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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
##### Keywords:
harmonic oscillator; delay; stability; characteristic equation
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