and uniqueness of a solution with some fixed return period. For a system in the canonical form with diagonal matrix and with feedback vector of a special form, we obtain conditions for the solvability of a system of transcendental equations for the first switching instant for a given return period and formulas for the switching points. For a three-dimensional system, we give a numerical example to illustrate the theoretical results. </p>","PeriodicalId":18294,"journal":{"name":"Mathematical Notes","volume":null,"pages":null},"PeriodicalIF":0.6000,"publicationDate":"2024-07-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Mathematical Notes","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1134/s0001434624050080","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
We consider an \(n\)-dimensional system of first-order ordinary differential equations with a constant matrix having real, simple, and nonzero eigenvalues, with a discontinuous nonlinearity of two-position relay type with positive hysteresis and a continuous bounded perturbation function. We study continuous two-point oscillatory solutions with a certain period for the representative point to be returned to the switching hyperplane in the state space. When solving the Cauchy problem with initial condition at the switching point, we use the fitting method. We construct a system of transcendental equations for the switching instants and points. We prove a criterion for the existence and uniqueness of a solution with some fixed return period. For a system in the canonical form with diagonal matrix and with feedback vector of a special form, we obtain conditions for the solvability of a system of transcendental equations for the first switching instant for a given return period and formulas for the switching points. For a three-dimensional system, we give a numerical example to illustrate the theoretical results.
期刊介绍：
Mathematical Notes is a journal that publishes research papers and review articles in modern algebra, geometry and number theory, functional analysis, logic, set and measure theory, topology, probability and stochastics, differential and noncommutative geometry, operator and group theory, asymptotic and approximation methods, mathematical finance, linear and nonlinear equations, ergodic and spectral theory, operator algebras, and other related theoretical fields. It also presents rigorous results in mathematical physics.
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