A perturbation procedure for nonlinear oscillations (The dynamics of two oscillators with weak nonlinear coupling)

Abstract

This thesis considers in detail the dynamics of two oscillators with weak nonlinear coupling. There are three classes of such problems: non-resonant, where the Poincare procedure is valid to the order considered; weakly resonant, where the Poincare procedure breaks down because small divisors appear (but do not affect the O(1) term) and strongly resonant, where small divisors appear and lead to O(1) corrections. A perturbation method based on Cole's two-timing procedure is introduced. It avoids the small divisor problem in a straightforward manner, gives accurate answers which are valid for long times, and appears capable of handling all three types of problems with no change in the basic approach. One example of each type is studied with the aid of this procedure: for the nonresonant case the answer is equivalent to the Poincare result; for the weakly resonant case the analytic form of the answer is found to depend (smoothly) on the difference between the initial energies of the two oscillators; for the strongly resonant case we find that the amplitudes of the two oscillators vary slowly with time as elliptic functions of ϵ t, where ϵ is the (small) coupling parameter. Our results suggest that, as one might expect, the dynamical behavior of such systems varies smoothly with changes in the ratio of the fundamental frequencies of the two oscillators. Thus the pathological behavior of Whittaker's adelphic integrals as the frequency ratio is varied appears to be due to the fact that Whittaker ignored the small divisor problem. The energy sharing properties of these systems appear to depend strongly on the initial conditions, so that the systems not ergodic. The perturbation procedure appears to be applicable to a wide variety of other problems in addition to those considered here.