Periodic Orbits and Scaling Laws for a Driven Damped Quartic Oscillator

In this paper we investigate the conditions under which periodic solutions of the nonlinear oscillator ẍ + x3 = 0 persist when the differential equation is perturbed so as to become ẍ + x3 + εx3 cos t + γẋ = 0. We conjecture that for any periodic orbit, characterized by its frequency ω, there exists a threshold for the damping coefficient γ, above which the orbit disappears, and that this threshold is infinitesimal in the perturbation parameter, with integer order depending on the frequency ω. Some rigorous analytical results toward the proof of these conjectures are provided. Moreover the relative size and shape of the basins of attraction of the existing stable periodic orbits are investigated numerically, giving further support to the validity of the conjectures.