Oscillations within oscillations

摘要

Previously we have shown that for a particular choice of origin point data, the nonintegrable Aesthetic Field Equations collapse into a simpler set of equations, called the A, B, J, L equations. These equations lead to sinusoidal variation along any path segment for all the field quantities. In this paper we find, for a different choice of origin point data, that the Aesthetic Field Equations simply into a nonlinear system that describes a sine curve within a sine curve along any path segment. These results are obtained from visual inspection of the computer plots as well as numerical fitting to the data. We examine two dimensional maps, first when we specify an integration path, and then when we make use of a superposition principle at each point. The superposition principle arises as a consequence of the theory of nonintegrable systems, which we have developed on an earlier occasion. We find two dimensional maps in both instances that do not appear regular in the small. However, in both cases, we were able to observe large scale regularities. In our studies, we make use of computer techniques that focus on the large wavelength oscillations (big picture) as well as techniques that focus on the small wavelength oscillations (small picture). When the superposition principle is used, the small as well as big oscillations no longer have a sinusoidal appearance for y equals constant lines. This contrasts with the case when we specify a path where we see a sine curve within a sine curve along these y equals constant lines. In addition, we obtain a set of linear equations that describes a sine curve within a sine curve that can be considered in its own right. In this case, in addition to numerical integrations, we show analytically that a sine curve within a sine curve is a solution to the equations.