Spatially Random Processes in One-Dimensional Maps: The Logistic Map and The Arnold Circle Map.

摘要

One way to model in-situ remediation of contaminated groundwater is to consider spatially random processes in nonlinear systems. Groundwater remediation often requires injecting an aquifer with treatment solution, where degradation reactions break down the toxins. As the treatment solution and contaminated water flow through the aquifer, their movement is limited by the types of sediment found in the aquifer, which act as spatial barriers to mixing. The onset of chaos in this system implies the two solutions are well mixed, and thus the contaminants are rendered inert. The spatially random processes explored in this thesis are meant to mimic the distribution of sediment in the aquifer. These processes were constructed using uniform random variables and normal random variables, and incorporate an exponentially decaying spatial correlation.;The three-dimensional model of the fluid flow in the aquifer has been simplified to an in depth study of two one-dimensional maps: the logistic map and the Arnold circle map. Injection of the treatment solution in the aquifer may be thought of as the initial condition imposed on the map. Numerical simulations of the one-dimensional maps lay the groundwork for future studies of higher-dimensional systems.;Simulations indicate evidence of newly stabilized regions of the randomized logistic map, as well as a breakdown of symmetry and stable behavior in the Arnold circle map. The combination of bifurcation diagrams and Lyapunov exponents from the randomized logistic map lead us to hypothesize the spatially random process may stabilize the map in regions previously unstable. In the random circle map, analysis of the Arnold tongues, devil's staircases, and Lyapunov exponents suggest the random processes incur chaotic behavior in typically stable regions.