On dynamics of second degree polinomial functions

In this thesis, after giving some basic concepts about chaotic dynamical systems, logistic map, which is one of the well-known chaotic dynamic systems, is investigated when the restrictions are lifted on the domain set and the parameter, in what, whether the system formed is chaotic or not depending on the range. Furthermore, an example of the logistic map is given that is not chaotic on an interval but chaotic on an uncountably infinite set of points. In the second part of the study, topological conjugate functions with some special homeomorphisms are found for logistic map and quadratic map, which are well-known second degree chaotic dynamic systems. Then the largest compact interval in which second degree polynomial functions can be defined is determined. With the help of this interval and the topological conjugate functions mentioned above, a chaotic family of second degree polynomial functions, which constitute the main part of the thesis, is given. Finally, functions that are not chaotic in the largest defined compact interval, but are also chaotic in a subinterval, are determined with the help of computer, and several findings are given about the bifurcation diagram of the logistic map found with the help of computer.