Non-linear normal modes for systems with discrete symmetry

Normal modes in linear mechanical systems with a discrete symmetry group in their equilibrium state can be classified by irreducible representations ( irreps) of this group. In non-linear dynamical systems, excitation of a giv en mode spreads to a number of other modes associated with different irreps , a nd this collection of modes was called a "bush" of modes in previous pa pers. There are some special cases where, because of symmetry restrictions, a bush is "irreducible" - it contains modes associated with a single irrep only. We looked for all irreducible bushes of vibrational modes for N-part icle mechanical systems with the symmetry of any of the 230 space groups an d, for the case of analytical potentials, found that there exist only 19 cl asses of such hushes. As a result, all modal subspaces to which symmetry de termined similar non-linear normal modes (introduced by Rosenberg) belong, were found, as well as all analytical mechanical systems whose dynamics, wi th a certain mode being initially excited, strictly reduces to only one res onance subspace corresponding to a single irrep. We found that the dimensio nality of such resonance subspaces does not exceed four. (C) 1999 Elsevier Science Ltd. All rights reserved.