Statistics of Arnold’s Diffusion.

Abstract

Vladimir Arnold, a Russian mathematician, developed a series of dependent equations that can be used to describe the stability of a system. To illustrate, Arnold’s ideas were used in developing a mixing diagnostic tool based on time theory for incompressible fluid flow. This was used in predicting the timing and location of oil washing ashore in Plaquemines Parish and Grand Isle, Louisiana, and Pensacola, Florida, in May 2010 and the flow of oil toward Panama City Beach, Florida, in June 2010 (Mezic 2010). This system of equations has important physical applications. The focus of this paper is an in depth analysis of the components that make up Arnold’s system of equations, which will lend a deeper understanding to the experiments that follow. Arnold’s system behaves like a pendulum in two dimensions, and a harmonic oscillator in the other two dimensions. The computer experiments generate a series of initial conditions, and integrate them using Arnold’s di↵usion equations. The Taylor Method is used to perform the integration. The final conditions are then analyzed for their di↵usion along the I2 variable of the harmonic oscillator. The hypothesis is that with increasing time of integration and as the coupling constant increases, the system will have a larger variance from the mean. In order to determine the behavior of the I2 dimension of the system, histograms are generated, and the mean and standard deviation are computed. Increasing the value of the coupling constant is e↵ectively increasing the magnitude of the disturbance to the system, while when analyzing the e↵ect of time, the system is being disturbed by a force of equal strength, over various lengths of time. The hypothesis is in fact reflected in the histograms and calculated standard deviations, however, to truly confirm this one would have to integrate many more times with varying parameters, a more extensive investigation than was performed for the sake of this paper.