Periodic solutions of planar Hamiltonian systems with asymmetric nonlinearities

Abstract

Abstract In this paper, we look for periodic solutions of planar Hamiltonian systems { x ′ = f ( y ) + p 1 ( t , y ) , y ′ = − g ( x ) + p 2 ( t , x ) . $$\left \{ \textstyle\begin{array}{l} x'=f(y)+p_{1}(t,y),\\ y'=-g(x)+p_{2}(t,x). \end{array}\displaystyle \right . $$ By using the Poincaré-Birkhoff twist theorem, we prove the existence and multiplicity of periodic solutions of the given system when f satisfies an asymmetric condition and the related time map satisfies an oscillating condition.

Keywords

• asymmetric nonlinearity
• periodic solution
• Poincaré-Birkhoff twist theorem