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Intermode interactions in one-dimensional nonlinear periodic structures havebeen studied by many authors, starting with the classic work by Fermi, Pasta,and Ulam (FPU) in the middle of the last century. However, symmetry selectionrules for the energy transfer between nonlinear vibrational modes of differentsymmetry, which lead to the possibility of excitation of some bushes of suchmodes, were not revealed. Each bush determines an exact solution of nonlineardynamical equations of the considered system. The collection of modes of agiven bush does not change in time, while there is a continuous energy exchangebetween these modes. Bushes of nonlinear normal modes (NNMs) are constructedwith the aid of group-theoretical methods and therefore they can exist forcases with large-amplitude atomic vibrations and for any type of interatomicinteractions. In most publications, bushes of NNMs or similar dynamical objectsin one-dimensional systems are investigated under periodic boundary conditions.In this paper, we present a study of bushes of NNMs in monoatomic chains forthe case of fixed boundary conditions, which sheds light on a series of newproperties of intermode interactions in such systems, as well as a method forconstructing bushes of NNMs by continuation of conventional normal modes to thecase of large atomic oscillations. The present study was carried out for thechains with the Lennard-Jones interatomic potential, but methods developed inthis paper are valid for arbitrary monoatomic chains.

Resonant excitation of the bushes of nonlinear vibrational modes in monoatomic chains