Citation

Xf. Liao et Gr. Chen, Local stability, Hopf and resonant codimension-two bifurcation in a harmonic oscillator with two time delays, INT J B CH, 11(8), 2001, pp. 2105-2121

Citations number

26

Language

INGLESE

art.tipo

Article

Categorie Soggetti

Multidisciplinary

Journal title

INTERNATIONAL JOURNAL OF BIFURCATION AND CHAOS

ISSN journal

0218-1274
→ ACNP

Volume

11

Issue

8

Year of publication

2001

Pages

2105 - 2121

Database

ISI

SICI code

0218-1274(200108)11:8<2105:LSHARC>2.0.ZU;2-D

Abstract

A harmonic oscillator with two discrete time delays is considered. The loca
l stability of the zero solution of this equation is investigated by analyz
ing the corresponding transcendental characteristic equation of its lineari
zed equation and employing the Nyquist criterion. Some general stability cr
iteria involving the delays and the system parameters are derived. By choos
ing one of the delays as a bifurcation parameter, the model is found to und
ergo a sequence of Hopf bifurcation. The direction and stability of the bif
urcating periodic solutions are determined by using the normal form theory
and the center manifold theorem. Resonant codimension-two bifurcation is al
so found to occur in this model. A complete description is given to the loc
ation of points in the parameter space at which the transcendental characte
ristic equation possesses two pairs of pure imaginary roots, +/-i omega (1)
, +/-i omega (2) with omega (1) : omega (2) = m : n, where m and n are posi
tive integers. Some numerical examples are finally given for justifying the
theoretical results.