Existence and multiplicity of rotating periodic solutions for resonant Hamiltonian systems

In the present paper, we consider a class of resonant Hamiltonian systems x′=JHx(t,x) in R2N. 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∈R with T>0 and Q an symplectic orthogonal matrix. In the case: Qk≠I2N 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.