Subharmonic Solutions of Indefinite Hamiltonian Systems via Rotation Numbers

Abstract

We investigate the multiplicity of subharmonic solutions for indefinite planar Hamiltonian systems J⁢z′=∇⁡H⁢(t,z){Jz^{\prime}=\nabla H(t,z)} from a rotation number viewpoint. The class considered is such that the behaviour of its solutions near zero and infinity can be compared two suitable positively homogeneous systems. Our approach can be used to deal with the problems in absence of the sign assumption on ∂⁡H∂⁡x⁢(t,x,y){\frac{\partial H}{\partial x}(t,x,y)}, uniqueness and global continuability for the solutions of the associated Cauchy problems. These systems may also be resonant. By the use of an approach of rotation number, the phase-plane analysis of the spiral properties of large solutions and a recent version of Poincaré–Birkhoff theorem for Hamiltonian systems, we are able to extend previous multiplicity results of subharmonic solutions for asymptotically semilinear systems to indefinite planar Hamiltonian systems.

Keywords

• subharmonic solution
• indefinite hamiltonian system
• spiral function
• rotation number
• poincaré–birkhoff theorem
• 34c25
• 34b15
• 34c15
• 34d15