| Most physical systems, being inherently nonlinear, cannot be studied in detail using traditional methods of linear dynamical system theory. Numerical integration methods are commonly used since relatively few of the governing nonlinear ordinary differential equations have closed form solutions.;This dissertation will present some of the most useful and enlightening geometric and statistical methods which are used to study nonlinear dynamics. In addition to proven techniques, a new approach--the Hyper-Volume Method--will be presented. It will be shown that the Hyper-Volume Method provides new information about nonlinear dynamics previously unknown or not quantified.;To demonstrate the utility of the Hyper-Volume Method, it will be used to analyze the dynamics of several physical models. The first will be the Duffing Oscillator which, in its various forms, can be used to study such things as the motion of forced buckled beams and some electrical circuits. This oscillator is significant in that it is deterministic yet yields random-like behavior known as chaos. Next the dynamics of supersonic panel flutter will be discussed. The dynamics here are interesting because the panel may exhibit periodic and chaotic responses without harmonic excitation.;Finally, to verify experimentally the predictions made by the Hyper-Volume Method, both a one and a two degree-of-freedom harmonically forced pendulum are analyzed. Using these devices it is shown that this new method can be used to make accurate predictions of the location of boundaries between qualitatively different steady-state responses as the forcing amplitude is varied. |
