On the stability of periodic orbits of high dimensional autonomous Hamiltonian systems

We study the stability of periodic orbits of autonomous Hamiltonian systems with N+1 degrees of freedom or equivalently of 2N-dimensional symplectic maps, with N≥1. We classify the different stability types, introducing a new terminology which is perfectly suited for systems with many degrees of freedom, since it clearly reflects the configuration of the eigenvalues of the corresponding monodromy matrix, on the complex plane. The different stability types correspond to different regions of the N-dimensional parameter space S, defined by the coefficients of the characteristic polynomial of the monodromy matrix. All the possible direct transitions between different stability types are classified, and the corresponding transition hypersurface in S is determined. The dimension of the transition hypersurface is an indicator of how probable to happen is the corresponding transition. As an application of the general results we consider the well-known cases of Hamiltonian systems with two and three degrees of freedom. We also describe in detail the different stability regions in the three-dimensional parameter space S of a Hamiltonian system with four degrees of freedom or equivalently of a six-dimensional symplectic map.