Calculating the Poincaré map for two-dimensional periodic systems and Riccati equations
type: Item
instance of: scholarly article
title: Calculating the Poincaré map for two-dimensional periodic systems and Riccati equations
zbMATH Open document ID: 1492.34040
DOI: 10.1134/S0012266121100062
author: V. I. Mironenko
author: V. V. Mironenko
published in: Differential Equations
publication date: 2022-01-05T00:00:00Z
review text: For the study of periodic solutions of non-autonomous periodic systems \(dx/dt=X(t,x)\), A.I. Mironov introduced in 1984 the so-called reflecting function which can be used to represent the corresponding Poincaré map. In the present paper, the authors consider two-dimensional linear non-autonomous systems \[ \frac{dx}{dt} = P(t)x. \tag{1} \] They derive two groups of conditions on the entries of the matrix \(P\) such that the reflecting function can be determined explicitely. In case that \(P\) is periodic, the corresponding Poincaré map can also be determined explicitely, which yields analytic conditions for the existence of periodic solutions of (1). Additionally, the authors show that the same approach can be applied to the Riccati equation \[ \frac{dx}{dt} = a(t)+b(t)x + c(t)x^2. \]
Mathematics Subject Classification ID: 34C25
Mathematics Subject Classification ID: 34A30
Mathematics Subject Classification ID: 34B30
Mathematics Subject Classification ID: 37C60
zbMATH DE Number: 7452233
zbMATH Keywords: reflecting function
zbMATH Keywords: two-dimensional non-autonomous systems
zbMATH Keywords: Riccati equation
zbMATH Keywords: Poincaré map
zbMATH Keywords: periodic solutions
Wikidata QID: Q115249180
reviewed by: Klaus R. Schneider
MaRDI profile type: Publication
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full work available at URL: s0012266121100062
OpenAlex ID: W4200084425
description: scientific article; zbMATH DE number 7452233