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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.

MSC:
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
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