Stability and bifurcation in a harmonic oscillator with delays (Q1771635)

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.