Bifurcation of the Equilibrium of an Oscillator with a Velocity-Dependent Restoring Force under Periodic Perturbations

Abstract

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 perturbations.