A necessary and sufficient condition for the stability of the equilibrium (Q1184664)

The authors obtain a necessary and sufficient condition for the stability of the zero solution of the system (1) \(\ddot x=-xf(x)\), (2) \(\ddot y=- yg(x)\) with \(f(0)>0\), \(g(0)>0\), exploiting the fact that equation (1) is independent of \(y\) and its origin is a center, i.e., equation (2) is in fact a family of Hill's equations \(\ddot y=-yg(x(t;x_ 0))\) with periodic functions \(x(t;x_ 0)\) parametrized by \(x_ 0\).