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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 \(n\geq 2\), \(c\) 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 \(x=0\).

\[ x''+b(t)x'+a(t)x=0. \]

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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\textit{J. Chu} and \textit{T. Xia}, Abstr. Appl. Anal. 2010, Article ID 286040, 12 p. (2010; Zbl 1204.34073)

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