An infinite class of periodic solutions of periodically perturbed Duffing equations at resonance

Abstract

In this paper, by using a generalized form of the Poincaré-Birkhoff Theorem, we demonstrate that the Duffing equation \[ d 2 x d t 2 + g ( x ) = p ( t ) ( ≡ p ( t + 2 π ) ) \frac {{{d^2}x}} {{d{t^2}}} + g(x) = p(t)\quad ( \equiv p(t + 2\pi )) \] may also admit an infinite number of 2 π 2\pi -periodic solutions even in a resonance case.