Periodic Solutions, Stability and Non-Integrability in a Generalized Henon-Heiles Hamiltonian System

We consider the Hamiltonian function defined by the cubic polynomial H - 1/2 (p(x)(2) + p(y)(2))+ 1/2 (x(2) + y(2))+ A/3x(3) + Bxy(2) + Dx(2)y, where A, B, D is an element of R are parameters and so H is an extension of the well known Henon-Heiles problem. Our main contribution for D not equal 0, A + B not equal 0 and other technical restrictions are in three aspects: existence of periodic solutions, stability and instability of these periodic solutions and the problem of non-integrability of the system associated to H. Initially we give sufficient conditions on the three parameters of these generalized Henon-Heiles systems, which guarantees that at any positive energy level, the Hamiltonian system has periodic orbits. After that, we prove that its stability changes with the values of the parameters. Finally, we show that the generalized Henon-Heiles systems cannot have any second first integral of class l(1) in the sense of Liouville-Arnol'd. In fact, the parameters where our problem is not integrable in the sense of Liouville-Arnol'd are the same where the periodic orbits were analytically found through averaging theory.