Local stability, Hopf and resonant codimension-two bifurcation in a harmonic oscillator with two time delays

Authors
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.