Asymptotic techniques and complex dynamics in weakly non-linear forced mechanical systems

Abstract

The applicability of asymptotic techniques and their ability to predict complex dynamical motions in weakly non-linear forced mechanical systems is investigated. The basic theorems in the method of averaging and integral manifolds are reviewed. A two degrees-of-freedom system, representing one mode truncation of general non-planar equations of a harmonically excited string, is used as the example. It is shown that the averaged equations, for small enough damping, possess non-planar constant solutions which become unstable and give rise to limit cycles, period-doublings, and isolated periodic solutions, as well as chaotic attractors. The truncated string equations are also directly integrated for small forcing amplitudes. There are non-planar periodic responses that bifurcate into amplitude-modulated motions on a two-torus. Changes in damping and frequency detuning result in torus-doubling, coexisting torus branches, and merging as well as destruction of the torus, leading to chaotic amplitude modulations. The bifurcation values of parameters are found to exhibit a scaling behavior and the results of averaged equations are found to be in qualitative agreement with the actual response.