Characteristic equation for autonomous planar half-linear differential systems
type: Item
instance of: scholarly article
title: Characteristic equation for autonomous planar half-linear differential systems
zbMATH Open document ID: 1389.34165
published in: Acta Mathematica Hungarica
publication date: 2017-11-10T00:00:00Z
review text: The authors investigate the planar half-linear differential system \[ x'=a(t)x+b(t)\phi_{p^\ast}(y),\, y'=c(t)\phi_p(x)+d(t)y,\leqno(S) \] where \(a\), \(b\), \(c\), \(d\in C(\mathbb R)\), \(p\), \(p^\ast>1\) with \(\frac{1}{p}+\frac{1}{p^\ast}=1\), \(\phi_q(s)=| s| ^{q-2}s\) if \(s\not=0\), and \(\phi_q(s)=0\) if \(s=0\), for \(q>1\), and the corresponding autonomous system \((S_0)\) with \(a\), \(b\), \(c\), \(d\) real constants. First they introduce and study the characteristic equation of \((S_0)\). Then, by using the generalized Prüfer transformation, they prove the existence and uniqueness of solutions to \((S)\) with the initial conditions \((x(t_0),y(t_0))=(x_0,y_0)\). A detailed asymptotic behavior of solutions for \((S_0)\) is also established by roots of the characteristic equation, and a classification of the origin \((0,0)\) for \((S_0)\) is finally addressed.
Mathematics Subject Classification ID: 34D05
Mathematics Subject Classification ID: 34A12
Mathematics Subject Classification ID: 34C05
Mathematics Subject Classification ID: 34D20
zbMATH DE Number: 6806136
zbMATH Keywords: half-linear system
zbMATH Keywords: characteristic equation
zbMATH Keywords: eigenvalue
zbMATH Keywords: asymptotic behavior
zbMATH Keywords: generalized Prüfer transformation
Wikidata QID: Q115384312
author: Masakazu Onitsuka
author: Satoshi Tanaka
reviewed by: Rodica Luca
MaRDI profile type: Publication
full work available at URL: s10474-017-0722-6
OpenAlex ID: W2614830031
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DOI: 10.1007/S10474-017-0722-6
description: scientific article; zbMATH DE number 6806136