Existence and multiplicity of rotating periodic solutions for resonant Hamiltonian systems

In the present paper, we consider a class of resonant Hamiltonian systems x' = J H-x (t, x) in R-2N. We will use saddle point reduction, Morse theory combining the technique of penalized functionals to obtain the existence of nontrivial rotating periodic solutions, i.e., x(t + T) = Qx(t) for any t is an element of R with T > 0 and Q an symplectic orthogonal matrix. In the case: Q(k) not equal I-2N for any positive integer k, such a rotating periodic solution is just a quasi-periodic solution. Moreover, if H is even in x, we will give the multiplicity of nontrivial rotating periodic solutions by using two abstract critical theorems and previous techniques. (C) 2018 Elsevier Inc. All rights reserved.