Least Square Method for Finding Absorbing Areas of Planar Quadratic Maps

Planar noninvertible maps have been studied recently by several authors such as Mira [32], Gardini [17], and Cathala [9], much of their work has been concentrated on analyzing some examples and making some conclusions on the properties of the maps.
     The main purpose of this thesis can be divided into three objectives:
First objective: introduce the mathematical background of the main notions and proposition on the theory of the dynamical system. Specifically we shall foucus our study on planar nonivertiable continuously differentiable maps T:
Definition of critical curves and some different types of noninvertible maps related to their critical curves and some properties of critical curves are presented. Second objective: we have studied some properties of such kind of maps in particular absorbing areas, invariant areas of such maps. Also, we give proposed algorithm to approximate the equations of the critical curves LC  which cause find an approximated absorbing and invariant areas such as least square method. Third objective: give some illustrative examples that use the proposed algorithm to find an approximated absorbing and invariant areas.
     In our work, we have made use of the Matlab version 7.0 software to solve the discussed examples.