Bifurcation of an Oscillatory Mode under a Periodic Perturbation of a Special Oscillator

Abstract

We study a bifurcation from the zero solution of the differential equation ẍ + xp/q = 0, where p > q > 1 are odd coprime numbers, under periodic (in particular, time-invariant) perturbations depending on a small positive parameter e. The motion separation method is used to derive the bifurcation equation. To each positive root of this equation, there corresponds an invariant two-dimensional torus (a closed trajectory in the time-invariant case) shrinking to the equilibrium position x = 0 as e → 0. The proofs use methods of the Krylov-Bogolyubov theory to study time-periodic perturbations and the implicit function theorem in the case of time-invari ant perturbations.