The joining of local expansions in the theory of non-linear oscillations

Abstract

The behaviour of normal modes of oscillation in non-linear conservative systems with a finite number of degrees of freedom, when the amplitude changes from zero to infinity is studied. In the non-linear case, the normal oscillations represent a generalization of the normal oscillations of linear conservative systems (see /1/). It is assumed that the potential of a non-linear system is a polynomial of even degree in all positional variables. One can construct the trajectories of the normal oscillations in configuration space both for sufficiently small amplitude (a quasi-linear expansion), and for sufficiently large amplitude, using the fact that in these cases the system is close to a uniform system (see /2, 3/). The local expansions obtained are joined using rational-fractional Pade representations (see /4/) which enables the behaviour of oscillation modes to be followed when the amplitude changes continuously.