Stability of the equilibrium of an oscillator with an infinitely high natural oscillation frequency (Q2282755)

This paper is concerned with the question of the stability of the zero solution of the following differential equation \[ \ddot{x}+x^{\frac{p}{q}}=X\left( x,\dot{x},t\right) \] where \(\frac{p}{q}\) is an irreducible fraction, \(p\) and \(q\) are odd numbers, and \(p<q\); \(X\left(x,y,t\right)\) is a real analytical function of the variables \(x\) and \(y\) in a neighborhood of the point \(\left(x=0,y=0\right)\), which is continuous and periodic in \(t\) with period \(T\); \(X\left(0,0,t\right)=0\).