We study the bifurcation of an oscillator whose restoring force depends on the velocity of motion under periodic perturbations. Separation of variables is used to derive a bifurcation equation. To each positive root of this equation, there corresponds an invariant twodimensional torus (a closed trajectory in the case of a time-independent perturbation) shrinking to the equilibrium position as the small parameter tends to zero. The proofs use methods of the Krylov-Bogolyubov theory for the case of periodic perturbations or the implicit function theorem for the case of time-independent.