Abstract:

We generalized the stability criteria for the equilibrium of the periodic system to those for that of quasiperiodic system, applying the method of main integration. Concretely, we showed the stability for the equilibrium x=x′=0 of the quasiperiodic second order differential equation x″+h(t)x′+a(t)x²ⁿ⁺¹+e(t,x)=0, n≥1, where h(t),a(t),e(t,x) are quasi-periodic coefficients, whose frequency vectors meet the requirements proposed by Diophantine. And moreover, |e(t,x)|=O(x²ⁿ⁺²) near x=x′=0. The results we obtained also imply that, under some conditions, the equilibrium of the quasiperiodic oscillator with damping changing sign can still be stable.

Key words: quasiperiodic, Diophantine condition, stability of the equilibrium