Stability of forced nonlinear oscillators via Poincaré map

Abstract

The behavior of nonlinear oscillators x(t) driven by a periodic external force is completely determined by the corresponding Poincaré map, which loses stability only in certain well-known ways. These translate into different classes of perturbations ξ(t) of x(t) that must be considered. By choosing simple representatives in each class, the stability of approximate solutions can be studied analytically. The Duffing equation is considered as an example. An extra island of stability is predicted for a range of driving forces and this is confirmed by numerical computation.