Phase asymptotic semiflows, Poincaré's condition, and the existence of stable limit cycles
type: Item
instance of: scholarly article
title: Phase asymptotic semiflows, Poincaré's condition, and the existence of stable limit cycles
zbMATH Open document ID: 0847.34059
author: Michael Y. Li
author: James S. Muldowney
published in: Journal of Differential Equations
publication date: 1996-10-08T00:00:00Z
review text: The paper deals with phase asymptotic semiflows. The authors prove that any stable orbit at which the semiflow is phase asymptotic limits to a stable periodic orbit. The basic results (Poincaré-Bendixson-type theorems) are introduced in Section 4 of the paper. Let \(\varphi= \varphi (t)\), \(t\in {\mathbb{R}}\) be the solution of the differential equation \(\dot x= f(x)\), where \(f= f(x)\) is a \(C^1\)-smooth function, \(\Gamma_+= \{\varphi (t): t\geq 0\}\), \(\Omega\) is the \(\omega\)-limit set of \(\Gamma_+\), and let the differential equation \[ \dot z= {{\partial f^2} \over {\partial x}} (\varphi (t))z \] be uniformly asymptotically stable, where \( {{\partial f^2} \over {\partial x}} (x)\) is the second additive compound of \({{\partial f} \over {\partial x}} (x)\). Then if \(\Omega\) contains no equilibrium, then it is a periodic orbit; if \(\Omega\) contains an equilibrium \(x_0\), then either \(x_0\) is asymptotically stable or it has an \((n-1)\)-dimensional stable manifold together with a 1-dimensional unstable manifold; if \(\Omega\) contains an equilibrium \(x_0\) and the differential equation \(\dot y= {{\partial f} \over {\partial x}} y\) is uniformly stable, then \(\Omega= \{x_0\}\).
Mathematics Subject Classification ID: 34D20
Mathematics Subject Classification ID: 34C05
zbMATH DE Number: 842914
zbMATH Keywords: Poincaré-Bendixson-type theorems
zbMATH Keywords: phase asymptotic semiflows
reviewed by: S. I. Nenov
MaRDI profile type: Publication
OpenAlex ID: W1970889639
DOI: 10.1006/JDEQ.1996.0018
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description: scientific article; zbMATH DE number 842914