Applications of the Poincaré–Birkhoff theorem to impulsive Duffing equations at resonance

Many dynamical systems possess an impulsive dynamical behavior due to abrupt changes at certain instants during the evolution process. The mathematical description of these phenomena leads to impulsive differential equations. In this paper, we present a new approach via the well-known Poincaré–Birkhoff theorem to obtain the existence of periodic solutions to impulsive problems. We consider an impulsive Duffing equation, and find the possibility of applying a generalized form of the Poincaré–Birkhoff theorem due to Ding to construct infinitely many periodic solutions of the impulsive Duffing equation even in a resonance case.