Oscillations in a second-order discontinuous system with delay

Eugenii Shustin; Emilia Fridman; Leonid Fridman

doi:10.3934/dcds.2003.9.339

Article Contents

2003, Volume 9, Issue 2: 339-358. Doi: 10.3934/dcds.2003.9.339

This issue Previous Article Quasi-invariant attractors of piecewise isometric systems Next Article Asymptotic measures and distributions of Birkhoff averages with respect to Lebesgue measure

Oscillations in a second-order discontinuous system with delay

1.
School of Mathematical Sciences, Tel Aviv University, Ramat Aviv, 69978 Tel Aviv
2.
Department of Electrical Engineering & Systems, Tel Aviv University, Ramat Aviv, 69978 Tel Aviv
3.
Division of Postgraduate and Investigation, Chihuahua Institute of Technology, Chihuahua, Chi, C.P. 31160

Received: February 28, 2001

Revised: January 31, 2002

Published: February 2003

Abstract Related Papers Cited by

Abstract

We consider the equation

$\alpha x''(t)=-x'(t)+F(x(t),t)-$sign$x(t-h),\quad\alpha=$const$>0,\ $ $h=$const$>0,$

which is a model for a scalar system with a discontinuous negative delayed feedback, and study the dynamics of oscillations with emphasis on the existence, frequency and stability of periodic oscillations. Our main conclusion is that, in the autonomous case $F(x,t)\equiv F(x)$, for $|F(x)|<1$, there are periodic solutions with different frequencies of oscillations, though only slowly-oscillating solutions are (orbitally) stable. Under extra conditions we show the uniqueness of a periodic slowly-oscillating solution. We also give a criterion for the existence of bounded oscillations in the case of unbounded function $F(x,t)$. Our approach consists basically in reducing the problem to the study of dynamics of some discrete scalar system.

Keywords:

Mathematics Subject Classification: 34K11, 34K13, 34K35.

Citation:

Article Metrics

HTML views() PDF downloads(358) Cited by(0)

Oscillations in a second-order discontinuous system with delay

Abstract

Access History

Article Metrics

Access History

Other Articles By Authors

Catalog

Oscillations in a second-order discontinuous system with delay

Abstract

Access History

SHARE

Article Metrics

Access History

Other Articles By Authors

Catalog

Export File

Citation

Format

Content