In the present tutorial we address a problem with a long history, which remains of great interest to date due to its many important applications: It concerns the existence and stability of periodic and quasiperiodic orbits in N-degree of freedom Hamiltonian systems and their connection with discrete symmetries. Of primary importance in our study is what we call nonlinear normal modes (NNMs), i.e. periodic solutions which represent continuations of the system's linear normal modes in the nonlinear regime. We examine questions concerning the existence of such solutions and discuss different methods for constructing them and studying their stability under fixed and periodic boundary conditions. In the periodic case, we find it particularly useful to approach the problem through the discrete symmetries of many models, employing group theoretical concepts to identify a special type of NNMs which we call one-dimensional "bushes". We then describe how to use linear combinations of s ≥ 2 such NNMs to construct s-dimensional bushes of quasiperiodic orbits, for a wide variety of Hamiltonian systems including particle chains, a square molecule and octahedral crystals in 1, 2 and 3 dimensions. Next, we exploit the symmetries of the linearized equations of motion about these bushes to demonstrate how they may be simplified to study the destabilization of these orbits, as a result of their interaction with NNMs not belonging to the same bush. Applying this theory to the famous Fermi Pasta Ulam (FPU) chain, we review a number of interesting results concerning the stability of NNMs and higher-dimensional bushes, which have appeared in the recent literature. We then turn to a newly developed approach to the analytical and numerical construction of quasiperiodic orbits, which does not depend on the symmetries or boundary conditions of our system. Using this approach, we demonstrate that the well-known "paradox" of FPU recurrences may in fact be explained in terms of the exponential localization of the energies Eq of NNM's being excited at the low part of the frequency spectrum, i.e. q = 1, 2, 3, …. These results indicate that it is the stability of these low-dimensional compact manifolds called q-tori, that is related to the persistence or FPU recurrences at low energies. Finally, we discuss a novel approach to the stability of orbits of conservative systems, expressed by a spectrum of indices called GALI k, k = 2, …, 2N, by means of which one can determine accurately and efficiently the destabilization of q-tori, leading, after very long times, to the breakdown of recurrences and, ultimately, to the equipartition of energy, at high enough values of the total energy E.
A proof that convection in a porous vertical slab may be unstable