Bushes of normal modes represent the exact excitations in the nonlinear physical systems with discrete symmetries [Physica D 117 (1998) 43]. The present paper is the continuation of our previous paper [Physica D 166 (2002) 208], where these dynamical objects of new type were discussed for the monoatomic nonlinear chains. Here, we develop a simple crystallographic method for finding bushes in nonlinear chains and investigate stability of one-dimensional and two-dimensional vibrational bushes for both FPU-α and FPU-β models, in particular, of those revealed recently in [Physica D 175 (2003) 31].

We present a theorem that allows to simplify linear stability analysis of periodic and quasiperiodic nonlinear regimes in N -particle mechanical systems (both conservative and dissipative) with different kinds of discrete symmetry. This theorem suggests a decomposition of the linearized system arising in the standard stability analysis into a number of subsystems whose dimensions can be considerably less than that of the full system. As an example of such simplification, we discuss the stability of bushes of modes (invariant manifolds) for the Fermi-Pasta-Ulam chains and prove another theorem about the maximal dimension of the above mentioned subsystems.