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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 0 T 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 n2, 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:
34D20Stability of ODE
34C20Transformation and reduction of ODE and systems, normal forms