Amplitude Instability and Ergodic Behavior for Conservative Nonlinear Oscillator Systems

Abstract

Several earlier computer studies of nonlinear oscillator systems have revealed an amplitude instability marking a sharp transition from conditionally periodic to ergodic-type motion, and several authors have explained the observed instabilities in terms of a mathematical theorem due to Kolmogorov, Arnol'd, and Moser. In view of the significance of these results to several diverse fields, especially to statistical mechanics, this paper attempts to provide an elementary introduction to Kolmogorov-Arnol'd-Moser amplitude instability and to provide a verifiable scheme for predicting the onset of this instability. This goal is achieved by demonstrating that amplitude instability can occur even in simple oscillator systems which admit to a clear and detailed analysis. The analysis presented here is related to several earlier studies. Special attention is given to the relevance of amplitude instability for statistical mechanics.