The stability of the equilibrium for a perturbed asymmetric oscillator

Abstract

In this paper we will derive some stability criteria for theequilibrium of a perturbed asymmetric oscillator$\ddot x + a^+ x^+ - a^-$$ x^-$ $+ b(t)x^2+r(t,x)=0,$where $a^+,a^-$ are two different positive numbers, $b(t)$ is a$2\pi$-periodic function, and the remaining term $r(t,x)$ is$2\pi$-periodic with respect to the time $t$ and dominated by thepower $x^3$ in a neighborhood of the equilibrium $x=0$.