Existence and multiplicity of rotating periodic solutions for Hamiltonian systems with a general twist condition

Abstract

In this paper, we consider rotating periodic solutions of the Hamiltonian systemx˙=JH′(t,x),x∈R2N, having the form x(t+T)=Qx(t),∀t∈R, for some T>0 and a symplectic orthogonal matrix Q. We study the system under a general twist condition: the nonlinear term H′(t,x) is required to be of linear growth but not necessarily to be asymptotically linear at infinity. The twist is reflected in the difference of the generalized Morse index at the origin and at infinity. By combining a finite dimensional reduction method, Morse theory and minimax principle, we establish the existence and multiplicity of nontrivial rotating periodic solutions.