zbMATH — the first resource for mathematics

Stability and bifurcation in a harmonic oscillator with delays. (English) Zbl 1078.34050
The subject of the paper is the harmonic oscillator with delays \[ m\ddot x (t) + c \dot x(t) + k x(t) = s_1 f(x(t-\tau_1)) + s_2 f(\dot x(t-\tau_2)), \] where \(m>0\), \(c\geq 0\), and \(k\geq 0\) are constants, \(\tau_1, \tau_2>0\) are constant delays, and \(s_1,s_2\) feedback parameters. The function \(f\) is assumed to satisfy \(f\in C^4(\mathbb{R})\), \(f(0)=0\), and \(uf(u)<0\) for \(u\neq 0\). Under these conditions, zero is the unique equilibrium of the system.
By considering the characteristic equation, the authors provide a detailed stability analysis of the equilibrium. In particular, they give conditions for the Hopf bifurcation to occur. Using the normal form theory, the direction of the Hopf bifurcation is studied.

34K18 Bifurcation theory of functional-differential equations
34K20 Stability theory of functional-differential equations
34K60 Qualitative investigation and simulation of models involving functional-differential equations
Full Text: DOI
[1] Chow, S.-N.; Hale, J., Methods of bifurcation theory, (1982), Springer-Verlag New York
[2] Cooke, K.L.; Grossman, Z., Discrete delay, distributed delay and stability switches, J. math. anal. appl., 86, 592-627, (1982) · Zbl 0492.34064
[3] Faria, T., Stability and bifurcation for a delayed predator – prey model and the effect of diffusion, J. math. anal. appl., 254, 433-463, (2001) · Zbl 0973.35034
[4] Faria, T., On a planar system modelling a neuron network with memory, J. differen. equat., 168, 129-149, (2000) · Zbl 0961.92002
[5] Faria, T.; Magalhães, L.T., Normal form for retarded functional differential equations and applications to bogdanov – akens singularity, J. differen. equat., 122, 201-224, (1995) · Zbl 0836.34069
[6] Faria, T.; Magalhães, L.T., Normal form for retarded functional differential equations with parameters and applications to Hopf bifurcation, J. differen. equat., 122, 181-200, (1995) · Zbl 0836.34068
[7] Hale, J.; Magalhães, L.T.; Oliva, W.M., Dynamics in infinite dimensions, Applied mathematical sciences, vol. 47, (2002), Springer New York
[8] Hale, J.; Verduyn Lunel, S., Introduction to functional differential equations, (1993), Springer New York · Zbl 0787.34002
[9] Kuang, Y., Delay differential equations with applications in population dynamics, (1993), Academic Press New York · Zbl 0777.34002
[10] Liao, X.; Chen, G., Local stability, Hopf and resonant codimension-two bifurcation in a harmonic oscillator with two time delays, Int. J. bifurcat. chaos, 11, 2105-2121, (2001) · Zbl 1091.70502
[11] Meng, X.; Wei, J., Stability and bifurcation of mutual system with time delay, Chaos, solitons & fractals, 21, 729-740, (2004) · Zbl 1048.34122
[12] Wei, J.; Ruan, S., Stability and bifurcation in a neural network model with two delays, Physica D, 130, 255-272, (1999) · Zbl 1066.34511
[13] Xiao, D.; Ruan, S., Multiple bifurcations in a delayed predator – prey system with nonmonotonic functional response, J. differen. equat., 176, 494-510, (2001) · Zbl 1003.34064
[14] Yuan, S.; Han, M., Bifurcation analysis of a chemostat model with two distributed delays, Chaos, solitons & fractals, 20, 995-1004, (2004) · Zbl 1059.34059
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.