Existence and multiplicity of rotating periodic solutions for resonant Hamiltonian systems

Abstract In the present paper, we consider a class of resonant Hamiltonian systems x ′ = J H x ( t , x ) in R 2 N . 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 ) = Q x ( t ) for any t ∈ R with T > 0 and Q an symplectic orthogonal matrix. In the case: Q k ≠ I 2 N 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.

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