Periodic perturbations of a nonlinear oscillator (Q304629)

In this paper, small time periodic perturbations of an oscillator with a power-law odd restoring force is studied. The equation of the considered oscillator is: \[ \ddot{x}+x^{2n-1}=X(x,\dot{x},t,\varepsilon), \quad n \geq 2, \] where the time-periodic perturbations \(X(x,\dot{x},t,\varepsilon)\) are depending upon the small parameter \(\varepsilon \geq 0\). Two problems are treated: a) the Lyapunov stability of the equilibrium for \(\varepsilon=0\) and b) the bifurcation of an invariant two dimensional torus from the equilibrium for \(\varepsilon >0\). A focal quantity and a bifurcation equation that describes the character of the stability and branching of the equilibrium are constructed.