Selection rules for periodic orbits and scaling laws for a driven damped quartic oscillator

Abstract

In this paper we investigate the conditions under which periodic solutions of the nonlinear oscillator x ¨ + x 3 = 0 persist when the differential equation is perturbed so as to become x ¨ + x 3 + ε x 3 cos t + γ x ˙ = 0 . For any frequency ω , there exists a threshold for the damping coefficient γ , above which there is no periodic orbit with period 2 π / ω . We conjecture that this threshold is infinitesimal in the perturbation parameter, with integer order depending on the frequency ω . Some rigorous analytical results towards the proof of this conjecture are given: these results would provide a complete proof if we could rule out the possibility that other periodic solutions arise besides subharmonic solutions. Moreover the relative size and shape of the basins of attraction of the existing stable periodic orbits are investigated numerically, showing that all attractors different from the origin are subharmonic solutions and hence giving further support to the validity of the conjecture. The method we use is different from those usually applied in bifurcation theory, such as Mel ′ nikov's method or that of Chow and Hale, and allows us to investigate situations in which the non-degeneracy assumptions on the perturbation are violated.