Localization of periodic oscillations in discrete non-linear systems

Qualitative analysis of periodic oscillations of non-linear n degrees-of-freedom systems is carried out; the results cover both autonomous (conservative and non-conservative) and non-autonomous systems. Sufficient conditions for strong localization (n - 1 oscillation amplitudes do not exceed the given small values) are obtained. Another set of conditions provide for the existence of a one-parameter family of periodic solutions such that some amplitudes tend to infinity while the rest tend to zero as the parameter increases (asymptotic localization). It is shown that in an autonomous system strongly localized oscillations are, as a rule, orbitally stable to a first approximation. The results are illustrated by two connected oscillators and an irregular chain. (C) 1996 Academic Press Limited