The stability of the equilibrium for a perturbed asymmetric oscillator (Q5920418)

The stability of the equilibrium of the perturbed differential equation \[ \ddot{x}(t) + a^{+} x^{+}(t) - a^{-}x^{-}(t) + p(t,x(t)) = 0,\quad t\in\mathbb{R}, \] is studied, where \(x^{+}(t):=\max(x(t),0)\) and \(x^{-}(t):=\max(-x(t),0)\) and \(a^{+},a^{-}>0\). These equations arise, e.g., when modelling one-sided springs. The perturbation \(p(t,x)\) is assumed to be periodic in \(t\) and has the form \(p(t,x)=b(t)x^2 + r(t,x)\), where \(r(t,0)=0\), \(\partial_x r(t,0)=0\), and \(\partial_x^2 r(t,0)\equiv 0\). In the ``resonant'' case, i.e. \(1/\sqrt{a^{+}} + 1/\sqrt{a^{-}} \in \mathbb{Q}\), conditions for stability as well as for instability of the equilibrium are formulated. These conditions involve the solution of the unperturbed differential equation. In the nonresonant case conditions for the stability of the equilibrium are given, in this case the conditions only involve properties of the perturbation. This paper is a newer version of [\textit{Xiong Li}, Commun. Pure Appl. Anal. 5, No.~3, 515--528 (2006; Zbl 1147.34041), reviewed below] with some minor corrections.