The existence of periodic orbits for nonlinear impulsive differential systems
type: Item
instance of: scholarly article
title: The existence of periodic orbits for nonlinear impulsive differential systems
zbMATH Open document ID: 0940.35206
DOI: 10.1016/S1007-5704(99)90056-4
author: Xilin Fu
author: Yansheng Liu
author: Jiangang Qi
published in: Communications in Nonlinear Science and Numerical Simulation
publication date: 2000-05-18T00:00:00Z
review text: The authors consider a nonlinear autonomous impulsive differential system, \(x'= f(x)\), \(t\neq t_i\) \((t>0)\) under impulsive condition \(x(t_i+ 0)- x(t_i- 0)= I_i(x)\), where the impulses act in a set of fixed points \(\{t_i\}^{+\infty}_{i= 0}\), \(0= t_0< t_1<\cdots< t_n<\cdots\) (\(t_n\to+\infty\) as \(n\to+\infty\)). The functions \(f,I_i: \Omega\to \mathbb{R}\) \((\Omega\subset\mathbb{R})\) are continuous and \(x+ I_i(x)\in\Omega\) for \(x\in\Omega\) and every \(i\). The initial value is \(x(0)= x_0\). The necessary and sufficient condition for existence of periodic solutions is investigated. Stability properties are discussed as well. The main result is that the considered problem possesses a periodic orbit if \(G(x)= \tau+ \int^{h(x)}_x f^{-1}(s) ds\) has a zero point in \((0,+\infty)\) (\(h\in C(\mathbb{R}^+)\) is increasing in \(\mathbb{R}^+\) and \(h(0)= 0\)); \(\tau\) is the minimal positive period of any periodic orbit of the considered impulsive problem.
Mathematics Subject Classification ID: 35R12
Mathematics Subject Classification ID: 35B10
Mathematics Subject Classification ID: 35K45
zbMATH DE Number: 1359981
zbMATH Keywords: impulsive differential equations
zbMATH Keywords: periodic solutions
zbMATH Keywords: stability criteria
reviewed by: Dimitar A. Kolev
MaRDI profile type: Publication
cites work: Stability criteria for impulsive differential equations in terms of two measures
cites work: Uniform boundedness and stability criteria in terms of two measures for impulsive integro-differential equations
cites work: Q4312240
cites work: Q3995625
description: scientific article; zbMATH DE number 1359981