INTERACTIONS BETWEEN NORMAL-MODES IN NONLINEAR DYNAMICAL-SYSTEMS WITHDISCRETE SYMMETRY - EXACT RESULTS

We discuss in detail the concept of bush of modes introduced by us ear lier for classical nonlinear systems with discrete (point or space) sy mmetry. Each bush comprises all modes singled out by the symmetry grou p of an initial excitation and may be considered as a geometrical and dynamical object. We prove theorems that describe structure and some p roperties of bushes for Hamiltonian and for a wide class of non-Hamilt onian systems. Theorems 2(a) and (b) permit one to introduce new varia bles nonlinearly connected with normal modes, which, in a sense, are i ndependent of each other, incase they are associated with different ir reducible representations of the symmetry group of a system in equilib rium. Such independence provides a possibility of singling out specifi c dynamical regimes of an essentially lesser dimensionality. Since man y different bushes are described by the same differential equations, t hey may be classified by certain classes of universality. Possible phy sical applications of vibrational bushes are suggested. Copyright (C) 1998 Published by Elsevier Science B.V.