Interactions between normal modes in nonlinear dynamical systems with discrete symmetry. Exact results

We discuss in detail the concept of bush of modes introduced by us earlier for classical nonlinear systems with discrete (point or space) symmetry. Each bush comprises all modes singled out by the symmetry group of an initial excitation and may be considered as a geometrical and dynamical object. We prove theorems that describe structure and some properties of bushes for Hamiltonian and for a wide class of non-Hamiltonian systems. Theorems 2(a) and (b) permit one to introduce new variables nonlinearly connected with normal modes, which, in a sense, are independent of each other, incase they are associated with different irreducible representations of the symmetry group of a system in equilibrium. Such independence provides a possibility of singling out specific dynamical regimes of an essentially lesser dimensionality. Since many different bushes are described by the same differential equations, they may be classified by certain classes of universality. Possible physical applications of vibrational bushes are suggested. Copyright (C) 1998 Published by Elsevier Science B.V.