## The Lyapunov stability for the linear and nonlinear damped oscillator with time-periodic parameters.(English)Zbl 1204.34073

Summary: Let $$a(t),b(t)$$ be continuous $$T$$-periodic functions with $$\int^T_0 b(t)\,dt = 0$$. We establish a stability criterion for the linear damped oscillator
$x''+b(t)x'+a(t)x=0.$
Moreover, based on the computation of the corresponding Birkhoff normal forms, we present a sufficient condition for the stability of the equilibrium of the nonlinear damped oscillator
$x''+b(t)x'+a(t)x+c(t)x^{2n-1}+e(t,x)=0,$
where $$n\geq 2$$, $$c$$ is a continuous $$T$$-periodic function, $$e(t,x)$$ is continuous $$T$$-periodic in $$t$$ and dominated by the power $$x^{2n}$$ in a neighborhood of $$x=0$$.

### MSC:

 34D20 Stability of solutions to ordinary differential equations 34C20 Transformation and reduction of ordinary differential equations and systems, normal forms
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### References:

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