The elusive Liapunov periodic solutions (Q904885)

The authors study the existence of families of periodic solutions for Hamiltonian systems when the classical Liapunov center theorem fails due to a resonance. In the proof, regular and singular reduction methods are used, together with classical perturbation theory. The starting point is the classic 1892 Liapunov center theorem and the test problem is a two-degree-of freedom Hamiltonian system in \(\mathbb{R}^4\) which is studied by a Hamiltonian system on a two dimensional real algebraic surface. Several examples are discussed in detail with graphical illustrations: resonance \(2:-1\), resonance \(3:-1\), resonance \(k:-1\), \(k \geq 4\) and resonance \(1:-1\).