Derivation Of Generalized Lorenz Systems To Study The Onset Of Chaos In High Dimension

Abstract

This thesis provides a new method to derive high dimensional generalized Lorenz systems. Lorenz system is a celebrated nonlinear dynamical dissipative system which was originally derived by Lorenz to study chaos in weather pattern. The classical two dimensional and dissipative Rayleigh-Benard convection can be approximated by Lorenz model, which was originally derived by taking into account only the lowest three Fourier modes. Numerous attempts have been made to generalize this 3D Lorenz model as the study of this high dimensional model will pave the way to better understand the onset of chaos in high dimensional systems of current interest in various disciplines. In this thesis a new method to extend this 3D Lorenz model to high dimension is developed and used to construct generalized Lorenz systems. These models are constructed by selecting vertical modes, horizontal modes and finally by both vertical and horizontal modes. The principle based on which this construction is carried out is the conservation of energy in the dissipationless limit and the requirement that the models are bounded. Finally the routes to chaos of these constructed models have been studied in great detail and an overall comparison is provided.