Estimation of basın detection and representation from non-linear system data

Abstract

In this thesis, we are interested in a good qualititive representation of a nonlinear autonomous flow. It is assumed that the equations governing a given nonchaotic nonlinear autonomous flow are unknown. The only prior knowledge about the system is its dimension, that is to say the number of state variables. As a realistic assumption this analysis will be confined to a part of the state space where the initial conditions are gathered. On the basis of a set of trajectory recordings gathered for a sufficiently large set of initial conditions our task is to identify the possible long term behaviour alternatives and to determine the set of initial conditions starting from which the system trajectories exhibit a specified long term behaviour. We will use different mathematical tools such as kernel estimation algorithms or image processing filters to visualize the long term behaviour of the given system. After the geometrical identification of important characteristic behaviours in our nonlinear system, we will define the basin of attractions and basins of some other phenomena with the help of algorithms which are originally developed. Proposed algorithms will be used first on two dimensional non-linear phase portraits and then will be extended to the third dimension with restrictions and limitations. Chaotic systems will be omitted due to their complicated strange attractor phenomena. Multivariate kernel estimators will be used among many places in the thesis. We will restrict ourself to diagonal bandwidth matrices since the optimal multivariate kernel estimators, especially if the system bandwidth varies according to the data position, are too complicated to find a place in this study. These estimation techniques and our algorithmic contribution will give a better insight about the long term behaviour of non-linear systems and will help us accomplish the aim of the thesis which is to identify the attractors of a continuous time, autonomous, non-chaotic system and to provide an approximate description of the basins of attractions to be used later for control purposes.