Abstract. In this paper we study the periodic orbits of the Hamiltonian system with the Armburster-Guckenheimer-Kim potential and its non-integrability in the sense of LiouvilleArnold.

Abstract. This paper deals with the dynamics of time-reversible Hamiltonian vector fields with 2 and 3 degrees of freedom around an elliptic equilibrium point in presence of symplectic involutions. The main results discuss the existence of one-parameter families of reversible periodic solutions terminating at the equilibrium. The main techniques used are Birkhoff and Belitskii normal forms combined with the LiapunovSchmidt reduction.

The classical Hill's problem is a simplified version of the restricted three-body problem where the distance of the two massive bodies (say, primary for the largest one and secondary for the smallest one) is made infinity through the use of Hill's variables. The Levi-Civita regularization takes the Hamiltonian of the Hill lunar problem into the form of two uncoupled harmonic oscillators perturbed by the Coriolis force and the Sun action, polynomials of degree 4 and 6, respectively. In this paper, we study periodic orbits of the planar Hill problem using the averaging theory. Moreover, we provide information about the C 1 integrability or non-integrability of the regularized Hill lunar problem. C

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We provide sufficient conditions for the existence of periodic solutions in the class of Duffing differential equationswhere the functions a(t), b(t), c(t) and h(t, x) are C 2 and T -periodic in the variable t.