Generalized fuzzy Mandelbrot sets

GENERALIZED FUZZY MANDELBROT SETS SUMMARY Keywords: Fuzzy logic, fuzzy set, Mandelbrot set This thesis consists of five chapters. The first chapter is devoted to the introduction. In the second chapter, the basic definitions and theorems related to the Julia set, the orbits of the Julia set, Mandelbrot set, Mandelbar set, the generalized Mandelbrot sets, and the generalized Mandelbar sets are given. In the third chapter, the basic definitions and theorems of fuzzy sets, fuzzy real numbers, fuzzy complex numbers, and fuzzy topological spaces are introduced. In addition to these, the necessary and sufficient conditions are expressed for any fuzzy set to be a fuzzy number. The fourth and fifth chapters constitute the original parts of this thesis. In the introduction part of the fourth chapter, a new definition is given for the fuzzy Mandelbrot set. It is expressed both visually and algebraically in detail how this new definition removes the inadequacy of the existing definition in the literature. In addition, the formation stages of the fuzzy Mandelbrot set are examined separately for different iteration steps. The topological properties of the newly defined fuzzy Mandelbrot set are investigated. In the second subsection of this section, the fuzzy Mandelbar set is introduced and analyzed from geometric, algebraic, and topological aspects. The similarities and differences between this set and the fuzzy Mandelbrot set are also investigated. In the fifth chapter, in accordance with the main purpose of this thesis, the generalized fuzzy Mandelbrot and generalized fuzzy Mandelbar sets are introduced under consideration of the new information acquired in the fourth chapter. By stating the basic definitions and theorems related to their properties and the relevant algorithms are given. Detailed images of these sets are also presented. Consequently, suggestions are expressed for new research to be done in the future with the obtained data.