RESONANT EXCITATION OF BUSH NONLINEAR NORMAL MODES IN TWO-DIMENSIONAL LATTICES
Abstract
This paper addresses the problem of resonant excitation of bushes in two-dimensional lattices with discrete symmetry. A bush is defined as a set of vibrational modes that persists over time and constitutes an exact solution to the nonlinear equations of motion, which cannot be obtained within the framework of perturbation theory. The relevance of this study lies in the fact that the proposed scheme for bush excitation, in contrast to previous mathematical models aimed at exciting bushes of nonlinear normal modes, can be implemented in a physical experiment. The above-mentioned ensembles of vibrational modes are considered within the framework of the theory of Chechin and Sakhnenko, based on group-theoretic research methods, which makes it possible to use any models of interatomic interaction potentials with equal success. This, in turn, allows us to consider dynamic systems of multiple scales, from molecular vibrations to the movement of macroscopic crystal structures. The aim of the work is to present a mathematical model of excitation of stable nonlinear dynamic objects in two-dimensional molecular structures suitable for both computational and physical experiments. Providing the results of a computational experiment with a description of the detected dynamic objects. Unlike traditional methods, Bush theory provides a solid theoretical framework based on group theory that overcomes the limitations of perturbation theory, offering results of exceptional reliability. At the same time, the symmetry properties used in Bush theory can provide significant advantages for numerical modeling, significantly reducing computational complexity. Now, this issue is being investigated by many authors around the world. It is worth noting, however, that the works in which the problems of excitation of dynamic objects close to the concept of bouches are discussed relate only to one-dimensional cases and that group-theoretic methods are not considered in them, with very rare exceptions
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