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

Dante Carrasco, Claudio Vidal
2013 Journal of Nonlinear Mathematical Physics  
We consider the Hamiltonian function defined by the cubic polynomial H = 1 2 (p 2 x + p 2 y ) + 1 2 (x 2 + y 2 ) + A 3 x 3 + Bxy 2 + Dx 2 y, where A, B, D ∈ R are parameters and so H is an extension of the well known Hénon-Heiles problem. Our main contribution for D = 0, A + B = 0 and other technical restrictions are in three aspects: existence of periodic solutions, stability and instability of these periodic solutions and the problem of nonintegrability of the system associated to H.
more » ... we give sufficient conditions on the three parameters of these generalized Hénon-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 Hénon-Heiles systems cannot have any second first integral of class C 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.
doi:10.1080/14029251.2013.805567 fatcat:7pma6ffesjfafjbxx7m5ka4t5y