Integrability and Non-integrability of Hamiltonian Normal Forms

This paper summarizes the present state of integrability of Hamiltonian normal forms and it aims at characterizing non-integrable behaviour in higher-dimensional systems. Non-generic behaviour in Hamiltonian systems can be a sign of integrability, but it is not a conclusive indication. We will discuss a few degenerations and briefly review the integrability of Hamiltonian normal forms in two and three degrees of freedom. In addition we discuss two integrable normal form Hamiltonian chains, FPU and 1:2:2:2:2:2, and three non-integrable normal form chains, with emphasis on the 1:2:3:3:3:3 resonance. To distinguish between various forms of non-integrability is a major issue; time-series and projections based on the presence of a universal quadratic integral of the normal forms can be a useful predictor.