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The research group from the Rostov State University has been developing thetheory of bushes of nonlinear normal modes (NNMs) in Hamiltonian systems withdiscrete symmetry since the late 90s of the last century. Group-theoreticalmethods for studying large-amplitude atomic vibrations in molecular and crystalstructures were developed. Each bush represents a certain collection ofvibrational modes, which do not change in time despite the time evolution ofthese modes, and the energy of the initial excitation remains trapped in thebush. Any bush is characterized by its symmetry group, which is a subgroup ofthe system's symmetry group. The modes contained in the given bush aredetermined by symmetry-related methods and do not depend on the interatomicinteractions in the considered system. The irreducible representations of thepoint and space groups are essentially used in the theory of the bushes ofNNMs, and this theory can be considered as a generalization of the well-knownWigner classification of the small-amplitude vibrations in molecules andcrystals for the case of large-amplitudes vibrations. Since using of theirreducible representations of the symmetry groups can be an obstacle to aninitial familiarization with the bush theory, in the present review, we explainthe basic concepts of this theory only with the aid of the ordinary normalmodes, which is well known from the standard textbooks considering the theoryof small atomic vibrations in mechanical systems. Our description is based onthe example of describing plane nonlinear atomic vibrations of a simple squaremolecule.

Properties of large-amplitudes vibrations in dynamical systems with discrete symmetry. Geometrical aspects