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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 $n\ge 2$, $c$ 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 ODE 34C20 Transformation and reduction of ODE and systems, normal forms
Full Text:
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